Average Error: 0.0 → 0.0
Time: 3.6s
Precision: 64
\[x - \frac{2.30753 + x \cdot 0.27061000000000002}{1 + \left(0.992290000000000005 + x \cdot 0.044810000000000003\right) \cdot x}\]
↓
\[\frac{-\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + x\]
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)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + x
double f(double x) {
        double r102737 = x;
        double r102738 = 2.30753;
        double r102739 = 0.27061;
        double r102740 = r102737 * r102739;
        double r102741 = r102738 + r102740;
        double r102742 = 1.0;
        double r102743 = 0.99229;
        double r102744 = 0.04481;
        double r102745 = r102737 * r102744;
        double r102746 = r102743 + r102745;
        double r102747 = r102746 * r102737;
        double r102748 = r102742 + r102747;
        double r102749 = r102741 / r102748;
        double r102750 = r102737 - r102749;
        return r102750;
}
↓
double f(double x) {
        double r102751 = 0.27061;
        double r102752 = x;
        double r102753 = 2.30753;
        double r102754 = fma(r102751, r102752, r102753);
        double r102755 = -r102754;
        double r102756 = 0.04481;
        double r102757 = 0.99229;
        double r102758 = fma(r102756, r102752, r102757);
        double r102759 = 1.0;
        double r102760 = fma(r102752, r102758, r102759);
        double r102761 = r102755 / r102760;
        double r102762 = r102761 + r102752;
        return r102762;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 +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)))))