Average Error: 18.5 → 0.1
Time: 5.2s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -96388745.0459894985 \lor \neg \left(y \le 31980879.8987302519\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -96388745.0459894985 \lor \neg \left(y \le 31980879.8987302519\right):\\
\;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\

\end{array}
double f(double x, double y) {
        double r414730 = 1.0;
        double r414731 = x;
        double r414732 = y;
        double r414733 = r414731 - r414732;
        double r414734 = r414730 - r414732;
        double r414735 = r414733 / r414734;
        double r414736 = r414730 - r414735;
        double r414737 = log(r414736);
        double r414738 = r414730 - r414737;
        return r414738;
}
↓
double f(double x, double y) {
        double r414739 = y;
        double r414740 = -96388745.0459895;
        bool r414741 = r414739 <= r414740;
        double r414742 = 31980879.898730252;
        bool r414743 = r414739 <= r414742;
        double r414744 = !r414743;
        bool r414745 = r414741 || r414744;
        double r414746 = 1.0;
        double r414747 = x;
        double r414748 = 2.0;
        double r414749 = pow(r414739, r414748);
        double r414750 = r414747 / r414749;
        double r414751 = 1.0;
        double r414752 = r414751 / r414739;
        double r414753 = r414750 - r414752;
        double r414754 = r414746 * r414753;
        double r414755 = r414747 / r414739;
        double r414756 = r414754 + r414755;
        double r414757 = log(r414756);
        double r414758 = r414746 - r414757;
        double r414759 = r414747 - r414739;
        double r414760 = r414746 - r414739;
        double r414761 = cbrt(r414760);
        double r414762 = r414761 * r414761;
        double r414763 = r414759 / r414762;
        double r414764 = r414763 / r414761;
        double r414765 = r414746 - r414764;
        double r414766 = log(r414765);
        double r414767 = r414746 - r414766;
        double r414768 = r414745 ? r414758 : r414767;
        return r414768;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.5
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -96388745.0459895 or 31980879.898730252 < y

    1. Initial program 47.3

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]

    2. Taylor expanded around inf 0.1

      \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)}\]

    3. Simplified0.1

      \[\leadsto 1 - \log \color{blue}{\left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)}\]

    if -96388745.0459895 < y < 31980879.898730252

    1. Initial program 0.1

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt0.1

      \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]

    4. Applied associate-/r*0.1

      \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -96388745.0459894985 \lor \neg \left(y \le 31980879.8987302519\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))