Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 14.2s

Precision: 64

Math

\[1 - x \cdot \left(0.2530000000000000026645352591003756970167 + x \cdot 0.1199999999999999955591079014993738383055\right)\]

↓

\[1 - \mathsf{fma}\left(0.1199999999999999955591079014993738383055, x, 0.2530000000000000026645352591003756970167\right) \cdot x\]

C

double f(double x) {
        double r89979 = 1.0;
        double r89980 = x;
        double r89981 = 0.253;
        double r89982 = 0.12;
        double r89983 = r89980 * r89982;
        double r89984 = r89981 + r89983;
        double r89985 = r89980 * r89984;
        double r89986 = r89979 - r89985;
        return r89986;
}

↓

double f(double x) {
        double r89987 = 1.0;
        double r89988 = 0.12;
        double r89989 = x;
        double r89990 = 0.253;
        double r89991 = fma(r89988, r89989, r89990);
        double r89992 = r89991 * r89989;
        double r89993 = r89987 - r89992;
        return r89993;
}

Error

Bits error versus x

Derivation

1. Initial program 0.1

   \[1 - x \cdot \left(0.2530000000000000026645352591003756970167 + x \cdot 0.1199999999999999955591079014993738383055\right)\]

2. Simplified 0.1

   \[\leadsto \color{blue}{1 - \mathsf{fma}\left(0.1199999999999999955591079014993738383055, x, 0.2530000000000000026645352591003756970167\right) \cdot x}\]

3. Final simplification 0.1

   \[\leadsto 1 - \mathsf{fma}\left(0.1199999999999999955591079014993738383055, x, 0.2530000000000000026645352591003756970167\right) \cdot x\]

Reproduce

herbie shell --seed 2019208 +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)))))