accurpy 0.2.3

Ultra-accurate and fast FM function evaluator with <4.5e-16 relative error

AccurPy

Ultra-accurate and fast FM function evaluator with <4.5e-16 relative error (~2x machine epsilon).

Installation

pip install accurpy

Usage

from accurpy import syncF, fm_scaled

# Compute FM(x) * exp(x) - the numerically stable form
result = syncF(1.0)

# Compute FM(x) = exp(-x) * integral of t^(-1/2) * exp(-t) / (1 + t/x) dt
result_scaled = fm_scaled(1.0)

# Works with numpy arrays
import numpy as np
x = np.linspace(0.001, 100, 1000)
y = syncF(x)

Performance

The C extension uses METH_O calling convention with PyFloat_AS_DOUBLE for minimal overhead:

x range	syncF(x) time	Notes
Small (1e-8)	~38 ns	Cube-root expansion
Wide (1.0)	~45 ns	Rational approximation
Large (1e10)	~32 ns	Asymptotic expansion

Comparable to math.sin (~38 ns) - only ~20 ns above the minimum possible CPython C-extension overhead (~16 ns for extract+create).

Array throughput: ~10 ns/element.

Accuracy

• Relative error < 4.5e-16 across entire domain (x > 0)
• Three-region algorithm:
  • Small x (< 1e-7): Cube-root expansion
  • Wide domain (1e-7 to 1e6): 63-segment rational approximation
  • Large x (> 1e6): Asymptotic expansion

License

MIT