Average Error: 0.0 → 0.0
Time: 7.4s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[x - \sqrt[3]{{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.044810000000000003, 0.992290000000000005\right), 1\right)}\right)}^{3}}\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
x - \sqrt[3]{{\left(\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.044810000000000003, 0.992290000000000005\right), 1\right)}\right)}^{3}}
double f(double x) {
        double r87879 = x;
        double r87880 = 2.30753;
        double r87881 = 0.27061;
        double r87882 = r87879 * r87881;
        double r87883 = r87880 + r87882;
        double r87884 = 1.0;
        double r87885 = 0.99229;
        double r87886 = 0.04481;
        double r87887 = r87879 * r87886;
        double r87888 = r87885 + r87887;
        double r87889 = r87888 * r87879;
        double r87890 = r87884 + r87889;
        double r87891 = r87883 / r87890;
        double r87892 = r87879 - r87891;
        return r87892;
}
↓
double f(double x) {
        double r87893 = x;
        double r87894 = 0.27061;
        double r87895 = 2.30753;
        double r87896 = fma(r87894, r87893, r87895);
        double r87897 = 1.0;
        double r87898 = 0.04481;
        double r87899 = 0.99229;
        double r87900 = fma(r87893, r87898, r87899);
        double r87901 = 1.0;
        double r87902 = fma(r87893, r87900, r87901);
        double r87903 = r87897 / r87902;
        double r87904 = r87896 * r87903;
        double r87905 = 3.0;
        double r87906 = pow(r87904, r87905);
        double r87907 = cbrt(r87906);
        double r87908 = r87893 - r87907;
        return r87908;
}

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-cbrt-cube0.0

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

  4. Applied add-cbrt-cube21.7

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

  5. Applied cbrt-undiv21.7

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

  6. Simplified0.0

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

  8. Applied div-inv0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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)))))