Average Error: 0.1 → 0.1
Time: 4.2s
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 r82021 = 1.0;
        double r82022 = x;
        double r82023 = 0.253;
        double r82024 = 0.12;
        double r82025 = r82022 * r82024;
        double r82026 = r82023 + r82025;
        double r82027 = r82022 * r82026;
        double r82028 = r82021 - r82027;
        return r82028;
}
↓
double f(double x) {
        double r82029 = x;
        double r82030 = -r82029;
        double r82031 = 0.12;
        double r82032 = 0.253;
        double r82033 = fma(r82031, r82029, r82032);
        double r82034 = 1.0;
        double r82035 = fma(r82030, r82033, r82034);
        return r82035;
}

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 2020060 +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)))))