Average Error: 18.8 → 0.1
Time: 5.2s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -325586491895.521545 \lor \neg \left(y \le 36683682.5736824796\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -325586491895.521545 \lor \neg \left(y \le 36683682.5736824796\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 r373725 = 1.0;
        double r373726 = x;
        double r373727 = y;
        double r373728 = r373726 - r373727;
        double r373729 = r373725 - r373727;
        double r373730 = r373728 / r373729;
        double r373731 = r373725 - r373730;
        double r373732 = log(r373731);
        double r373733 = r373725 - r373732;
        return r373733;
}
↓
double f(double x, double y) {
        double r373734 = y;
        double r373735 = -325586491895.52155;
        bool r373736 = r373734 <= r373735;
        double r373737 = 36683682.57368248;
        bool r373738 = r373734 <= r373737;
        double r373739 = !r373738;
        bool r373740 = r373736 || r373739;
        double r373741 = 1.0;
        double r373742 = x;
        double r373743 = 2.0;
        double r373744 = pow(r373734, r373743);
        double r373745 = r373742 / r373744;
        double r373746 = 1.0;
        double r373747 = r373746 / r373734;
        double r373748 = r373745 - r373747;
        double r373749 = r373741 * r373748;
        double r373750 = r373742 / r373734;
        double r373751 = r373749 + r373750;
        double r373752 = log(r373751);
        double r373753 = r373741 - r373752;
        double r373754 = r373742 - r373734;
        double r373755 = r373741 - r373734;
        double r373756 = cbrt(r373755);
        double r373757 = r373756 * r373756;
        double r373758 = r373754 / r373757;
        double r373759 = r373758 / r373756;
        double r373760 = r373741 - r373759;
        double r373761 = log(r373760);
        double r373762 = r373741 - r373761;
        double r373763 = r373740 ? r373753 : r373762;
        return r373763;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.8
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 < -325586491895.52155 or 36683682.57368248 < y

    1. Initial program 47.2

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -325586491895.52155 < y < 36683682.57368248

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

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

    4. Applied associate-/r*0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -325586491895.521545 \lor \neg \left(y \le 36683682.5736824796\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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))))))