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

Average Error: 0.1 → 0.1

Time: 15.5s

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 r3456333 = 1.0;
        double r3456334 = x;
        double r3456335 = 0.253;
        double r3456336 = 0.12;
        double r3456337 = r3456334 * r3456336;
        double r3456338 = r3456335 + r3456337;
        double r3456339 = r3456334 * r3456338;
        double r3456340 = r3456333 - r3456339;
        return r3456340;
}

↓

double f(double x) {
        double r3456341 = 1.0;
        double r3456342 = 0.12;
        double r3456343 = x;
        double r3456344 = 0.253;
        double r3456345 = fma(r3456342, r3456343, r3456344);
        double r3456346 = r3456345 * r3456343;
        double r3456347 = r3456341 - r3456346;
        return r3456347;
}

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 2019174 +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)))))