Average Error: 0.1 → 0.1
Time: 4.3s
Precision: 64
\[1 - x \cdot \left(0.2530000000000000026645352591003756970167 + x \cdot 0.1199999999999999955591079014993738383055\right)\]
↓
\[\mathsf{fma}\left(-x, \mathsf{fma}\left(0.1199999999999999955591079014993738383055, x, 0.2530000000000000026645352591003756970167\right), 1\right)\]
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)
double f(double x) {
        double r95487 = 1.0;
        double r95488 = x;
        double r95489 = 0.253;
        double r95490 = 0.12;
        double r95491 = r95488 * r95490;
        double r95492 = r95489 + r95491;
        double r95493 = r95488 * r95492;
        double r95494 = r95487 - r95493;
        return r95494;
}
↓
double f(double x) {
        double r95495 = x;
        double r95496 = -r95495;
        double r95497 = 0.12;
        double r95498 = 0.253;
        double r95499 = fma(r95497, r95495, r95498);
        double r95500 = 1.0;
        double r95501 = fma(r95496, r95499, r95500);
        return r95501;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020002 +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)))))