Average Error: 18.2 → 0.2
Time: 18.0s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9998974867815034:\\ \;\;\;\;1.0 - \log \left(1.0 - \frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\left(\frac{x}{y} \cdot \frac{1.0}{y} + \frac{x}{y}\right) - \frac{1.0}{y}\right)\\ \end{array}\]
1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9998974867815034:\\
\;\;\;\;1.0 - \log \left(1.0 - \frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}\right)\\

\mathbf{else}:\\
\;\;\;\;1.0 - \log \left(\left(\frac{x}{y} \cdot \frac{1.0}{y} + \frac{x}{y}\right) - \frac{1.0}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r20830754 = 1.0;
        double r20830755 = x;
        double r20830756 = y;
        double r20830757 = r20830755 - r20830756;
        double r20830758 = r20830754 - r20830756;
        double r20830759 = r20830757 / r20830758;
        double r20830760 = r20830754 - r20830759;
        double r20830761 = log(r20830760);
        double r20830762 = r20830754 - r20830761;
        return r20830762;
}
↓
double f(double x, double y) {
        double r20830763 = x;
        double r20830764 = y;
        double r20830765 = r20830763 - r20830764;
        double r20830766 = 1.0;
        double r20830767 = r20830766 - r20830764;
        double r20830768 = r20830765 / r20830767;
        double r20830769 = 0.9998974867815034;
        bool r20830770 = r20830768 <= r20830769;
        double r20830771 = cbrt(r20830767);
        double r20830772 = r20830771 * r20830771;
        double r20830773 = r20830765 / r20830772;
        double r20830774 = r20830773 / r20830771;
        double r20830775 = r20830766 - r20830774;
        double r20830776 = log(r20830775);
        double r20830777 = r20830766 - r20830776;
        double r20830778 = r20830763 / r20830764;
        double r20830779 = r20830766 / r20830764;
        double r20830780 = r20830778 * r20830779;
        double r20830781 = r20830780 + r20830778;
        double r20830782 = r20830781 - r20830779;
        double r20830783 = log(r20830782);
        double r20830784 = r20830766 - r20830783;
        double r20830785 = r20830770 ? r20830777 : r20830784;
        return r20830785;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.2
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.61947241:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9998974867815034

    1. Initial program 0.1

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt0.1

      \[\leadsto 1.0 - \log \left(1.0 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}\right) \cdot \sqrt[3]{1.0 - y}}}\right)\]

    4. Applied associate-/r*0.1

      \[\leadsto 1.0 - \log \left(1.0 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}}\right)\]

    if 0.9998974867815034 < (/ (- x y) (- 1.0 y))

    1. Initial program 60.0

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]

    2. Taylor expanded around inf 0.4

      \[\leadsto 1.0 - \log \color{blue}{\left(\left(\frac{x}{y} + 1.0 \cdot \frac{x}{{y}^{2}}\right) - 1.0 \cdot \frac{1}{y}\right)}\]

    3. Simplified0.4

      \[\leadsto 1.0 - \log \color{blue}{\left(\left(\frac{x}{y} + \frac{1.0}{y} \cdot \frac{x}{y}\right) - \frac{1.0}{y}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9998974867815034:\\ \;\;\;\;1.0 - \log \left(1.0 - \frac{\frac{x - y}{\sqrt[3]{1.0 - y} \cdot \sqrt[3]{1.0 - y}}}{\sqrt[3]{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\left(\frac{x}{y} \cdot \frac{1.0}{y} + \frac{x}{y}\right) - \frac{1.0}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))