Average Error: 18.2 → 0.1
Time: 11.4s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -51178879.934393361 \lor \neg \left(y \le 53143023.8704946265\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\frac{1 - y}{\sqrt[3]{x - y}}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -51178879.934393361 \lor \neg \left(y \le 53143023.8704946265\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r992979 = 1.0;
        double r992980 = x;
        double r992981 = y;
        double r992982 = r992980 - r992981;
        double r992983 = r992979 - r992981;
        double r992984 = r992982 / r992983;
        double r992985 = r992979 - r992984;
        double r992986 = log(r992985);
        double r992987 = r992979 - r992986;
        return r992987;
}
↓
double f(double x, double y) {
        double r992988 = y;
        double r992989 = -51178879.93439336;
        bool r992990 = r992988 <= r992989;
        double r992991 = 53143023.87049463;
        bool r992992 = r992988 <= r992991;
        double r992993 = !r992992;
        bool r992994 = r992990 || r992993;
        double r992995 = 1.0;
        double r992996 = exp(r992995);
        double r992997 = 1.0;
        double r992998 = r992995 / r992988;
        double r992999 = r992997 + r992998;
        double r993000 = x;
        double r993001 = r993000 / r992988;
        double r993002 = r992999 * r993001;
        double r993003 = r993002 - r992998;
        double r993004 = r992996 / r993003;
        double r993005 = log(r993004);
        double r993006 = r993000 - r992988;
        double r993007 = cbrt(r993006);
        double r993008 = r993007 * r993007;
        double r993009 = r992995 - r992988;
        double r993010 = r993009 / r993007;
        double r993011 = r993008 / r993010;
        double r993012 = r992995 - r993011;
        double r993013 = log(r993012);
        double r993014 = r992995 - r993013;
        double r993015 = r992994 ? r993005 : r993014;
        return r993015;
}

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

Derivation

  1. Split input into 2 regimes

  2. if y < -51178879.93439336 or 53143023.87049463 < y

    1. Initial program 47.0

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

    3. Applied add-log-exp47.0

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

    4. Applied diff-log47.0

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

    if -51178879.93439336 < y < 53143023.87049463

    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

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

    4. Applied associate-/l*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -51178879.934393361 \lor \neg \left(y \le 53143023.8704946265\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\frac{1 - y}{\sqrt[3]{x - y}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))))