Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D

Average Error: 0.0 → 0.0

Time: 13.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]

↓

\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(\sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x} \cdot \sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x}\right) \cdot \sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x}}\]

C

double f(double x) {
        double r236129 = x;
        double r236130 = 2.30753;
        double r236131 = 0.27061;
        double r236132 = r236129 * r236131;
        double r236133 = r236130 + r236132;
        double r236134 = 1.0;
        double r236135 = 0.99229;
        double r236136 = 0.04481;
        double r236137 = r236129 * r236136;
        double r236138 = r236135 + r236137;
        double r236139 = r236138 * r236129;
        double r236140 = r236134 + r236139;
        double r236141 = r236133 / r236140;
        double r236142 = r236129 - r236141;
        return r236142;
}

↓

double f(double x) {
        double r236143 = x;
        double r236144 = 2.30753;
        double r236145 = 0.27061;
        double r236146 = r236143 * r236145;
        double r236147 = r236144 + r236146;
        double r236148 = 1.0;
        double r236149 = 0.04481;
        double r236150 = 0.99229;
        double r236151 = fma(r236149, r236143, r236150);
        double r236152 = r236151 * r236143;
        double r236153 = cbrt(r236152);
        double r236154 = r236153 * r236153;
        double r236155 = r236154 * r236153;
        double r236156 = r236148 + r236155;
        double r236157 = r236147 / r236156;
        double r236158 = r236143 - r236157;
        return r236158;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

   \[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.0

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \color{blue}{\left(\sqrt[3]{\left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x} \cdot \sqrt[3]{\left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\right) \cdot \sqrt[3]{\left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}}}\]

4. Simplified 0.0

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x} \cdot \sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x}\right)} \cdot \sqrt[3]{\left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}}\]

5. Simplified 0.0

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(\sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x} \cdot \sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x}\right) \cdot \color{blue}{\sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x}}}\]

6. Final simplification 0.0

   \[\leadsto x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(\sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x} \cdot \sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x}\right) \cdot \sqrt[3]{\mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right) \cdot x}}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* (+ 0.99229 (* x 0.04481)) x)))))