Average Error: 0.0 → 0.0
Time: 17.8s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right)\right) \cdot \sqrt[3]{x}}\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right)\right) \cdot \sqrt[3]{x}}
double f(double x) {
        double r2954936 = x;
        double r2954937 = 2.30753;
        double r2954938 = 0.27061;
        double r2954939 = r2954936 * r2954938;
        double r2954940 = r2954937 + r2954939;
        double r2954941 = 1.0;
        double r2954942 = 0.99229;
        double r2954943 = 0.04481;
        double r2954944 = r2954936 * r2954943;
        double r2954945 = r2954942 + r2954944;
        double r2954946 = r2954945 * r2954936;
        double r2954947 = r2954941 + r2954946;
        double r2954948 = r2954940 / r2954947;
        double r2954949 = r2954936 - r2954948;
        return r2954949;
}
↓
double f(double x) {
        double r2954950 = x;
        double r2954951 = 2.30753;
        double r2954952 = 0.27061;
        double r2954953 = r2954950 * r2954952;
        double r2954954 = r2954951 + r2954953;
        double r2954955 = 1.0;
        double r2954956 = cbrt(r2954950);
        double r2954957 = r2954956 * r2954956;
        double r2954958 = 0.04481;
        double r2954959 = 0.99229;
        double r2954960 = fma(r2954950, r2954958, r2954959);
        double r2954961 = r2954957 * r2954960;
        double r2954962 = r2954961 * r2954956;
        double r2954963 = r2954955 + r2954962;
        double r2954964 = r2954954 / r2954963;
        double r2954965 = r2954950 - r2954964;
        return r2954965;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.0

    \[\leadsto x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}\]

  4. Applied associate-*r*0.0

    \[\leadsto x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \color{blue}{\left(\left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}}}\]

  5. Simplified0.0

    \[\leadsto x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right)\right)} \cdot \sqrt[3]{x}}\]

  6. Final simplification0.0

    \[\leadsto x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \mathsf{fma}\left(x, 0.04481000000000000260680366181986755691469, 0.992290000000000005364597654988756403327\right)\right) \cdot \sqrt[3]{x}}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))