Average Error: 0.0 → 0.0
Time: 8.7s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[x - \sqrt[3]{{\left(\mathsf{fma}\left(x, 0.27061000000000002, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}^{3}}\]
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(x, 0.27061000000000002, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}^{3}}
double f(double x) {
        double r90667 = x;
        double r90668 = 2.30753;
        double r90669 = 0.27061;
        double r90670 = r90667 * r90669;
        double r90671 = r90668 + r90670;
        double r90672 = 1.0;
        double r90673 = 0.99229;
        double r90674 = 0.04481;
        double r90675 = r90667 * r90674;
        double r90676 = r90673 + r90675;
        double r90677 = r90676 * r90667;
        double r90678 = r90672 + r90677;
        double r90679 = r90671 / r90678;
        double r90680 = r90667 - r90679;
        return r90680;
}
↓
double f(double x) {
        double r90681 = x;
        double r90682 = 0.27061;
        double r90683 = 2.30753;
        double r90684 = fma(r90681, r90682, r90683);
        double r90685 = 1.0;
        double r90686 = 0.04481;
        double r90687 = 0.99229;
        double r90688 = fma(r90686, r90681, r90687);
        double r90689 = 1.0;
        double r90690 = fma(r90681, r90688, r90689);
        double r90691 = r90685 / r90690;
        double r90692 = r90684 * r90691;
        double r90693 = 3.0;
        double r90694 = pow(r90692, r90693);
        double r90695 = cbrt(r90694);
        double r90696 = r90681 - r90695;
        return r90696;
}

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

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube21.7

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

  6. Applied cbrt-undiv21.7

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

  7. Simplified0.0

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

  9. Applied div-inv0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

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)))))