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 r73770 = 1.0;
        double r73771 = x;
        double r73772 = 0.253;
        double r73773 = 0.12;
        double r73774 = r73771 * r73773;
        double r73775 = r73772 + r73774;
        double r73776 = r73771 * r73775;
        double r73777 = r73770 - r73776;
        return r73777;
}
↓
double f(double x) {
        double r73778 = x;
        double r73779 = -r73778;
        double r73780 = 0.12;
        double r73781 = 0.253;
        double r73782 = fma(r73780, r73778, r73781);
        double r73783 = 1.0;
        double r73784 = fma(r73779, r73782, r73783);
        return r73784;
}

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 2020021 +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)))))