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

Average Error: 0.1 → 0.1

Time: 16.7s

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 r3434309 = 1.0;
        double r3434310 = x;
        double r3434311 = 0.253;
        double r3434312 = 0.12;
        double r3434313 = r3434310 * r3434312;
        double r3434314 = r3434311 + r3434313;
        double r3434315 = r3434310 * r3434314;
        double r3434316 = r3434309 - r3434315;
        return r3434316;
}

↓

double f(double x) {
        double r3434317 = 1.0;
        double r3434318 = 0.12;
        double r3434319 = x;
        double r3434320 = 0.253;
        double r3434321 = fma(r3434318, r3434319, r3434320);
        double r3434322 = r3434321 * r3434319;
        double r3434323 = r3434317 - r3434322;
        return r3434323;
}

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 2019171 +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)))))