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

Average Error: 0.1 → 0.1

Time: 4.1s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r112405 = 1.0;
        double r112406 = x;
        double r112407 = 0.253;
        double r112408 = 0.12;
        double r112409 = r112406 * r112408;
        double r112410 = r112407 + r112409;
        double r112411 = r112406 * r112410;
        double r112412 = r112405 - r112411;
        return r112412;
}

↓

double f(double x) {
        double r112413 = x;
        double r112414 = -r112413;
        double r112415 = 0.12;
        double r112416 = 0.253;
        double r112417 = fma(r112415, r112413, r112416);
        double r112418 = 1.0;
        double r112419 = fma(r112414, r112417, r112418);
        return r112419;
}

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

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019352 +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)))))