Average Error: 0.1 → 0.1
Time: 1.9s
Precision: binary64
\[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right) \]
1 - x \cdot \left(0.253 + x \cdot 0.12\right)
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))
↓
(FPCore (x) :precision binary64 (fma x (fma x -0.12 -0.253) 1.0))
double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}
↓
double code(double x) {
	return fma(x, fma(x, -0.12, -0.253), 1.0);
}

Error

Bits error versus x

Derivation

  1. Initial program 0.1

    \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right) \]

Reproduce

herbie shell --seed 2022077 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))