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

Average Error: 0.1 → 0.1

Time: 20.0s

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 r43125 = 1.0;
        double r43126 = x;
        double r43127 = 0.253;
        double r43128 = 0.12;
        double r43129 = r43126 * r43128;
        double r43130 = r43127 + r43129;
        double r43131 = r43126 * r43130;
        double r43132 = r43125 - r43131;
        return r43132;
}

↓

double f(double x) {
        double r43133 = 1.0;
        double r43134 = 0.12;
        double r43135 = x;
        double r43136 = 0.253;
        double r43137 = fma(r43134, r43135, r43136);
        double r43138 = r43137 * r43135;
        double r43139 = r43133 - r43138;
        return r43139;
}

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 2019323 +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)))))