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

Average Error: 0.1 → 0.1

Time: 10.7s

Precision: 64

Math

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

↓

\[1 - x \cdot \mathsf{fma}\left(\sqrt[3]{0.2530000000000000026645352591003756970167} \cdot \sqrt[3]{0.2530000000000000026645352591003756970167}, \sqrt[3]{0.2530000000000000026645352591003756970167}, 0.1199999999999999955591079014993738383055 \cdot x\right)\]

C

double f(double x) {
        double r56696 = 1.0;
        double r56697 = x;
        double r56698 = 0.253;
        double r56699 = 0.12;
        double r56700 = r56697 * r56699;
        double r56701 = r56698 + r56700;
        double r56702 = r56697 * r56701;
        double r56703 = r56696 - r56702;
        return r56703;
}

↓

double f(double x) {
        double r56704 = 1.0;
        double r56705 = x;
        double r56706 = 0.253;
        double r56707 = cbrt(r56706);
        double r56708 = r56707 * r56707;
        double r56709 = 0.12;
        double r56710 = r56709 * r56705;
        double r56711 = fma(r56708, r56707, r56710);
        double r56712 = r56705 * r56711;
        double r56713 = r56704 - r56712;
        return r56713;
}

Error

Bits error versus x

Derivation

1. Initial program 0.1

   \[1 - x \cdot \left(0.2530000000000000026645352591003756970167 + x \cdot 0.1199999999999999955591079014993738383055\right)\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.1

   \[\leadsto 1 - x \cdot \left(\color{blue}{\left(\sqrt[3]{0.2530000000000000026645352591003756970167} \cdot \sqrt[3]{0.2530000000000000026645352591003756970167}\right) \cdot \sqrt[3]{0.2530000000000000026645352591003756970167}} + x \cdot 0.1199999999999999955591079014993738383055\right)\]

4. Applied fma-def 0.1

   \[\leadsto 1 - x \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{0.2530000000000000026645352591003756970167} \cdot \sqrt[3]{0.2530000000000000026645352591003756970167}, \sqrt[3]{0.2530000000000000026645352591003756970167}, x \cdot 0.1199999999999999955591079014993738383055\right)}\]

5. Final simplification 0.1

   \[\leadsto 1 - x \cdot \mathsf{fma}\left(\sqrt[3]{0.2530000000000000026645352591003756970167} \cdot \sqrt[3]{0.2530000000000000026645352591003756970167}, \sqrt[3]{0.2530000000000000026645352591003756970167}, 0.1199999999999999955591079014993738383055 \cdot x\right)\]

Reproduce

herbie shell --seed 2019179 +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)))))