Average Error: 0.0 → 0.0
Time: 5.1s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[x - \sqrt[3]{{\left(\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, 0.992290000000000005 + x \cdot 0.044810000000000003, 1\right)}\right)}^{3}}\]
x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}
↓
x - \sqrt[3]{{\left(\frac{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, 0.992290000000000005 + x \cdot 0.044810000000000003, 1\right)}\right)}^{3}}
double f(double x) {
        double r127736 = x;
        double r127737 = 2.30753;
        double r127738 = 0.27061;
        double r127739 = r127736 * r127738;
        double r127740 = r127737 + r127739;
        double r127741 = 1.0;
        double r127742 = 0.99229;
        double r127743 = 0.04481;
        double r127744 = r127736 * r127743;
        double r127745 = r127742 + r127744;
        double r127746 = r127745 * r127736;
        double r127747 = r127741 + r127746;
        double r127748 = r127740 / r127747;
        double r127749 = r127736 - r127748;
        return r127749;
}
↓
double f(double x) {
        double r127750 = x;
        double r127751 = 0.27061;
        double r127752 = 2.30753;
        double r127753 = fma(r127751, r127750, r127752);
        double r127754 = 0.99229;
        double r127755 = 0.04481;
        double r127756 = r127750 * r127755;
        double r127757 = r127754 + r127756;
        double r127758 = 1.0;
        double r127759 = fma(r127750, r127757, r127758);
        double r127760 = r127753 / r127759;
        double r127761 = 3.0;
        double r127762 = pow(r127760, r127761);
        double r127763 = cbrt(r127762);
        double r127764 = r127750 - r127763;
        return r127764;
}

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-cube21.5

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

  5. Applied cbrt-undiv21.5

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 +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)))))