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

Average Error: 0.1 → 0.1

Time: 26.3s

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 r11562582 = 1.0;
        double r11562583 = x;
        double r11562584 = 0.253;
        double r11562585 = 0.12;
        double r11562586 = r11562583 * r11562585;
        double r11562587 = r11562584 + r11562586;
        double r11562588 = r11562583 * r11562587;
        double r11562589 = r11562582 - r11562588;
        return r11562589;
}

↓

double f(double x) {
        double r11562590 = 1.0;
        double r11562591 = 0.12;
        double r11562592 = x;
        double r11562593 = 0.253;
        double r11562594 = fma(r11562591, r11562592, r11562593);
        double r11562595 = r11562594 * r11562592;
        double r11562596 = r11562590 - r11562595;
        return r11562596;
}

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 2019173 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))