Average Error: 0.1 → 0.1
Time: 4.5s
Precision: 64
\[1 - x \cdot \left(0.253 + x \cdot 0.12\right)\]
↓
\[\mathsf{fma}\left(-x, \mathsf{fma}\left(0.12, x, 0.253\right), 1\right)\]
1 - x \cdot \left(0.253 + x \cdot 0.12\right)
↓
\mathsf{fma}\left(-x, \mathsf{fma}\left(0.12, x, 0.253\right), 1\right)
double f(double x) {
        double r96507 = 1.0;
        double r96508 = x;
        double r96509 = 0.253;
        double r96510 = 0.12;
        double r96511 = r96508 * r96510;
        double r96512 = r96509 + r96511;
        double r96513 = r96508 * r96512;
        double r96514 = r96507 - r96513;
        return r96514;
}
↓
double f(double x) {
        double r96515 = x;
        double r96516 = -r96515;
        double r96517 = 0.12;
        double r96518 = 0.253;
        double r96519 = fma(r96517, r96515, r96518);
        double r96520 = 1.0;
        double r96521 = fma(r96516, r96519, r96520);
        return r96521;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1 (* x (+ 0.253 (* x 0.12)))))