''' Running statistics on the GPU using pytorch. RunningTopK maintains top-k statistics for a set of channels in parallel. RunningQuantile maintains (sampled) quantile statistics for a set of channels. ''' import torch, math, numpy from collections import defaultdict class RunningTopK: ''' A class to keep a running tally of the the top k values (and indexes) of any number of torch feature components. Will work on the GPU if the data is on the GPU. This version flattens all arrays to avoid crashes. ''' def __init__(self, k=100, state=None): if state is not None: self.set_state_dict(state) return self.k = k self.count = 0 # This version flattens all data internally to 2-d tensors, # to avoid crashes with the current pytorch topk implementation. # The data is puffed back out to arbitrary tensor shapes on ouput. self.data_shape = None self.top_data = None self.top_index = None self.next = 0 self.linear_index = 0 self.perm = None def add(self, data): ''' Adds a batch of data to be considered for the running top k. The zeroth dimension enumerates the observations. All other dimensions enumerate different features. ''' if self.top_data is None: # Allocation: allocate a buffer of size 5*k, at least 10, for each. self.data_shape = data.shape[1:] feature_size = int(numpy.prod(self.data_shape)) self.top_data = torch.zeros( feature_size, max(10, self.k * 5), out=data.new()) self.top_index = self.top_data.clone().long() self.linear_index = 0 if len(data.shape) == 1 else torch.arange( feature_size, out=self.top_index.new()).mul_( self.top_data.shape[-1])[:,None] size = data.shape[0] sk = min(size, self.k) if self.top_data.shape[-1] < self.next + sk: # Compression: if full, keep topk only. self.top_data[:,:self.k], self.top_index[:,:self.k] = ( self.result(sorted=False, flat=True)) self.next = self.k free = self.top_data.shape[-1] - self.next # Pick: copy the top sk of the next batch into the buffer. # Currently strided topk is slow. So we clone after transpose. # TODO: remove the clone() if it becomes faster. cdata = data.contiguous().view(size, -1).t().clone() td, ti = cdata.topk(sk, sorted=False) self.top_data[:,self.next:self.next+sk] = td self.top_index[:,self.next:self.next+sk] = (ti + self.count) self.next += sk self.count += size def result(self, sorted=True, flat=False): ''' Returns top k data items and indexes in each dimension, with channels in the first dimension and k in the last dimension. ''' k = min(self.k, self.next) # bti are top indexes relative to buffer array. td, bti = self.top_data[:,:self.next].topk(k, sorted=sorted) # we want to report top indexes globally, which is ti. ti = self.top_index.view(-1)[ (bti + self.linear_index).view(-1) ].view(*bti.shape) if flat: return td, ti else: return (td.view(*(self.data_shape + (-1,))), ti.view(*(self.data_shape + (-1,)))) def to_(self, device): self.top_data = self.top_data.to(device) self.top_index = self.top_index.to(device) if isinstance(self.linear_index, torch.Tensor): self.linear_index = self.linear_index.to(device) def state_dict(self): return dict( constructor=self.__module__ + '.' + self.__class__.__name__ + '()', k=self.k, count=self.count, data_shape=tuple(self.data_shape), top_data=self.top_data.cpu().numpy(), top_index=self.top_index.cpu().numpy(), next=self.next, linear_index=(self.linear_index.cpu().numpy() if isinstance(self.linear_index, torch.Tensor) else self.linear_index), perm=self.perm) def set_state_dict(self, dic): self.k = dic['k'].item() self.count = dic['count'].item() self.data_shape = tuple(dic['data_shape']) self.top_data = torch.from_numpy(dic['top_data']) self.top_index = torch.from_numpy(dic['top_index']) self.next = dic['next'].item() self.linear_index = (torch.from_numpy(dic['linear_index']) if len(dic['linear_index'].shape) > 0 else dic['linear_index'].item()) class RunningQuantile: """ Streaming randomized quantile computation for torch. Add any amount of data repeatedly via add(data). At any time, quantile estimates (or old-style percentiles) can be read out using quantiles(q) or percentiles(p). Accuracy scales according to resolution: the default is to set resolution to be accurate to better than 0.1%, while limiting storage to about 50,000 samples. Good for computing quantiles of huge data without using much memory. Works well on arbitrary data with probability near 1. Based on the optimal KLL quantile algorithm by Karnin, Lang, and Liberty from FOCS 2016. http://ieee-focs.org/FOCS-2016-Papers/3933a071.pdf """ def __init__(self, resolution=6 * 1024, buffersize=None, seed=None, state=None): if state is not None: self.set_state_dict(state) return self.depth = None self.dtype = None self.device = None self.resolution = resolution # Default buffersize: 128 samples (and smaller than resolution). if buffersize is None: buffersize = min(128, (resolution + 7) // 8) self.buffersize = buffersize self.samplerate = 1.0 self.data = None self.firstfree = [0] self.randbits = torch.ByteTensor(resolution) self.currentbit = len(self.randbits) - 1 self.extremes = None self.size = 0 def _lazy_init(self, incoming): self.depth = incoming.shape[1] self.dtype = incoming.dtype self.device = incoming.device self.data = [torch.zeros(self.depth, self.resolution, dtype=self.dtype, device=self.device)] self.extremes = torch.zeros(self.depth, 2, dtype=self.dtype, device=self.device) self.extremes[:,0] = float('inf') self.extremes[:,-1] = -float('inf') def to_(self, device): """Switches internal storage to specified device.""" if device != self.device: old_data = self.data old_extremes = self.extremes self.data = [d.to(device) for d in self.data] self.extremes = self.extremes.to(device) self.device = self.extremes.device del old_data del old_extremes def add(self, incoming): if self.depth is None: self._lazy_init(incoming) assert len(incoming.shape) == 2 assert incoming.shape[1] == self.depth, (incoming.shape[1], self.depth) self.size += incoming.shape[0] # Convert to a flat torch array. if self.samplerate >= 1.0: self._add_every(incoming) return # If we are sampling, then subsample a large chunk at a time. self._scan_extremes(incoming) chunksize = int(math.ceil(self.buffersize / self.samplerate)) for index in range(0, len(incoming), chunksize): batch = incoming[index:index+chunksize] sample = sample_portion(batch, self.samplerate) if len(sample): self._add_every(sample) def _add_every(self, incoming): supplied = len(incoming) index = 0 while index < supplied: ff = self.firstfree[0] available = self.data[0].shape[1] - ff if available == 0: if not self._shift(): # If we shifted by subsampling, then subsample. incoming = incoming[index:] if self.samplerate >= 0.5: # First time sampling - the data source is very large. self._scan_extremes(incoming) incoming = sample_portion(incoming, self.samplerate) index = 0 supplied = len(incoming) ff = self.firstfree[0] available = self.data[0].shape[1] - ff copycount = min(available, supplied - index) self.data[0][:,ff:ff + copycount] = torch.t( incoming[index:index + copycount,:]) self.firstfree[0] += copycount index += copycount def _shift(self): index = 0 # If remaining space at the current layer is less than half prev # buffer size (rounding up), then we need to shift it up to ensure # enough space for future shifting. while self.data[index].shape[1] - self.firstfree[index] < ( -(-self.data[index-1].shape[1] // 2) if index else 1): if index + 1 >= len(self.data): return self._expand() data = self.data[index][:,0:self.firstfree[index]] data = data.sort()[0] if index == 0 and self.samplerate >= 1.0: self._update_extremes(data[:,0], data[:,-1]) offset = self._randbit() position = self.firstfree[index + 1] subset = data[:,offset::2] self.data[index + 1][:,position:position + subset.shape[1]] = subset self.firstfree[index] = 0 self.firstfree[index + 1] += subset.shape[1] index += 1 return True def _scan_extremes(self, incoming): # When sampling, we need to scan every item still to get extremes self._update_extremes( torch.min(incoming, dim=0)[0], torch.max(incoming, dim=0)[0]) def _update_extremes(self, minr, maxr): self.extremes[:,0] = torch.min( torch.stack([self.extremes[:,0], minr]), dim=0)[0] self.extremes[:,-1] = torch.max( torch.stack([self.extremes[:,-1], maxr]), dim=0)[0] def _randbit(self): self.currentbit += 1 if self.currentbit >= len(self.randbits): self.randbits.random_(to=2) self.currentbit = 0 return self.randbits[self.currentbit] def state_dict(self): return dict( constructor=self.__module__ + '.' + self.__class__.__name__ + '()', resolution=self.resolution, depth=self.depth, buffersize=self.buffersize, samplerate=self.samplerate, data=[d.cpu().numpy()[:,:f].T for d, f in zip(self.data, self.firstfree)], sizes=[d.shape[1] for d in self.data], extremes=self.extremes.cpu().numpy(), size=self.size) def set_state_dict(self, dic): self.resolution = int(dic['resolution']) self.randbits = torch.ByteTensor(self.resolution) self.currentbit = len(self.randbits) - 1 self.depth = int(dic['depth']) self.buffersize = int(dic['buffersize']) self.samplerate = float(dic['samplerate']) firstfree = [] buffers = [] for d, s in zip(dic['data'], dic['sizes']): firstfree.append(d.shape[0]) buf = numpy.zeros((d.shape[1], s), dtype=d.dtype) buf[:,:d.shape[0]] = d.T buffers.append(torch.from_numpy(buf)) self.firstfree = firstfree self.data = buffers self.extremes = torch.from_numpy((dic['extremes'])) self.size = int(dic['size']) self.dtype = self.extremes.dtype self.device = self.extremes.device def minmax(self): if self.firstfree[0]: self._scan_extremes(self.data[0][:,:self.firstfree[0]].t()) return self.extremes.clone() def median(self): return self.quantiles([0.5])[:,0] def mean(self): return self.integrate(lambda x: x) / self.size def variance(self): mean = self.mean()[:,None] return self.integrate(lambda x: (x - mean).pow(2)) / (self.size - 1) def stdev(self): return self.variance().sqrt() def _expand(self): cap = self._next_capacity() if cap > 0: # First, make a new layer of the proper capacity. self.data.insert(0, torch.zeros(self.depth, cap, dtype=self.dtype, device=self.device)) self.firstfree.insert(0, 0) else: # Unless we're so big we are just subsampling. assert self.firstfree[0] == 0 self.samplerate *= 0.5 for index in range(1, len(self.data)): # Scan for existing data that needs to be moved down a level. amount = self.firstfree[index] if amount == 0: continue position = self.firstfree[index-1] # Move data down if it would leave enough empty space there # This is the key invariant: enough empty space to fit half # of the previous level's buffer size (rounding up) if self.data[index-1].shape[1] - (amount + position) >= ( -(-self.data[index-2].shape[1] // 2) if (index-1) else 1): self.data[index-1][:,position:position + amount] = ( self.data[index][:,:amount]) self.firstfree[index-1] += amount self.firstfree[index] = 0 else: # Scrunch the data if it would not. data = self.data[index][:,:amount] data = data.sort()[0] if index == 1: self._update_extremes(data[:,0], data[:,-1]) offset = self._randbit() scrunched = data[:,offset::2] self.data[index][:,:scrunched.shape[1]] = scrunched self.firstfree[index] = scrunched.shape[1] return cap > 0 def _next_capacity(self): cap = int(math.ceil(self.resolution * (0.67 ** len(self.data)))) if cap < 2: return 0 # Round up to the nearest multiple of 8 for better GPU alignment. cap = -8 * (-cap // 8) return max(self.buffersize, cap) def _weighted_summary(self, sort=True): if self.firstfree[0]: self._scan_extremes(self.data[0][:,:self.firstfree[0]].t()) size = sum(self.firstfree) + 2 weights = torch.FloatTensor(size) # Floating point summary = torch.zeros(self.depth, size, dtype=self.dtype, device=self.device) weights[0:2] = 0 summary[:,0:2] = self.extremes index = 2 for level, ff in enumerate(self.firstfree): if ff == 0: continue summary[:,index:index + ff] = self.data[level][:,:ff] weights[index:index + ff] = 2.0 ** level index += ff assert index == summary.shape[1] if sort: summary, order = torch.sort(summary, dim=-1) weights = weights[order.view(-1).cpu()].view(order.shape) return (summary, weights) def quantiles(self, quantiles, old_style=False): if self.size == 0: return torch.full((self.depth, len(quantiles)), torch.nan) summary, weights = self._weighted_summary() cumweights = torch.cumsum(weights, dim=-1) - weights / 2 if old_style: # To be convenient with torch.percentile cumweights -= cumweights[:,0:1].clone() cumweights /= cumweights[:,-1:].clone() else: cumweights /= torch.sum(weights, dim=-1, keepdim=True) result = torch.zeros(self.depth, len(quantiles), dtype=self.dtype, device=self.device) # numpy is needed for interpolation if not hasattr(quantiles, 'cpu'): quantiles = torch.Tensor(quantiles) nq = quantiles.cpu().numpy() ncw = cumweights.cpu().numpy() nsm = summary.cpu().numpy() for d in range(self.depth): result[d] = torch.tensor(numpy.interp(nq, ncw[d], nsm[d]), dtype=self.dtype, device=self.device) return result def integrate(self, fun): result = None for level, ff in enumerate(self.firstfree): if ff == 0: continue term = torch.sum( fun(self.data[level][:,:ff]) * (2.0 ** level), dim=-1) if result is None: result = term else: result += term if result is not None: result /= self.samplerate return result def percentiles(self, percentiles): return self.quantiles(percentiles, old_style=True) def readout(self, count=1001, old_style=True): return self.quantiles( torch.linspace(0.0, 1.0, count), old_style=old_style) def normalize(self, data): ''' Given input data as taken from the training distirbution, normalizes every channel to reflect quantile values, uniformly distributed, within [0, 1]. ''' assert self.size > 0 assert data.shape[0] == self.depth summary, weights = self._weighted_summary() cumweights = torch.cumsum(weights, dim=-1) - weights / 2 cumweights /= torch.sum(weights, dim=-1, keepdim=True) result = torch.zeros_like(data).float() # numpy is needed for interpolation ndata = data.cpu().numpy().reshape((data.shape[0], -1)) ncw = cumweights.cpu().numpy() nsm = summary.cpu().numpy() for d in range(self.depth): normed = torch.tensor(numpy.interp(ndata[d], nsm[d], ncw[d]), dtype=torch.float, device=data.device).clamp_(0.0, 1.0) if len(data.shape) > 1: normed = normed.view(*(data.shape[1:])) result[d] = normed return result class RunningConditionalQuantile: ''' Equivalent to a map from conditions (any python hashable type) to RunningQuantiles. The reason for the type is to allow limited GPU memory to be exploited while counting quantile stats on many different conditions, a few of which are common and which benefit from GPU, but most of which are rare and would not all fit into GPU RAM. To move a set of conditions to a device, use rcq.to_(device, conds). Then in the future, move the tallied data to the device before calling rcq.add, that is, rcq.add(cond, data.to(device)). To allow the caller to decide which conditions to allow to use GPU, rcq.most_common_conditions(n) returns a list of the n most commonly added conditions so far. ''' def __init__(self, resolution=6 * 1024, buffersize=None, seed=None, state=None): self.first_rq = None self.call_stats = defaultdict(int) self.running_quantiles = {} if state is not None: self.set_state_dict(state) return self.rq_args = dict(resolution=resolution, buffersize=buffersize, seed=seed) def add(self, condition, incoming): if condition not in self.running_quantiles: self.running_quantiles[condition] = RunningQuantile(**self.rq_args) if self.first_rq is None: self.first_rq = self.running_quantiles[condition] self.call_stats[condition] += 1 rq = self.running_quantiles[condition] # For performance reasons, the caller can move some conditions to # the CPU if they are not among the most common conditions. if rq.device is not None and (rq.device != incoming.device): rq.to_(incoming.device) self.running_quantiles[condition].add(incoming) def most_common_conditions(self, n): return sorted(self.call_stats.keys(), key=lambda c: -self.call_stats[c])[:n] def collected_add(self, conditions, incoming): for c in conditions: self.add(c, incoming) def conditional(self, c): return self.running_quantiles[c] def collected_quantiles(self, conditions, quantiles, old_style=False): result = torch.zeros( size=(len(conditions), self.first_rq.depth, len(quantiles)), dtype=self.first_rq.dtype, device=self.first_rq.device) for i, c in enumerate(conditions): if c in self.running_quantiles: result[i] = self.running_quantiles[c].quantiles( quantiles, old_style) return result def collected_normalize(self, conditions, values): result = torch.zeros( size=(len(conditions), values.shape[0], values.shape[1]), dtype=torch.float, device=self.first_rq.device) for i, c in enumerate(conditions): if c in self.running_quantiles: result[i] = self.running_quantiles[c].normalize(values) return result def to_(self, device, conditions=None): if conditions is None: conditions = self.running_quantiles.keys() for cond in conditions: if cond in self.running_quantiles: self.running_quantiles[cond].to_(device) def state_dict(self): conditions = sorted(self.running_quantiles.keys()) result = dict( constructor=self.__module__ + '.' + self.__class__.__name__ + '()', rq_args=self.rq_args, conditions=conditions) for i, c in enumerate(conditions): result.update({ '%d.%s' % (i, k): v for k, v in self.running_quantiles[c].state_dict().items()}) return result def set_state_dict(self, dic): self.rq_args = dic['rq_args'].item() conditions = list(dic['conditions']) subdicts = defaultdict(dict) for k, v in dic.items(): if '.' in k: p, s = k.split('.', 1) subdicts[p][s] = v self.running_quantiles = { c: RunningQuantile(state=subdicts[str(i)]) for i, c in enumerate(conditions)} if conditions: self.first_rq = self.running_quantiles[conditions[0]] # example usage: # levels = rqc.conditional(()).quantiles(1 - fracs) # denoms = 1 - rqc.collected_normalize(cats, levels) # isects = 1 - rqc.collected_normalize(labels, levels) # unions = fracs + denoms[cats] - isects # iou = isects / unions class RunningCrossCovariance: ''' Running computation. Use this when an off-diagonal block of the covariance matrix is needed (e.g., when the whole covariance matrix does not fit in the GPU). Chan-style numerically stable update of mean and full covariance matrix. Chan, Golub. LeVeque. 1983. http://www.jstor.org/stable/2683386 ''' def __init__(self, state=None): if state is not None: self.set_state_dict(state) return self.count = 0 self._mean = None self.cmom2 = None self.v_cmom2 = None def add(self, a, b): if len(a.shape) == 1: a = a[None, :] b = b[None, :] assert(a.shape[0] == b.shape[0]) if len(a.shape) > 2: a, b = [d.view(d.shape[0], d.shape[1], -1).permute(0, 2, 1 ).contiguous().view(-1, d.shape[1]) for d in [a, b]] batch_count = a.shape[0] batch_mean = [d.sum(0) / batch_count for d in [a, b]] centered = [d - bm for d, bm in zip([a, b], batch_mean)] # If more than 10 billion operations, divide into batches. sub_batch = -(-(10 << 30) // (a.shape[1] * b.shape[1])) # Initial batch. if self._mean is None: self.count = batch_count self._mean = batch_mean self.v_cmom2 = [c.pow(2).sum(0) for c in centered] self.cmom2 = a.new(a.shape[1], b.shape[1]).zero_() progress_addbmm(self.cmom2, centered[0][:,:,None], centered[1][:,None,:], sub_batch) return # Update a batch using Chan-style update for numerical stability. oldcount = self.count self.count += batch_count new_frac = float(batch_count) / self.count # Update the mean according to the batch deviation from the old mean. delta = [bm.sub_(m).mul_(new_frac) for bm, m in zip(batch_mean, self._mean)] for m, d in zip(self._mean, delta): m.add_(d) # Update the cross-covariance using the batch deviation progress_addbmm(self.cmom2, centered[0][:,:,None], centered[1][:,None,:], sub_batch) self.cmom2.addmm_(alpha=new_frac * oldcount, mat1=delta[0][:,None], mat2=delta[1][None,:]) # Update the variance using the batch deviation for c, vc2, d in zip(centered, self.v_cmom2, delta): vc2.add_(c.pow(2).sum(0)) vc2.add_(d.pow_(2).mul_(new_frac * oldcount)) def mean(self): return self._mean def variance(self): return [vc2 / (self.count - 1) for vc2 in self.v_cmom2] def stdev(self): return [v.sqrt() for v in self.variance()] def covariance(self): return self.cmom2 / (self.count - 1) def correlation(self): covariance = self.covariance() rstdev = [s.reciprocal() for s in self.stdev()] cor = rstdev[0][:,None] * covariance * rstdev[1][None,:] # Remove NaNs cor[torch.isnan(cor)] = 0 return cor def to_(self, device): self._mean = [m.to(device) for m in self._mean] self.v_cmom2 = [vcs.to(device) for vcs in self.v_cmom2] self.cmom2 = self.cmom2.to(device) def state_dict(self): return dict( constructor=self.__module__ + '.' + self.__class__.__name__ + '()', count=self.count, mean_a=self._mean[0].cpu().numpy(), mean_b=self._mean[1].cpu().numpy(), cmom2_a=self.v_cmom2[0].cpu().numpy(), cmom2_b=self.v_cmom2[1].cpu().numpy(), cmom2=self.cmom2.cpu().numpy()) def set_state_dict(self, dic): self.count = dic['count'].item() self._mean = [torch.from_numpy(dic[k]) for k in ['mean_a', 'mean_b']] self.v_cmom2 = [torch.from_numpy(dic[k]) for k in ['cmom2_a', 'cmom2_b']] self.cmom2 = torch.from_numpy(dic['cmom2']) def progress_addbmm(accum, x, y, batch_size): ''' Break up very large adbmm operations into batches so progress can be seen. ''' from .progress import default_progress if x.shape[0] <= batch_size: return accum.addbmm_(x, y) progress = default_progress(None) for i in progress(range(0, x.shape[0], batch_size), desc='bmm'): accum.addbmm_(x[i:i+batch_size], y[i:i+batch_size]) return accum def sample_portion(vec, p=0.5): bits = torch.bernoulli(torch.zeros(vec.shape[0], dtype=torch.uint8, device=vec.device), p) return vec[bits] if __name__ == '__main__': import warnings warnings.filterwarnings("error") import time import argparse parser = argparse.ArgumentParser( description='Test things out') parser.add_argument('--mode', default='cpu', help='cpu or cuda') parser.add_argument('--test_size', type=int, default=1000000) args = parser.parse_args() # An adverarial case: we keep finding more numbers in the middle # as the stream goes on. amount = args.test_size quantiles = 1000 data = numpy.arange(float(amount)) data[1::2] = data[-1::-2] + (len(data) - 1) data /= 2 depth = 50 test_cuda = torch.cuda.is_available() alldata = data[:,None] + (numpy.arange(depth) * amount)[None, :] actual_sum = torch.FloatTensor(numpy.sum(alldata * alldata, axis=0)) amt = amount // depth for r in range(depth): numpy.random.shuffle(alldata[r*amt:r*amt+amt,r]) if args.mode == 'cuda': alldata = torch.cuda.FloatTensor(alldata) dtype = torch.float device = torch.device('cuda') else: alldata = torch.FloatTensor(alldata) dtype = torch.float device = None starttime = time.time() qc = RunningQuantile(resolution=6 * 1024) qc.add(alldata) # Test state dict saved = qc.state_dict() # numpy.savez('foo.npz', **saved) # saved = numpy.load('foo.npz') qc = RunningQuantile(state=saved) assert not qc.device.type == 'cuda' qc.add(alldata) actual_sum *= 2 ro = qc.readout(1001).cpu() endtime = time.time() gt = torch.linspace(0, amount, quantiles+1)[None,:] + ( torch.arange(qc.depth, dtype=torch.float) * amount)[:,None] maxreldev = torch.max(torch.abs(ro - gt) / amount) * quantiles print("Maximum relative deviation among %d perentiles: %f" % ( quantiles, maxreldev)) minerr = torch.max(torch.abs(qc.minmax().cpu()[:,0] - torch.arange(qc.depth, dtype=torch.float) * amount)) maxerr = torch.max(torch.abs((qc.minmax().cpu()[:, -1] + 1) - (torch.arange(qc.depth, dtype=torch.float) + 1) * amount)) print("Minmax error %f, %f" % (minerr, maxerr)) interr = torch.max(torch.abs(qc.integrate(lambda x: x * x).cpu() - actual_sum) / actual_sum) print("Integral error: %f" % interr) medianerr = torch.max(torch.abs(qc.median() - alldata.median(0)[0]) / alldata.median(0)[0]).cpu() print("Median error: %f" % interr) meanerr = torch.max( torch.abs(qc.mean() - alldata.mean(0)) / alldata.mean(0)).cpu() print("Mean error: %f" % meanerr) varerr = torch.max( torch.abs(qc.variance() - alldata.var(0)) / alldata.var(0)).cpu() print("Variance error: %f" % varerr) counterr = ((qc.integrate(lambda x: torch.ones(x.shape[-1]).cpu()) - qc.size) / (0.0 + qc.size)).item() print("Count error: %f" % counterr) print("Time %f" % (endtime - starttime)) # Algorithm is randomized, so some of these will fail with low probability. assert maxreldev < 1.0 assert minerr == 0.0 assert maxerr == 0.0 assert interr < 0.01 assert abs(counterr) < 0.001 print("OK")