Average Error: 18.7 → 0.1
Time: 14.6s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -118579514.584518134593963623046875 \lor \neg \left(y \le 32110906.8722298182547092437744140625\right):\\ \;\;\;\;1 - \log \left(\left(\sqrt[3]{\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}} \cdot \sqrt[3]{\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}}\right) \cdot \sqrt[3]{\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -118579514.584518134593963623046875 \lor \neg \left(y \le 32110906.8722298182547092437744140625\right):\\
\;\;\;\;1 - \log \left(\left(\sqrt[3]{\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}} \cdot \sqrt[3]{\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}}\right) \cdot \sqrt[3]{\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r294005 = 1.0;
        double r294006 = x;
        double r294007 = y;
        double r294008 = r294006 - r294007;
        double r294009 = r294005 - r294007;
        double r294010 = r294008 / r294009;
        double r294011 = r294005 - r294010;
        double r294012 = log(r294011);
        double r294013 = r294005 - r294012;
        return r294013;
}
↓
double f(double x, double y) {
        double r294014 = y;
        double r294015 = -118579514.58451813;
        bool r294016 = r294014 <= r294015;
        double r294017 = 32110906.87222982;
        bool r294018 = r294014 <= r294017;
        double r294019 = !r294018;
        bool r294020 = r294016 || r294019;
        double r294021 = 1.0;
        double r294022 = r294021 / r294014;
        double r294023 = 1.0;
        double r294024 = r294022 + r294023;
        double r294025 = x;
        double r294026 = r294025 / r294014;
        double r294027 = r294024 * r294026;
        double r294028 = r294027 - r294022;
        double r294029 = cbrt(r294028);
        double r294030 = r294029 * r294029;
        double r294031 = r294030 * r294029;
        double r294032 = log(r294031);
        double r294033 = r294021 - r294032;
        double r294034 = r294025 - r294014;
        double r294035 = r294021 - r294014;
        double r294036 = r294023 / r294035;
        double r294037 = r294034 * r294036;
        double r294038 = r294021 - r294037;
        double r294039 = log(r294038);
        double r294040 = r294021 - r294039;
        double r294041 = r294020 ? r294033 : r294040;
        return r294041;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.7
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -118579514.58451813 or 32110906.87222982 < y

    1. Initial program 47.0

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

    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

      \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}\right)}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt0.1

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

    if -118579514.58451813 < y < 32110906.87222982

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -118579514.584518134593963623046875 \lor \neg \left(y \le 32110906.8722298182547092437744140625\right):\\ \;\;\;\;1 - \log \left(\left(\sqrt[3]{\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}} \cdot \sqrt[3]{\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}}\right) \cdot \sqrt[3]{\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))))))