Average Error: 0.0 → 0.0
Time: 2.6s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[\frac{-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}} + x\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
\frac{-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)\right)}^{3}}} + x
double f(double x) {
        double r68929 = x;
        double r68930 = 2.30753;
        double r68931 = 0.27061;
        double r68932 = r68929 * r68931;
        double r68933 = r68930 + r68932;
        double r68934 = 1.0;
        double r68935 = 0.99229;
        double r68936 = 0.04481;
        double r68937 = r68929 * r68936;
        double r68938 = r68935 + r68937;
        double r68939 = r68938 * r68929;
        double r68940 = r68934 + r68939;
        double r68941 = r68933 / r68940;
        double r68942 = r68929 - r68941;
        return r68942;
}
↓
double f(double x) {
        double r68943 = 0.27061;
        double r68944 = x;
        double r68945 = 2.30753;
        double r68946 = fma(r68943, r68944, r68945);
        double r68947 = -r68946;
        double r68948 = 0.04481;
        double r68949 = 0.99229;
        double r68950 = fma(r68948, r68944, r68949);
        double r68951 = 1.0;
        double r68952 = fma(r68944, r68950, r68951);
        double r68953 = 3.0;
        double r68954 = pow(r68952, r68953);
        double r68955 = cbrt(r68954);
        double r68956 = r68947 / r68955;
        double r68957 = r68956 + r68944;
        return r68957;
}

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. Simplified0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020033 +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)))))