Average Error: 0.1 → 0.1
Time: 3.2s
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 r116902 = 1.0;
        double r116903 = x;
        double r116904 = 0.253;
        double r116905 = 0.12;
        double r116906 = r116903 * r116905;
        double r116907 = r116904 + r116906;
        double r116908 = r116903 * r116907;
        double r116909 = r116902 - r116908;
        return r116909;
}
↓
double f(double x) {
        double r116910 = x;
        double r116911 = -r116910;
        double r116912 = 0.12;
        double r116913 = 0.253;
        double r116914 = fma(r116912, r116910, r116913);
        double r116915 = 1.0;
        double r116916 = fma(r116911, r116914, r116915);
        return r116916;
}

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 2020081 +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)))))