Average Error: 0.1 → 0.1
Time: 5.0s
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 r100938 = 1.0;
        double r100939 = x;
        double r100940 = 0.253;
        double r100941 = 0.12;
        double r100942 = r100939 * r100941;
        double r100943 = r100940 + r100942;
        double r100944 = r100939 * r100943;
        double r100945 = r100938 - r100944;
        return r100945;
}
↓
double f(double x) {
        double r100946 = x;
        double r100947 = -r100946;
        double r100948 = 0.12;
        double r100949 = 0.253;
        double r100950 = fma(r100948, r100946, r100949);
        double r100951 = 1.0;
        double r100952 = fma(r100947, r100950, r100951);
        return r100952;
}

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 2020035 +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)))))