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>>> import math >>> z = complex(float.fromhex('0x1.47e9c711723f5p+81'), ... float.fromhex('0x1.38afd1168e49fp+85')) >>> ref = complex(float.fromhex('0x0.4000000000000p-1022'), ... float.fromhex('0x0.3ffffffffffffp-1022')) >>> z ** -12 0j >>> abs((z**-12 - ref).real) / math.ulp(ref.real) 1125899906842624.0 >>> 0.0j ** 0 (1+0j) 

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6 employee success story templates 6 bonus options thanks for pointing out i dived into this, these are some results with these scripts i used Example Of IMS About Blog Post

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# compare.py -- normwise error vs correctly rounded MPC results import math, random, statistics from gmpy2 import mpc, get_context get_context().precision = 200 def normwise(got, ref): # Eq. (1) of Caprioli, Innocente & Zimmermann: e = |delta| / ulp(|z|) if got is None: return float('inf') if math.isnan(got.real) or math.isnan(got.imag): return None if math.isinf(got.real) or math.isinf(got.imag): return None return float(abs(mpc(got) - ref)) / math.ulp(float(abs(ref))) def measure(pow_fn, samples): errs = [] for z, n in samples: ref = mpc(z) ** n try: got = pow_fn(z, n) except (OverflowError, ZeroDivisionError): got = None errs.append(normwise(got, ref)) excluded = sum(1 for e in errs if e is None) scored = [e for e in errs if e is not None] return (statistics.median(scored), statistics.quantiles(scored, n=100)[94], max(scored), excluded, errs) random.seed(0) samples = [] for _ in range(50000): n = -random.choice((1, 2, 3, 5, 7, 12, 40, 100)) z = complex(random.uniform(-1, 1), random.uniform(-1, 1)) z *= math.ldexp(1.0, random.randint(-1020, 1020)) if z == 0 or not math.isfinite(abs(z)): continue ref = mpc(z) ** n if abs(ref) == 0 or not math.isfinite(float(abs(ref))): continue samples.append((z, n)) med, p95, mx, bad, errs = measure(lambda z, n: z ** n, samples) print(f"n={len(samples)} median={med:.3f} p95={p95:.3f} max={mx:.4g} excluded={bad}") import json, sys if len(sys.argv) > 1: with open(sys.argv[1], "w") as f: json.dump([e if (e is not None and math.isfinite(e)) else None for e in errs], f) 

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 n median p95 max excluded (NaN/Inf) main 32094 0.492 1.993 9.537e+14 17687 patched 32094 0.000 1.250 63.22 0 

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import json, statistics a = json.load(open('errs_main.json')) b = json.load(open('errs_patched.json')) print(f"lengths: main={len(a)} patched={len(b)}") both = [(x, y) for x, y in zip(a, b) if x is not None and y is not None] A = [x for x, _ in both] B = [y for _, y in both] q = lambda L: statistics.quantiles(L, n=100)[94] print(f"common pool: {len(both)}") print(f" main median={statistics.median(A):.3f} p95={q(A):.3f} max={max(A):.4g}") print(f" patched median={statistics.median(B):.3f} p95={q(B):.3f} max={max(B):.4g}") print(f" identical: {100 * sum(1 for x, y in both if x == y) / len(both):.2f}%") 

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common pool: 14407 samples (both builds return a finite, non-NaN result) main median=0.492 p95=1.993 max=9.537e+14 patched median=0.492 p95=1.986 max=63.22 identical values: 99.97% 

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ref: n=29187 median=0.500 p95=2.358 max=9.537e+14 excluded=0 patch: n=29187 median=1.000 p95=4.854 max=120.4 excluded=0 

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# compare.py -- normwise error vs correctly rounded MPC results import cmath, math, random, statistics from gmpy2 import mpc, ieee, set_context set_context(ieee(64)) def normwise(got, ref): # Eq. (1) of Caprioli, Innocente & Zimmermann: e = |delta| / ulp(|z|) if got is None: return float('inf') if cmath.isnan(got): return if cmath.isinf(got): return diff = abs(got - ref) if not math.isfinite(diff): return return diff/math.ulp(abs(ref)) def measure(pow_fn, samples): errs = [] for z, n in samples: ref = complex(mpc(z) ** n) try: got = pow_fn(z, n) except (OverflowError, ZeroDivisionError): got = None errs.append(normwise(got, ref)) excluded = sum(1 for e in errs if e is None) scored = [e for e in errs if e is not None] return (statistics.median(scored), statistics.quantiles(scored, n=100)[94], max(scored), excluded, errs) random.seed(0) samples = [] for _ in range(100000): n = -random.choice((1, 2, 3, 5, 7, 12, 40, 100)) z = complex(random.uniform(-1, 1), random.uniform(-1, 1)) z *= math.ldexp(1.0, random.randint(-1020, 1020)) if z == 0 or not cmath.isfinite(z): continue ref = complex(mpc(z) ** n) if ref == 0 or not math.isfinite(abs(ref)): continue samples.append((z, n)) med, p95, mx, bad, errs = measure(lambda z, n: z ** n, samples) print(f"n={len(samples)} median={med:.3f} p95={p95:.3f} max={mx:.4g} excluded={bad}") import json, sys if len(sys.argv) > 1: with open(sys.argv[1], "w") as f: json.dump([e if (e is not None and math.isfinite(e)) else None for e in errs], f)

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