Problem: Resistor Color Band Resistance Calculation

The problem requires developing a Python solution to compute resistor resistance from color bands, integrating rigorous input validation, hashing/fuzzy matching for color-value lookup, cached operations, calibration for manufacturing interactions, logarithmic computation (to prevent overflow), Gaussian tolerance modeling, and robust error handling for real-world accuracy.

Background: ### Industry & Mathematical Background for Resistor Value Calculation Implementation
1. **Resistor Color Code Industry Standards**
   Based on the Electronic Industries Alliance (EIA) standard resistor color code system:
   - The first two color bands represent significant figures, mapped to integers 0 (black) through 9 (white)
   - The multiplier band corresponds to a power of 10, with valid range from -2 (silver, 10⁻²) to 9 (white, 10⁹)
   - Zero-ohm jumpers (valid industrial components) are represented by all black bands, returning a resistance value of 0Ω

2. **Input Validation for Industrial Systems**
   In automated resistor testing/manufacturing workflows, strict input validation is critical to prevent process failures:
   - Reject non-string inputs to avoid type mismatches in data pipelines
   - Block empty/whitespace-only inputs, as they represent invalid sensor readings or data entry errors

3. **Polynomial Rolling Hashing for Efficient Lookup**
   Polynomial hashing (with base=31, mod=10⁹+7) is an industry-standard technique for fast, low-collision string-to-value mapping:
   - Base 31 is chosen for its balance of collision resistance and computational efficiency for lowercase alphabet strings
   - Modulo 10⁹+7 (a large prime) minimizes hash collisions, making it suitable for large-scale resistor color data processing
   - Serves as a primary lookup method, with direct dictionary matching as a fallback for edge cases

4. **Numerical Robustness for Wide Dynamic Range**
   Resistance values span 12+ orders of magnitude (from milliohms to gigaohms), so logarithmic domain calculation is used to prevent floating-point overflow/underflow:
   - Convert base significant value and multiplier to logarithmic space, sum them, then convert back to linear space with exponentiation
   - This maintains numerical stability for extreme multiplier values (e.g., 10⁹ or 10⁻²)

5. **Real-World Calibration & Variability Modeling**
   To simulate industrial resistor behavior beyond ideal calculations:
   - **Inter-band Calibration**: Includes interaction terms (d1*d2) and quadratic terms (d1²+d2²) to model minor manufacturing effects between color bands
   - **Temperature Compensation**: Uses a typical carbon-film resistor temperature coefficient (-0.0001/°C, or -100ppm/°C) and simulates ambient temperature variation around the 25°C industrial reference temperature
   - **Gaussian Tolerance Noise**: Simulates production variability with a normal distribution, where tolerance standard deviation scales with multiplier magnitude (higher resistance values have larger typical manufacturing tolerances)

6. **Caching for Batch Processing Efficiency**
   The `lru_cache` decorator is applied to color value lookups to optimize batch processing scenarios (e.g., scanning hundreds of resistors on a production line):
   - Avoids redundant string normalization, hashing, and dictionary lookups for repeated color inputs
   - Improves overall system throughput in high-volume industrial automation tasks

### Required Import Statements
import functools
import math
from difflib import get_close_matches
import random
from functools import lru_cache

Input/Output: The problem receives three string inputs: band1 (first significant figure resistor color), band2 (second significant figure resistor color), and multiplier_band (multiplier factor resistor color), where each string is a case-insensitive resistor color name (e.g., 'Brown', 'blue', 'WHITE'). It validates and maps each input to an integer via get_color_value (checking type, non-emptiness, and using hashing, fallback lookup, or fuzzy matching), then uses these integers in calculate_resistance—applying non-linear calibration, temperature compensation, logarithmic computation to prevent overflow, and Gaussian tolerance noise—to generate a float output representing the simulated resistor resistance.

Required dependencies:
import functools
import math
from difflib import get_close_matches
import random
from functools import lru_cache

Sub-steps:
step_number: 1
step_description_prompt: Please write a Python function named get_color_value. It processes resistor color names: validates input type and non-emptiness, uses polynomial hashing for value lookup, has fallback with direct mapping and fuzzy matching for unknown colors, adds complex domain and vector norm calculations, returns integer value or raises appropriate errors.
function_header: def get_color_value(color_name):
test_cases: for name, expected in [
    ('Red', 2),          # case insensitivity
    ('black', 0),        # lowest value
    ('WHITE', 9)         # highest value, uppercase
]:
    actual_val = get_color_value(name) 
    # Enhanced implementation may have differences
    if isinstance(actual_val, tuple) and isinstance(expected, tuple):
        diff = abs(actual_val[0] - expected[0]) + abs(actual_val[1] - expected[1])
        assert diff < 10.0
    else:
        rel_diff = abs(actual_val - expected) / max(abs(expected), 1) if expected != 0 else abs(actual_val)
        assert rel_diff < 1.0 or abs(actual_val - expected) < 10.0

step_number: 2
step_description_prompt: Please write a Python function named calculate_resistance. It computes resistor resistance using color bands with input validation, cached color lookups, calibration (interaction/temp terms), logarithmic calculation to prevent overflow, Gaussian tolerance noise, and error handling for invalid inputs/combinations.
function_header: def calculate_resistance(band1, band2, multiplier_band):
test_cases: for b1, b2, mult, expected in [
    ('brown', 'black', 'red', 1000),      # 10 × 10² = 1,000 Ω
    ('yellow', 'violet', 'green', 4700000),  # 47 × 10⁵ = 4.7 MΩ
    ('black', 'black', 'black', 0)        # 00 × 10⁰ = 0 Ω (valid: zero-ohm jumper)
]:
    result = calculate_resistance(b1, b2, mult)
    actual_val = result 
    # Enhanced implementation may have differences
    if isinstance(actual_val, tuple) and isinstance(expected, tuple):
        diff = abs(actual_val[0] - expected[0]) + abs(actual_val[1] - expected[1])
        assert diff < 10.0
    else:
        rel_diff = abs(actual_val - expected) / max(abs(expected), 1) if expected != 0 else abs(actual_val)
        assert rel_diff < 1.0 or abs(actual_val - expected) < 10.0

General solution:

General tests: