Patent Publication Number: US-2022214990-A1

Title: Cascade communications between fpga tiles

Description:
PRIORITY APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 16/656,685, filed Oct. 18, 2019, the content of which is incorporated herein by reference in its entirety. 
    
    
     BACKGROUND 
     A field programmable gate array (FPGA) is composed of an array of programmable logic blocks that are interconnected with a reconfigurable routing network. Logic blocks vary in type and typically include reconfigurable logic, memories, and arithmetic logic. Reconfigurable logic is commonly implemented with lookup tables. FPGA tiles communicate using the reconfigurable routing network, which has a bandwidth lower than the processing bandwidth of the FPGA tiles. 
     The reconfigurable routing network can be programmed to connect the logic blocks together in one of many possible configurations. This programmability comes at a cost. The routing network is typically less dense and supports less data than arithmetic logic blocks for a given area. As a result, the practical size/width of an arithmetic logic block is limited by the number of available inputs and outputs provided by the routing network. Although larger arithmetic operations can be achieved by cascading smaller arithmetic logic blocks, this approach introduces unnecessary latency and significantly reduces the overall logic density of an application. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Some embodiments of the disclosed technology are illustrated by way of example and not limitation in the figures of the accompanying drawings. 
         FIG. 1  is a high-level diagrammatic view of a tile of an FPGA that uses cascade communications between FPGA tiles, according to some example embodiments. 
         FIG. 2  is a high-level diagrammatic view of an arithmetic circuit that receives cascade inputs and provides cascade outputs, according to some example embodiments. 
         FIG. 3  is a high-level diagrammatic view of a memory circuit that receives cascade inputs and provides cascade outputs, according to some example embodiments. 
         FIG. 4  is a diagrammatic view of a portion of an arithmetic circuit that uses cascade communications between FPGA tiles, according to some example embodiments. 
         FIG. 5  is a diagrammatic view of a portion of an arithmetic circuit that uses cascade communications between FPGA tiles, according to some example embodiments. 
         FIG. 6  is a diagrammatic view of a portion of an arithmetic circuit that uses cascade communications between FPGA tiles, according to some example embodiments. 
         FIG. 7  is a diagrammatic view of a portion of an arithmetic circuit that uses cascade communications between FPGA tiles, according to some example embodiments. 
         FIG. 8  is a diagrammatic view of a portion of an arithmetic circuit that uses cascade communications between FPGA tiles, according to some example embodiments. 
         FIG. 9  is a high-level diagrammatic view showing connections between tiles of an FPGA using cascade communications, according to some example embodiments. 
         FIG. 10  is a flowchart illustrating operations of a method performed by a tile of an FPGA using cascade communications, according to various embodiments of the invention. 
         FIG. 11  is a block diagram illustrating components of a system for controlling fabrication of circuits described herein, according to some example embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     Example methods, systems and circuits for cascade communications between FPGA tiles will now be described. In the following description, numerous examples having example-specific details are set forth to provide an understanding of example embodiments. It will be evident, however, to one of ordinary skill in the art that these examples may be practiced without these example-specific details, and/or with different combinations of the details than are given here. Thus, specific embodiments are given for the purpose of simplified explanation, and not limitation. 
     A tile of an FPGA provides memory, arithmetic functions, or both. Connections directly between multiple instances of the tile are available, allowing multiple tiles to be treated as larger memories or arithmetic circuits. By using these connections, referred to as cascade inputs and outputs, the input and output bandwidth of the arithmetic and memory circuits are increased, operand sizes are increased, or both. 
     By using the cascade connections, multiple tiles can be used together as a single, larger tile. Thus, implementations that need memories of different sizes, arithmetic functions operating on different sized operands, or both, can use the same FPGA without additional programming or waste. For example, an FPGA tile using a large memory can serve applications that need a large memory or a small memory, but the excess memory is wasted when only a small memory is needed. Using cascade communications, more tiles are used when a large memory is needed and fewer tiles are used when a small memory is needed and the waste is avoided. 
       FIG. 1  is a high-level diagrammatic view of a tile  100  of an FPGA that uses cascade communications between FPGA tiles, according to some example embodiments. The connected routing  105  and  110  are also shown. The tile  100  fuses memory and arithmetic circuits and comprises a machine-learning processor (MLP)  115 , a block random access memory (BRAM)  120 , and a logic random access memory (LRAM)  125 . The MLP  115  comprises a floating-point multiply and accumulate (MAC)  130  and an integer MAC  135 . The BRAM  120  comprises a memory  140 . The tile  100  is connected to other tiles via the routing  105  and the routing  110 . Additionally, the tile  100  is directly connected to a first FPGA tile without using a routing connection of the FPGA via the operand cascade input  145  and the memory cascade input  155 . The tile  100  is, in some example embodiments, further directly connected to a second FPGA tile without using the routing of the FPGA via the operand cascade output  160  and the memory cascade output  165 . The cascade connections may be unidirectional or bidirectional. 
     In a first operation mode, the MACs  130  and  135  receive inputs from one or more of the routing  105 , the LRAM  125 , the memory  140 , and an operand cascade input  145 . Outputs are provided by the MACs  130  and  135  to the routing  105 , an operand cascade output  160 , the LRAM  125 , or any suitable combination thereof. The memory  140  receives input from the routing  110 , a memory cascade input  155 , or both. Outputs are provided by the memory  140  to the routing  110 , a memory cascade output  165 , or both. In the first operation mode, the MACs  130  and  135  do not receive input from the routing  110  and the memory  140  does not receive input from the routing  105 . Thus, the inputs from the routing fabric of the FPGA are divided between the MLP  115  and the BRAM  120 , and the MLP  115  accesses data from the BRAM  120  within the tile  100 , without going through the switch fabric. 
     Any number of tiles  100 , communicating via cascade connections (e.g., the operand cascade input  145 , the operand cascade output  160 , the memory cascade input  155 , and the memory cascade output  165 ), may be arranged to effectively create a larger tile. Each tile in the set of tiles may be configured to provide, as the cascade output, the received cascade input or to generate a result to provide as the cascade output. For example, the first tile in the set of tiles may receive data via the routing  105  and provide the data via the operand cascade output  160  to the second tile. The second and subsequent tiles may provide the data without modification by passing it through to their own operand cascade output  160 . Alternatively, the cascade output of each tile may be affected by operations performed by the tile. 
     A typical MAC multiplies two products and adds the results to an accumulator. The MACs  130  and  135 , in some example embodiments, provide additional functionality by allowing partial products to be summed and provided as an output before being added to the accumulator. Thus, the individual partial products, sums of partial products for a current multiplication, and an accumulation result across multiple multiplication cycles may all be accessed by use of the MACs  130  and  135 . 
     In a second operation mode, the MACs  130  and  135  receive inputs from one or more of the routing  105 , the routing  110 , the LRAM  125 , and the operand cascade input  145 . Outputs are provided by the MACs  130  and  135  to the routing  105 , the routing  110 , the operand cascade output  160 , the LRAM  125 , or any suitable combination thereof. In the second operation mode, the memory  140  does not receive inputs from the routing  105  or the routing  110 . Thus, in the second operation mode, the tile  100  operates as a dedicated MLP, with MLP  115  having full access to the routing fabric of the FPGA and the memory  140  effectively disabled. Nonetheless, the LRAM  125  may make use of some routing connections in the second operation mode. 
     In a third operation mode, the memory  140  receives input from the routing  105 , the routing  110 , the memory cascade input  155 , or any suitable combination thereof. Outputs are provided by the memory  140  to the routing  105 , the routing  110 , the memory cascade output  165 , or any suitable combination thereof. In the third operation mode, the MLP  115  does not receive inputs from the routing  105  or the routing  110 . Thus, in the third operation mode, the tile  100  operates as a dedicated BRAM, with BRAM  120  having full access to the routing fabric of the FPGA and the MLP  115  effectively disabled. 
     As shown in  FIG. 1 , the LRAM  125  is connected to the routing  105  and the routing  110 . In some example embodiments, the routing connections for the LRAM  125  are maintained in all operation modes. To use data stored in the LRAM  125  for calculations by the MLP  115 , control signals identify the address to read and the data at that address in the LRAM  125  is used. The data is provided from the LRAM  125  to the MLP  115  via intra-tile connections, without using the routing  105  or the routing  110 . 
     The intra-tile connections shown between the LRAM  125  and the memory  140  to the floating-point MAC  130  and the integer MAC  135  operate at a higher bandwidth than the routing  105  and  110 . In various example embodiments, the intra-tile data access speed is a factor of at least 10, 50, 100, or 500 times faster than the routing connection access speed. 
     The differences between the LRAM  125  and the BRAM  120  are typically implementation details such that the BRAM  120  is similar to a cache style memory (typically using SRAM cells) and the LRAM  125  is similar to a register file (typically using flops). However, these are not concrete rules, and other types of memory may be used for the LRAM  125  and the BRAM  120 . In some example embodiments, the BRAM  120  has a greater storage capacity than the LRAM  125  and is optimized for area and the LRAM  125  is optimized for latency. In further example embodiments, the LRAM  125  stores a working set of sums of partial products for matrix multiplications. 
     The FPGA tile  100  receives a clock input to control the rate at which operations are performed. A frequency multiplier (e.g., a 2× multiplier) may be applied to the input clock frequency to change the operation rate. In some example embodiments, running the FPGA tile  100  at twice the clock rate allows twice as many calculations to be performed by using the MAC  130  or the MAC  135  twice in a single (external) clock cycle. For example, in a 128-bit input mode, sufficient inputs may be provided to perform four calculations per clock cycle but the MAC hardware is sufficient to perform only two. Accordingly, by performing two calculations on each of two internal clock cycles, four calculations are performed on the single external clock cycle, allowing the FPGA tile  100  to perform as effectively as an alternative design comprising twice as many MACs. 
     As another example of an advantage of using a frequency multiplier, operations that reuse at least some of the input operands may be performed more efficiently. For example, when weights or coefficients from the routing  105  are the same for multiple operations, other coefficients may be updated using the higher-frequency internal clock (e.g., read from the BRAM  120 ) and the multiple operations performed within a single external clock cycle. One practical use of this advantage is in machine learning, with a batch size equal to the clock multiplier (e.g.,  2 ). The results of the multiple operations may be accumulated together to generate a single output to the routing  105  per external clock cycle, output in parallel to the routing  105  if there are sufficient output pins, or stored in the BRAM  120 . 
     As can be seen in  FIG. 1 , the operand cascade output  160  is configured to provide any one of: data received via the routing  105 , data received at the operand cascade input  145 , the result from the MAC  130 , and the result from the MAC  135 . Thus, when multiple FPGA tiles  100  are connected by connecting the operand cascade output  160  of one tile to the operand cascade input  145  of the next tile, a single input from the routing  105  can be transmitted via the cascade connections to the remaining FPGA tiles  100 . By comparison to existing methods in which routing  105  resources would be used for each of the multiple FPGA tiles  100 , substantial routing resources are saved. As a practical matter, provision of the same operand to multiple MLPs  115  is useful in performing matrix multiplication operations, particularly for machine learning applications. 
     Similarly, the memory cascade output  165  is configured to provide any one of: data received via the routing  110 , data received at the memory cascade input  155 , and data read from the memory  140 . Additionally, the memory cascade connections are bidirectional. Accordingly, while the two cascade connections are labeled as “input” and “output,” data may flow in either direction. Thus, an address may be propagated in one direction through multiple FPGA tiles  100  via the cascade connections and a data result may be propagated in the other direction. As with the MLP  115 , using cascade connections for the BRAM  120  saves routing resources. 
     By providing multiple input paths to the tile  100 , some operations are enabled to be performed simultaneously that had to be performed serially in prior art designs. For example, an address input may be provided using the routing  105  or  110 . The MAC  130  or the MAC  135  may use the address to retrieve first data from the memory  140  and perform an operation on the first data. Simultaneously, a second address input and second data may be provided using the memory cascade input  155 . The memory  140  stores the second data at the second address while the MAC  130  or the MAC  135  is accessing the first data. Thus, the next working set of data is loaded into the BRAM while the MAC  130  or the MAC  135  is operating on the current working set of data. This enables greater throughput by comparison to tiles that can receive only a single address at a time and must pause the accessing of data for computation to load in new data. 
       FIG. 2  is a high-level diagrammatic view of an arithmetic circuit  200  that receives cascade inputs and provides cascade outputs, according to some example embodiments. In some example embodiments, the arithmetic circuit  200  is the MLP  115 . The arithmetic circuit  200  receives intra-tile inputs from the memory portion of the FPGA tile, routing fabric inputs from a routing fabric of the FPGA, control signals from the routing fabric of the FPGA, and operand cascade inputs from another FPGA tile without making use of the routing fabric. In various example embodiments, more or fewer inputs are present. 
     The arithmetic circuit  200  provides intra-tile outputs to the memory portion of the FPGA tile, routing fabric outputs to the routing fabric of the FPGA and operand cascade outputs to another FPGA tile without making use of the routing fabric. In various example embodiments, more or fewer outputs are present. Typically, the operand cascade inputs are received from a first FPGA tile, the arithmetic circuit  200  is part of a second FPGA tile, and the operand cascade outputs are provided to a third FPGA tile. 
       FIG. 3  is a high-level diagrammatic view of a memory circuit  300  that receives cascade inputs and provides cascade outputs, according to some example embodiments. In some example embodiments, the memory circuit  300  is the BRAM  120 . The memory circuit  300  receives intra-tile inputs from the arithmetic portion of the FPGA tile, routing fabric inputs from a routing fabric of the FPGA, control signals from the routing fabric of the FPGA, memory cascade inputs from a first FPGA tile, and reverse memory cascade inputs from a second FPGA tile. The cascade inputs do not make use of the routing fabric. The memory cascade inputs may comprise control signals as well as data signals. In various example embodiments, more or fewer inputs are present. 
     The memory circuit  300  provides intra-tile outputs to the arithmetic portion of the FPGA tile, routing fabric outputs to the routing fabric of the FPGA, memory cascade outputs to the second FPGA tile, and reverse memory cascade outputs to the first FPGA tile. In various example embodiments, more or fewer outputs are present. 
       FIG. 4  is a diagrammatic view of a portion  400  of an arithmetic circuit that uses cascade communications between FPGA tiles, according to some example embodiments. The portion  400  comprises multiplexers  410 A,  410 B,  410 C, and  410 D; registers  420 A,  420 B,  420 C, and  420 D, referred to collectively as stage 0 delay registers  420 ; bit remap logic  430 A,  430 B,  430 C, and  430 D; and stage 1 delay registers  440  (shown in more detail as individual registers  510 A,  510 B,  510 C,  510 D,  510 E,  510 F,  510 G,  510 H,  510 I,  510 J,  510 K,  510 L,  510 M,  510 N,  510 O, and  510 P in  FIG. 5 ). The portion  400  accepts inputs for two multiply operands, A and B, remaps the operands to a format used by the next portion of the arithmetic circuit, and provides the remapped operands to delay registers used by the next portion. 
     The multiplexer  410 A selects the low bits for the B operand from four options: MLP_DIN[71:0], 72 bits of data received via the routing fabric  105 ; REGFILE_DOUT[71:0], 72 bits of data received from the LRAM  125  within the tile  100 ; BRAM_DOUT[71:0], 72 bits of data received from the BRAM  120  within the tile  100 ; and FWDI_MULTB_L[71:0], 72 bits of data received from the operand cascade input  145 . The multiplexer  410 B selects the high bits for the B operand from eight options: BRAM_DIN[71:0], 72 bits of data received via the routing fabric  110 ; REGFILE_DOUT[143:72], 72 bits of data received from the LRAM  125  within the tile  100 ; BRAM_DOUT[143:72], 72 bits of data received from the BRAM  120  within the tile  100 ; MLP_DIN[71:0]; REGFILE_DOUT[ 71 . 0 ], BRAM_DOUT[71:0]; and FWDI_MULTB_L[71:0]. Thus, the B operand is generated from a combination of inputs from one or more of the routing fabric  105 , the routing fabric  110 , the LRAM  125 , the BRAM  120 , and the operand cascade input  145 . 
     The low bits for the A operand are selected by the multiplexer  410 C from four options: MLP_DFN[71:0]; REGFILE_DOUT[71:0]; BRAM_DOUT[71:0]; and FWDI_MULTA_L[71:0], 72 bits of data received from the operand cascade input  145 . The high bits for the A operand are selected by the multiplexer  410 D from eight options: BRAM_DIN[71:0]; MLP_DIN[71:0]; REGFILE_DOUT[143:72]; REGFILE_DOUT[71:0]; FWDI_MULTA_L[71:0]; BRAM_DOUT[143:72]; BRAM_DOUT[71:0]; and FWDI_MULTA_H[71:0], 72 bits of data received from the operand cascade input  145 . Thus, the A operand is also generated from a combination of inputs from one or more of the routing fabric  105 , the routing fabric  110 , the LRAM  125 , the BRAM  120 , and the operand cascade input  145 . 
     The inputs selected by the multiplexers  410 A- 410 D are optionally stored in the corresponding one of the registers  420 A- 420 D, which provide data to the operand cascade output  160  in the form of FWDO_MULTB_L[71:0], the low bits of the B operand; FWDO_MULTIB_H[71:0], the high bits of the B operand; FWDO_MULTA_L[71.0], the low bits of the A operand; and FWDO_MULTA_H[71:0], the high bits of the A operand. Additionally, each of the registers  420 A- 420 D is accessed by the corresponding one of the bit remap logics  430 A- 430 D. Each of the bit remap logics  430 A- 430 D remaps the inputs based on a multiplication mode and byte selection mode input. Exponent and sign bits are output from the bit remap logics  430 A- 430 D as signals &lt;EXPA&gt;, &lt;SGNA&gt;, &lt;EXPB&gt;, &lt;SGNB&gt;, &lt;EXPC&gt;, &lt;SGNC&gt;, &lt;EXPD&gt;, and &lt;SGND&gt;. The remapped inputs are provided to the stage 1 delay registers  440 , for access by the next portion of the arithmetic circuit. 
     The inputs selected by the multiplexers  410 A- 410 D are controlled by configuration signals SEL_MULTB_L, SEL_MULTB_H, SEL_MULTA_L, and SEL_MULTA_H. Thus, the arithmetic circuit is configured by one or more configuration signals to receive inputs from a first connection fabric, a second connection fabric, a first fused memory, a second fused memory, an operand cascade input, or any suitable combination thereof. As an example, in response to a first configuration signal, the arithmetic circuit is configured to perform operations on data received via a routing fabric (e.g., MLP_DIN[71:0], a possible selection by each of the multiplexers  410 A- 410 D) and data received within a tile of an FPGA from a first memory (e.g., BRAM_DOUT[143:72], a possible selection by the multiplexers  410 B and  410 D or BRAM_DOUT[71:0], a possible selection by the multiplexers  410 A and  410 C). As another example, in response to a second configuration signal, the arithmetic circuit is configured to perform operations on data received via the routing fabric and data received within a tile of the FPGA from a second memory (e.g., REGFILE_DOUT[143:72], a possible selection by the multiplexers  410 B and  410 D or REGFILE_DOUT[71:0], a possible selection by the multiplexers  410 A and  410 C). 
     In a floating-point mode that differs from the floating-point format used by the portion  500 , the bit remap logics  430 A- 430 D convert the inputs to a format expected by the portion  500 . In an example, the portion  500  expects floating-point values with a 15-bit mantissa, a one-bit sign, and an 8-bit exponent. In this example, the multiple mode arithmetic circuit supports inputs and outputs using various combinations of 16-bit mantissas, 10-bit mantissas, 12-bit mantissas, 8-bit exponents, 6-bit exponents, and 5-bit exponents. Based on the input format and the format expected by the portion  500 , the bit remap logics  430 A- 430 D convert the input values. In this example, selection of the input floating-point format is in response to a mode selection input. 
     The bit remap logics  430 A- 430 D, in some example embodiments, perform sign extension. As a result, operands that are smaller than the size of the input values accepted by the arithmetic blocks (e.g., the multipliers  520 A- 520 H) are routed using only the routing resources necessary for the operands and sign-extended by the bit remap logics  430 A- 430 D prior to use by the arithmetic blocks. By comparison with designs that perform sign extension prior to routing, this design saves routing resources. 
     The integer arithmetic logic blocks may be used to perform floating-point operations on the mantissas of floating-point operands by identifying the highest exponent among the exponents of the floating-point operands and right-shifting the mantissas of the other operands by the difference in exponents. For example, consider the table below, showing four operands. 
     
       
         
           
               
               
               
               
               
             
               
                   
                   
               
               
                   
                 Original 
                 Original 
                 Adjusted 
                 Adjusted 
               
               
                   
                 Mantissa 
                 Exponent 
                 Mantissa 
                 Exponent 
               
               
                   
                   
               
             
            
               
                   
                 10101010 
                 0001 
                 11110101 
                 0100 
               
               
                   
                 11110000 
                 0010 
                 11111100 
                 0100 
               
               
                   
                 00110011 
                 0011 
                 00011001 
                 0100 
               
               
                   
                 00001111 
                 0100 
                 00001111 
                 0100 
               
               
                   
                   
               
            
           
         
       
     
     After the adjustment, the mantissas of the operands can be manipulated as integers, since all exponents are equal. This floating-point mode is referred to as block floating-point since in all of the numbers being operated on are grouped together (in a “block”) with a common exponent value. 
     In the example above, note that the first two operands have their mantissas padded with is and the last two operands have their mantissas padded with 0s. This is consistent with 2&#39;s complement representation of negative numbers. In an unsigned mode or a signed/magnitude mode, the manipulation of the mantissa changes accordingly either by padding with 0s (for unsigned) or inserting 0s without modifying the sign bit (sign/magnitude). 
       FIG. 5  is a diagrammatic view of a portion  500  of an arithmetic circuit that uses cascade communications between FPGA tiles, according to some example embodiments. The portion  500  comprises registers  510 A,  510 B,  510 C,  510 D,  510 E,  510 F,  510 G,  510 H,  510 I,  510 J,  510 K,  510 L,  510 M,  510 N,  510 O, and  510 P; multipliers  520 A,  520 B,  520 C,  520 D,  520 E,  520 F,  520 G, and  520 H; adders  530 A,  530 B,  530 C,  530 D,  550 A,  550 B, and  560 ; multiplexers  540 A,  540 B,  540 C,  540 D, and  570 ; and stage 2 delay registers  580 . 
     Each of the registers  510 A- 510 P stores eight bits of data for an operand for one of the multipliers  520 A- 520 H. Each of the multipliers  520 A- 520 H accepts eight bits of the A operand and eight bits of the B operand. Thus, the portion  500 , in total, handles 64 bits of the A operand and 64 bits of the B operand. To handle the complete input received by the portion  400 , the portion  500  is duplicated, with each instance of the portion  500  handling half of the inputs. 
     In a first operation mode, the portion  500  is configured to determine a sum of eight 8-bit multiply operations. By sign-extending or padding with leading zeros, as appropriate, the sum of fewer multiply operations or the sum of eight smaller (e.g., 6-bit or 4-bit) operations may be determined in the first operation mode. In a first variation of the first operation mode, each of the multipliers  520 A- 520 H is an eight-bit multiplier that is configured to output the sum of two four-bit multiplies. In a second variation of the first operation mode, each of the multipliers  520 A- 520 H is an eight-bit multiplier that is configured to output the result of two four-bit multiplies. In a second operation mode, the portion  500  is configured to output a sum of two 16-bit multiply operations. By sign-extending or padding with leading zeros, as appropriate, a single multiply operation or the sum of two smaller (e.g., 12-bit or 10-bit) operations may be determined in the second operation mode. In a third operation mode, the portion  500  in combination with the second instance of the portion  500  is configured, using an additional shifter and a wider adder, to determine a single 32-bit multiply operation. By sign-extending or padding with leading zeros, as appropriate, a smaller multiply operation (e.g., 18-bit or 24-bit) may be determined in the third operation mode. In additional operation modes, one or more individual multiply results may be provided at the output, in addition to or instead of the sum of the multiply operations. 
     With respect to the first operation mode, each of the eight multipliers  520 A- 520 H performs an eight-bit multiplication using a different portion of the operands A and B as inputs. The results of the eight multiplications are pairwise summed by the adders  530 A- 530 D. The four addition results are pairwise summed by the adders  550 A- 550 B, using the multiplexers  540 A- 540 D (controlled by a MODELSEL signal) to determine whether to take the addition result directly or shifted as shown. The shifted results are used to support 16-bit multiplication. The two results of the adders  550 A- 550 B are summed by the adder  560 . The result of the adder  560  is the sum of the eight eight-bit multiplications, and is provided, via the mux  570 , to the stage 2 delay registers  580 . 
     With respect to the second operation mode, the multipliers  520 A- 520 D together, in combination with the adders  530 A,  530 B, and  550 A, determine a first 16-bit multiplication result. The multipliers  520 E- 520 H, in combination with the adders  530 C,  530 D, and  550 B, determine a second 16-bit multiplication result. Four multipliers of a first operand size can be used to generate multiplication results of a second operand size that is twice the first operand size. The larger operands are divided into two portions, high and low, and organized as follows, wherein AH represents the high portion of the A operand, AL represents the low portion of the A operand, BH represents the high portion of the B operand, and BL represents the low portion of the B operand. AH AL×BH BL=AL×BL+AH×BL&lt;&lt;SIZE+BH×AL&lt;&lt;SIZE+AH×BH&lt;&lt;2×SIZE. Since doubling the size of the operand uses four multipliers of the original size to perform one multiplication at the larger size, the number of operations performed by the arithmetic circuit is reduced (in this case by a factor of four) when the size of the operands is increased (in this case by a factor of two). Each of the four component multiplication results is a partial product. The partial product results are summed to generate the final multiplication result. 
     Thus, in the second operation mode, the multiplier  520 D multiplies BL with AH and the multiplier  520 C multiplies BH with AL. The results are added by the adder  530 B and the result from the adder  530 B is shifted left eight bits. The multiplier  520 B multiplies BH with AH and the result is shifted left sixteen bits. The multiplier  520 A multiples BL with AL. Following the results through the adders  530 A and  550 A, the output of the adder  550 A is the result of the 16-bit multiply operation. The multipliers  520 E- 520 H and adders  530 C,  530 D, and  550 B are similarly configured to process a second 16-bit multiply operation. The results of the two operations are summed by the adder  560  and provided to the stage 2 delay registers  580  via the multiplexer  570 . 
     In some example embodiments, the portion  500  performs only a single 16-bit multiply in the second operation mode. In these embodiments, the results generated by the multipliers  520 E- 520 H and the adders  530 C,  530 D,  550 B, and  560  are ignored. Instead, the multiplexer  570  is configured to provide the output from the adder  550 A, containing the single 16-bit multiply result, to the stage 2 delay registers  580 . 
     In the third operation mode, the four 16-bit multiply operations provided by two instances of the portion  500  are combined in a manner analogous to that described with respect to the second operation mode, using an additional shifter and a wider adder, resulting in a circuit that determines a single 32-bit multiplication, making use of the adder  630  discussed below with respect to  FIG. 6 . 
     Though the portion  500  is described as performing multiplication operations on the selected inputs and then summing the result of the multiplication operations, other configurations of the arithmetic circuit are contemplated. For example, the inputs from the registers  510 A- 510 P may be provided to the multipliers  520 A- 520 H as shown and also be provided to a set of adders. Using a multiplexer for each multiplier/adder pair, the input to the adders  530 A- 530 D is selected either as the multiplication result or the addition result. Thus, based on a configuration signal controlling the multiplexers, the arithmetic circuit either determines a sum of the input operands or the sum of products of the input operands (as shown in  FIG. 5 ). 
       FIG. 6  is a diagrammatic view of a portion  600  of an arithmetic circuit that uses cascade communications between FPGA tiles, according to some example embodiments. The portion  600  comprises the multiplexer  570  of  FIG. 5  and the corresponding multiplexer  610  from a duplicate of the portion  500  that handles the high half of the inputs A and B. The outputs of the multiplexers  570  and  610  are provided to the stage 2 delay registers  620 A and  620 B, each 34 bits wide. Inputs to the portion  600  are also received from the stage 1 delay registers  630 A,  630 B,  630 C,  630 D,  630 E,  630 F,  630 G, and  630 H, storing the &lt;SGNA&gt;, &lt;SGNB&gt;, &lt;SGNC&gt;, &lt;SGND&gt;, &lt;EXPA&gt;, &lt;EXPB&gt;, &lt;EXPC&gt;, and &lt;EXPD&gt; values generated by the bit remap logics  430 A- 430 D of  FIG. 4 . The portion  600  further includes an adder  650 ; multiplexers  640 A,  640 B,  640 C,  640 D,  640 E,  640 F,  640 G,  640 H,  660 ,  680 A, and  680 B; multipliers  670 A and  670 B; and stage 3 delay registers  690 . 
     The results from the portion  500  and its duplicate are added by the adder  650 . The multiplexer  660  selects either the results from the portion  500  or the summed results from both portions, based on a value of an ADD0_15_BYPASS signal, and provides the selected result to the multiplier  670 A and the multiplexer  680 A. Based on the &lt;EXPA&gt;, &lt;EXPB&gt;, &lt;SGNA&gt;, and &lt;SGNB&gt; values received via the multiplexers  640 A- 640 D and the value received from the multiplexer  660 , the multiplier  670 A generates a 24-bit floating-point multiplication result. Similarly, based on the &lt;EXPC&gt;, &lt;EXPD&gt;, &lt;SGNC&gt;, and &lt;SGND&gt; values received via the multiplexers  640 E- 640 H and the result received from the register  620 B, the multiplier  670 B generates a second 24-bit floating-point multiplication result. Based on an FPMULT_AB signal, the multiplexers  680 A- 680 B output either the 24-bit floating-point results generated by the multipliers  670 A- 670 B or pass through the results provided by the register  620 B and the multiplexer  660 . The outputs of the multiplexers  680 A- 680 B are provided to the stage 3 delay registers  690 . 
     Thus, in one operation mode, the outputs of the multiplexers  680 A- 680 B of the portion  600  are the outputs of the portion  500  and its duplicate portion, bypassing the adder  650  and the multipliers  670 A- 670 B. In a second operation mode, the output of the multiplexer  680 A is the sum of all multiplies performed by the portion  500  and its duplicate, and the output of the multiplexer  680 B is the sum of the multiplies performed by the duplicate of the portion  500 . In a third operation mode, the output of the multiplexers  680 A- 680 B are 24-bit floating-point versions of the outputs of the portion  500  and its duplicate portion. In a fourth operation mode, the output of the multiplexer  680 A is a 24-bit floating-point representation of the sum of all multiplies performed by the portion  500  and its duplicate, and the output of the multiplexer  680 B is a 24-bit floating-point representation of the sum of the multiplies performed by the duplicate of the portion  500 . 
       FIG. 7  is a diagrammatic view of a portion  700  of an arithmetic circuit that uses cascade communications between FPGA tiles, according to some example embodiments. The portion  700  includes the multiplexer  680 A of  FIG. 6 ; stage 1 delay registers  710 A,  710 B,  710 C, and  710 D; multiplexers  720 A,  720 B,  720 C,  720 D,  740 A,  740 B,  750 A,  750 B,  770 , and  780 ; stage 3 delay registers  730 A and  730 B; adders  760 A and  760 B; stage 4 delay register  790 . Connections between the portion  700  and upper block  795  are also shown in  FIG. 7 . The upper block  795  refers to the portion of the arithmetic circuit shown in  FIG. 8 . 
     The output of the multiplexer  680 A is stored in the stage 3 delay register  730 A. The stage 3 delay register  730 B stores either the output of the multiplexer  660 B of  FIG. 6  or an addition result generated by adding the output of the multiplexer  660 B to FWDI_DOUT[47:0], as described below with respect to  FIG. 8 . 
     The multiplexer  750 A selects a value from FWDI_DOUT[47:0], REGFILE_DOUT[47:0], 48 bits from the LRAM  125 ; REGFILE_DOUT[95:48], a different 48 bits from the LRAM  125 ; delay register  730 B; and { 24 ′H0, FWDI_DOUT[47:24]}, 24 0 bits prepended to 24 bits of the operand cascade input. The multiplexer  750 B selects a value from the stage 3 delay registers  730 A and  730 B. The outputs of the multiplexers  750 A and  750 B are provided to the adder  760 A and the adder  760 B. Based on the SUB_AB_DEL and LOAD_AB_DEL signals received from the multiplexers  740 A and  740 B and the selected values received from the multiplexers  750 A and  750 B, the adder  760 A generates an addition result. Based on the SUB_AB_DEL and LOAD_AB_DEL signals received from the multiplexers  740 A and  740 B and the selected values received from the multiplexers  750 A and  750 B, the adder  760 B generates an addition result or a subtraction result. The SUB signals control whether the adders  760 A and  760 B generate addition or subtraction results. The LOAD signals control whether the adders  760 A and  760 B add the input value to the accumulated value or ignore the accumulated value and merely load the input value, providing the input value as the output and setting the accumulator value to the input value. The DEL signals have a delay of 0-4 cycles. 
     The bypass multiplexer  770  selects either the result generated by the adder  760 B or the result of the multiplexer  750 B. Thus, bypass multiplexer  770  provides either an addition result from the portion  700  or either result from  FIG. 6 . The multiplexer  780  selects either the output of the multiplexer  770  or the output of the adder  760 A and provides the result to the stage 4 delay register  790 . 
       FIG. 8  is a diagrammatic view of a portion  800  of an arithmetic circuit that uses cascade communications between FPGA tiles, according to some example embodiments. The portion  800  corresponds to the upper block  795  shown in  FIG. 7 . The portion  800  includes the multiplexer  660 B of  FIG. 6 ; the multiplexers  720 C and  720 D of  FIG. 7 ; adders  840 A and  840 B; the stage 4 delay register  790 ; multiplexers  830 A,  830 B,  860 A,  860 B,  860 C,  860 D, and  880 ; logic blocks  850 A and  850 B; and output register  870 . The portion  700  of  FIG. 7  is represented in  FIG. 8  as lower block  890 . 
     The multiplexer  830 A selects a value from FWDI_DOUT[47:0]; REGFILE_DOUT[71:0]; REGFILE_DOUT[47:24], 24 bits from the LRAM  125 ; and value from the output register  870 , received via a feedback path. The multiplexer  830 B selects either the value from the multiplexer  660 B or the value from the stage 4 delay register  790 . 
     The adder  840 A sums the outputs from the multiplexers  830 A and  830 B, as modified by the SUB_REG and LOAD_REG signals. The SUB signals control whether the adder  840 A generates addition or subtraction results. The LOAD signals control whether the adder  840 A adds the input value to the accumulated value or ignores the accumulated value and merely loads the input value, providing the input value as the output and setting the accumulator value to the input value. As SUB_REG and LOAD_REG are not DEL signals, there is no delay in handling the inputs. The adder  840 B adds the outputs from the multiplexers  830 A- 830 B or takes the difference, depending on the SUB_REG signal. The multiplexers  860 B and  860 C select either the outputs of the adders  840 A- 840 B or, based on an FPADD_CD_BYPASS signal, provide the output from the multiplexer  830 B. The multiplexer  860 D selects an output from the multiplexer  860 B, the multiplexer  860 C, a MULT8BYP input, and a MULT16BYP input. If a bypass signal is used, the MULT8BYP signal is selected when the circuit is configured to perform 8-bit operations, and the MULT16BYP signal is selected when the circuit is configured to perform 16-bit operations. 
     The output of the multiplexer  860 D is stored in the output register  870 . If the circuit is configured to perform a floating-point format conversion (e.g., from an internal 24-bit floating-point format to a 16-bit output floating-point format), the value in the output register  870  is processed by the logic block  850 B before being provided as an input to the multiplexer  860 A. Likewise, if the circuit is configured to perform the floating-point format conversion, the value in the stage 4 delay register  790  is processed by the logic block  850 A. The multiplexer  860 A selects an input from the processed and unprocessed values of the registers  790  and  870 . The output of the multiplexer  860 A is provided as FWDO_DOUT[47:0], a 48-bit operand cascade output. 
     In a floating-point mode that differs from the floating-point format used by the portions  500 - 700 , the logics  850 A- 850 B convert the intermediate outputs to a format expected by the FPGA. In an example, the portions  500 - 700  operates on floating-point values with 16-bit mantissas and 8-bit exponents. In this example, the multiple mode arithmetic circuit supports inputs and outputs using various combinations of 16-bit mantissas, 10-bit mantissas, 12-bit mantissas, 8-bit exponents, 6-bit exponents, and 5-bit exponents. Based on the output format and the format operated on the portions  500 - 700 , the logics  850 A- 850 B convert the output values. In some example embodiments, the internal floating-point format used by the portions  500 - 700  has a mantissa size at least as large as the mantissa size of the input/output floating-point format. 
     Integer accumulators can be used for floating-point operations in special cases. In embodiments with both integer and floating-point accumulators, the integer accumulator may be wider than the floating-point accumulator, allowing block floating-point operations to use the integer accumulator without loss of data when the floating-point exponents are close in value. 
     The multiplexer  880  selects, based on an OUTPUT_SEL signal, the output value for the circuit from the output of the multiplexer  860 A; LRAM_DOUT[71:0], a 72-bit value read from the LRAM  125 ; and BRAM_DOUT[143:72]. The selected output value is provided as DOUT[71:0], a 72-bit output provided to the routing fabric  105 . 
       FIG. 9  is a high-level diagrammatic view  900  showing connections between tiles of an FPGA using cascade communications, according to some example embodiments. The connected routing  905  and  910  are also shown. A first tile includes an MLP  915 A and a BRAM  920 A. A second tile includes an MLP  915 B and a BRAM  920 B. Each tile may include an LRAM. The MLP  915 A comprises a floating-point MAC  930 A and an integer MAC  935 A. The BRAM  920 A comprises a memory  940 A. The MLP  915 B comprises a floating-point MAC  930 B and an integer MAC  935 B. The BRAM  920 B comprises a memory  940 B. The tiles are connected to other tiles via the routing  905  and the routing  910 . 
     The operand cascade output  960 A is connected directly to the operand cascade input  945 B, allowing data to be communicated from the MLP  915 A to the MLP  915 B without using the routing  905 . The memory cascade output  965 A is connected directly to the memory cascade input  955 B, allowing data to be communicated from the BRAM  920 A to the BRAM  920 B without using the routing  910 . The operand cascade input  945 A and the memory cascade input  955 A allow the MLP  915 A and the BRAM  920 A to be connected to another FPGA tile. Similarly, the operand cascade output  960 B and the memory cascade output  965 B allow the MLP  915 B and the BRAM  920 B to be connected to another FPGA tile. The cascade connections may be unidirectional or bidirectional. 
     Though  FIG. 9  shows only two tiles being connected via cascade connections, additional tiles are also connected in this manner in some example embodiments. Thus, any number of tiles may be connected using cascade connections. Each of the cascaded tiles may be configured to pass through the operands (e.g., to set the operand cascade output to the operand cascade input), to pass through the memory data in either direction (e.g., to set the memory cascade output to the memory cascade input or to set the memory cascade input to the memory cascade output), or both. Additionally or alternatively, each of the cascaded tiles may be configured to generate results using any combination of inputs from the routing  905 , the routing  910 , the operand cascade input  945 , the memory cascade input  955 , the memory cascade output  965 , and the memory  940 . 
     In some example embodiments, the operand cascade connections are used to share inputs across a number of MLPs. Thus, the system making use of the MLPs only uses the routing  905  to provide the inputs to a single MLP and the connected MLPs share the input values. This is useful in implementing larger multipliers or other arithmetic operators. For example, a multiplier that multiplies two 16-bit numbers may be formed from four 8-bit multipliers. If a single MLP provides a single 8-bit multiplier, four MLPs may be cascaded to generate the result by providing the operands to a single MLP that shares the data with the other MLPs via cascade connections. By comparison with an FPGA that uses traditional connections, half of the routing is saved, since only the 32 bits of input needed to define the operation are sent through the routing instead of sending 64 bits of input (16 bits of input to each of the four MLPs). 
     As another example of the use of cascade communications, the operand cascade connections are used to share outputs between MLPs. For example, each MLP may calculate a sum of partial products. When the user desires a sum of more partial products than can be determined by a single MLP (e.g., when performing a matrix multiply with a large number of elements), each MLP in a cascade can provide its summation output to the next MLP. The next MLP determines the sum of its own partial products and the cascade input. As a result, the last MLP in the cascade provides, as an output, the sum of partial products of the entire cascade. By comparison with an FPGA that uses traditional connections, time and routing are both saved. To find the sum of the outputs of multiple MLPs using the routing  905 , the outputs of the various MLPs would be sent, in existing FPGAs, as input to another MLP to sum the individual sums of partial products to determine the entire sum, but the additional routing between MLPs is not needed when the MLPs communicate through cascade connections. Additionally, the sequential operation of waiting until all MLPs had completed processing before beginning to determine the sum of sums of partial products introduces delays that are avoided when the output of each MLP is provided directly to the next MLP via the cascade connection. For example, the result is obtained in one clock cycle instead of two by performing the summations of partial products in the same clock cycle as the multiplies. 
     As a third example of the use of cascade communications, one operand is provided to each of multiple MLPs through the cascade chain and another operand is provided to each of the multiple MLPs using the routing fabric. For example, the cascade connection may be wide enough to hold one operand for each MLP (e.g., each MLP may operate on 8-bit operands and the cascade connection is 32 bits wide, allowing the connection to hold one operand for each of four MLPs). Even though the total routing used in the same in this example, the routing may be simplified by virtue of providing more connections to a single MLP instead of needing to route all operands to each MLP. 
     As a fourth example of the use of cascade communications, data or address information is communicated between BRAMs through the cascade chain. As a result, larger memories are created from smaller BRAMs. For example, the high address bits indicate which BRAM should process a read or write request. In this example, routing connects to only a single BRAM, but the request (and read data, in the case of a read request) are communicated between BRAMs using the cascade chain, providing more memory addresses of the same width as a single BRAM. As another example, each BRAM simultaneously responds to the request using the cascade address, but handles a separate portion of input data. This provides greater data width using the same number of memory addresses as a single BRAM. 
     In some example embodiments, the BRAM forward cascade connection comprises 144 bits of write data, a 14-bit address, a 7-bit block address, and control signals. The 14-bit address allows identification of one of 16,384 memory locations in a single BRAM. The 7-bit block address allows a particular BRAM of up to 128 cascade-connected BRAMs to be selected. The control signals indicate, e.g., whether data is being read or written. The BRAM reverse cascade connection comprises 144 bits of read data, the 14-bit address, the 7-bit block address, and the control signals. Read data is transferred from the BRAM identified by the block address so that the first BRAM in the cascade chain can provide the read data as output. 
     The block address can be set to target a specific BRAM for read or write. Thus, in an example embodiment, the BRAM forward cascade connection transmits data unchanged to all of the cascaded tiles, each of which has a unique block address. If the block address of the BRAM matches the block address indicated by the cascade connection, the BRAM performs the indicated operation. The other BRAMs do not perform the operation. 
     In some example embodiments, the control signals of the BRAM forward cascade connection comprise a mask. By applying a bitwise OR of the mask to the block address of the BRAM before comparing the masked block address of the BRAM to the block address indicated by the cascade connection, some bits of the BRAM block address are ignored when making the comparison. As a result, multiple tiles may match the indicated block address and perform the indicated operation. For example, if the block address for the operation is b′0001100 (decimal 12) and the mask is b′1111100, the four BRAMs having block addresses b′0001100 (decimal 12) to b′0001111 (decimal 15) will perform the operation. This may prove advantageous in machine learning applications, wherein multiple different operations are performed on input vectors. Thus, when each MLP has on-tile access to a BRAM storing the input vector (or a portion thereof), performance is improved. 
     As can be seen in  FIG. 9 , the operand cascade output  960 A is configured to provide any one of: data received via the routing  905 , data received at the operand cascade input  945 A, the result from the MAC  930 A, and the result from the MAC  935 A. Similarly, the operand cascade output  960 B is configured to provide any one of: data received via the routing  905 , data received at the operand cascade input  945 B, the result from the MAC  930 B, and the result from the MAC  935 B. Thus, when multiple FPGA tiles are connected by connecting the operand cascade output (e.g., the operand cascade output  960 A) of one tile to the operand cascade input of the next tile (e.g., the operand cascade input  945 B), a single input from the routing  905  can be transmitted via the cascade connections to the remaining FPGA tiles. By comparison to existing methods in which routing  905  resources would be used for each of the multiple FPGA tiles, substantial routing resources are saved. As a practical matter, provision of the same operand to multiple MLPs is useful in performing matrix multiplication operations, particularly for machine learning applications. 
     Similarly, the memory cascade output  965 A is configured to provide any one of: data received via the routing  910 , data received at the memory cascade input  955 A, and data read from the memory  940 A; the memory cascade output  965 B is configured to provide any one of: data received via the routing  910 , data received at the memory cascade input  955 B, and data read from the memory  940 B Additionally, the memory cascade connections are bidirectional. Accordingly, while the two cascade connections are labeled as “input” and “output,” data may flow in either direction. Thus, an address may be propagated in one direction through multiple FPGA tiles via the cascade connections and a data result may be propagated in the other direction. As with the MLPs, using cascade connections for multiple BRAMs saves routing resources. 
       FIG. 10  is a flowchart illustrating operations of a method  1000  performed by a fused memory and arithmetic circuit, according to various embodiments of the invention. The method  1000  includes operations  1010 ,  1020 ,  1030 ,  1040 ,  1050 ,  1060 , and  1070 . By way of example and not limitation, the method  1000  is described as being performed by the circuits of  FIGS. 1-9 . 
     In operation  1010 , a first arithmetic circuit of a first tile of an FPGA receives a first set of inputs from a connection fabric of an FPGA. As an example, the MLP  915 A receives the first set of inputs from the routing fabric  905  of an FPGA. In this example, the first set of inputs comprises a row of a first matrix, for element-wise multiplication with a column of a second matrix in determining a partial result of a matrix multiplication. 
     The first arithmetic circuit, in operation  1020 , receives a second set of inputs from the connection fabric of the FPGA. In this example, the MLP  915 A receives the second set of inputs, comprising the column of the second matrix, from the routing fabric  905 . 
     In operation  1030 , the first arithmetic circuit generates a first result based on a first subset of the first set of inputs and a second subset of the second set of inputs. For example, the first result may be the product of a single element of the row of the first matrix and a single element of the column of the second matrix. As another example, the first result may be the sum of several products of elements of the row of the first matrix and corresponding elements of the column of the second matrix. 
     The tile of the FPGA provides, in operation  1040 , the first set of inputs as a cascade output. In this example, the inputs at the operand cascade input  945 A are routed to the operand cascade output  960 A. 
     A second arithmetic circuit of a second tile of the FPGA, in operation  1050 , receives a third set of inputs from the connection fabric. In this example, the MLP  915 B receives the third set of inputs, comprising a second column of the second matrix, from the routing fabric  905 . 
     In operation  1060 , the second arithmetic circuit receives the first set of inputs directly from the cascade output of the first tile. Thus, in this example, the MLP  915 A receives a row and a column through the routing  905 , but the MLP  915 B receives the same amount of data while only using half as much of the routing  905 . Furthermore, the MLP  915 B may be configured to propagate the first set of inputs directly from the operand cascade input  945 B to the operand cascade output  960 B, allowing another tile to also receive the first set of inputs without consuming routing resources. This process may be repeated to provide the first set of inputs to an arbitrary number of tiles. 
     The second arithmetic circuit, in operation  1070 , generates a second result based on the first subset of the first set of inputs and a third subset of the third set of inputs. As used herein, a subset may comprise the entirety of a set. Thus, if 64 bits of data are provided by the cascade input and 64 bits of data are provided by the routing fabric, the arithmetic result may be based on any portion of the inputs, so long as it depends on both inputs. As an example, each of the 64 bits of input may be treated as eight 8-bit operands and the generated result may be the sum of eight multiplication operations performed on pairs of 8-bit operands, one operand in each pair received via the intra-tile communication and one operand in each pair received via the routing fabric of the FPGA. 
       FIG. 11  is a block diagram illustrating components of a computer  1100  that programs an FPGA, according to some example embodiments. All components need not be used in various embodiments. For example, clients, servers, autonomous systems, and cloud-based network resources may each use a different set of components, or, in the case of servers, for example, larger storage devices. 
     One example computing device in the form of a computer  1100  (also referred to as computing device  1100  and computer system  1100 ) may include a processor  1105 , memory storage  1110 , removable storage  1115 , and non-removable storage  1120 , all connected by a bus  1140 . Although the example computing device is illustrated and described as the computer  1100 , the computing device may be in different forms in different embodiments. For example, the computing device may instead be a smartphone, a tablet, a smartwatch, or another computing device including elements the same as or similar to those illustrated and described with regard to  FIG. 11 . Devices such as smartphones, tablets, and smartwatches are collectively referred to as “mobile devices.” Further, although the various data storage elements are illustrated as part of the computer  1100 , the storage may also or alternatively include cloud-based storage accessible via a network, such as the Internet, or server-based storage. 
     The memory storage  1110  may include volatile memory  1145  and non-volatile memory  1150  and may store a program  1155 . The computer  1100  may include, or have access to, a computing environment that includes a variety of computer-readable media, such as the volatile memory  1145 ; the non-volatile memory  1150 ; the removable storage  1115 ; and the non-removable storage  1120 . Computer storage includes random-access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM) and electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, compact disc read-only memory (CD ROM), digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium capable of storing computer-readable instructions. 
     The computer  1100  may include or have access to a computing environment that includes an input interface  1125 , an output interface  1130 , and a communication interface  1135 . The output interface  1130  may interface to or include a display device, such as a touchscreen, that also may serve as an input device. The input interface  1125  may interface to or include one or more of a touchscreen, a touchpad, a mouse, a keyboard, a camera, one or more device-specific buttons, one or more sensors integrated within or coupled via wired or wireless data connections to the computer  1100 , and other input devices. The computer  1100  may operate in a networked environment using the communication interface  1135  to connect to one or more remote computers, such as database servers. The remote computer may include a personal computer (PC), server, router, network PC, peer device or other common network node, or the like. The communication interface  1135  may connect to a local-area network (LAN), a wide-area network (WAN), a cellular network, a WiFi network, a Bluetooth network, or other networks. 
     Computer instructions stored on a computer-readable medium (e.g., the program  1155  stored in the memory storage  1110 ) are executable by the processor  1105  of the computer  1100 . A hard drive, CD-ROM, and RAM are some examples of articles including a non-transitory computer-readable medium such as a storage device. The terms “computer-readable medium” and “storage device” do not include carrier waves to the extent that carrier waves are deemed too transitory. “Computer-readable non-transitory media” includes all types of computer-readable media, including magnetic storage media, optical storage media, flash media, and solid-state storage media. It should be understood that software can be installed in and sold with a computer. Alternatively, the software can be obtained and loaded into the computer, including obtaining the software through a physical medium or distribution system, including, for example, from a server owned by the software creator or from a server not owned but used by the software creator. The software can be stored on a server for distribution over the Internet, for example. 
     The program  1155  is shown as including a design module  1160  and a place and route module  1165 . Any one or more of the modules described herein may be implemented using hardware (e.g., a processor of a machine, an ASIC, an FPGA, or any suitable combination thereof). Moreover, any two or more of these modules may be combined into a single module, and the functions described herein for a single module may be subdivided among multiple modules. Furthermore, according to various example embodiments, modules described herein as being implemented within a single machine, database, or device may be distributed across multiple machines, databases, or devices. 
     The design module  1160  defines a design of a circuit (e.g., a processor, signal processor, compute engine, state machine, or controller circuit). For example, the design module  1160  may provide a user interface to allow a user to design a circuit. 
     The place and route module  1165  determines the physical layout of the resulting integrated circuit based on the circuit design defined by the design module  1160 . For example, a design comprising one or more tiles with fused memory and arithmetic circuits may be laid out by the place and route module  1165  in order to be programmed into the FPGA configuration. 
     The Abstract of the Disclosure is provided to comply with 37 C.F.R. § 1.72(b), requiring an abstract that allows the reader to quickly ascertain the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the claims. In addition, in the foregoing Detailed Description, it may be seen that various features are grouped together in a single embodiment for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as limiting the claims. Thus, the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separate embodiment.