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

Average Error: 0.1 → 0.1

Time: 11.9s

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 r54449 = 1.0;
        double r54450 = x;
        double r54451 = 0.253;
        double r54452 = 0.12;
        double r54453 = r54450 * r54452;
        double r54454 = r54451 + r54453;
        double r54455 = r54450 * r54454;
        double r54456 = r54449 - r54455;
        return r54456;
}

↓

double f(double x) {
        double r54457 = 1.0;
        double r54458 = 0.12;
        double r54459 = x;
        double r54460 = 0.253;
        double r54461 = fma(r54458, r54459, r54460);
        double r54462 = r54461 * r54459;
        double r54463 = r54457 - r54462;
        return r54463;
}

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 2019235 +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)))))