Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}\]
↓
\[\frac{-\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)\right)}^{3}}} + x\]
x - \frac{2.307529999999999859028321225196123123169 + x \cdot 0.2706100000000000171951342053944244980812}{1 + \left(0.992290000000000005364597654988756403327 + x \cdot 0.04481000000000000260680366181986755691469\right) \cdot x}
↓
\frac{-\mathsf{fma}\left(0.2706100000000000171951342053944244980812, x, 2.307529999999999859028321225196123123169\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.04481000000000000260680366181986755691469, x, 0.992290000000000005364597654988756403327\right), 1\right)\right)}^{3}}} + x
double f(double x) {
        double r84944 = x;
        double r84945 = 2.30753;
        double r84946 = 0.27061;
        double r84947 = r84944 * r84946;
        double r84948 = r84945 + r84947;
        double r84949 = 1.0;
        double r84950 = 0.99229;
        double r84951 = 0.04481;
        double r84952 = r84944 * r84951;
        double r84953 = r84950 + r84952;
        double r84954 = r84953 * r84944;
        double r84955 = r84949 + r84954;
        double r84956 = r84948 / r84955;
        double r84957 = r84944 - r84956;
        return r84957;
}
↓
double f(double x) {
        double r84958 = 0.27061;
        double r84959 = x;
        double r84960 = 2.30753;
        double r84961 = fma(r84958, r84959, r84960);
        double r84962 = -r84961;
        double r84963 = 0.04481;
        double r84964 = 0.99229;
        double r84965 = fma(r84963, r84959, r84964);
        double r84966 = 1.0;
        double r84967 = fma(r84959, r84965, r84966);
        double r84968 = 3.0;
        double r84969 = pow(r84967, r84968);
        double r84970 = cbrt(r84969);
        double r84971 = r84962 / r84970;
        double r84972 = r84971 + r84959;
        return r84972;
}

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. Simplified0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019346 +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)))))