Average Error: 18.7 → 0.1
Time: 18.2s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -118579514.584518134593963623046875 \lor \neg \left(y \le 32110906.8722298182547092437744140625\right):\\ \;\;\;\;1 - \log \left(\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]
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(\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 r258000 = 1.0;
        double r258001 = x;
        double r258002 = y;
        double r258003 = r258001 - r258002;
        double r258004 = r258000 - r258002;
        double r258005 = r258003 / r258004;
        double r258006 = r258000 - r258005;
        double r258007 = log(r258006);
        double r258008 = r258000 - r258007;
        return r258008;
}
↓
double f(double x, double y) {
        double r258009 = y;
        double r258010 = -118579514.58451813;
        bool r258011 = r258009 <= r258010;
        double r258012 = 32110906.87222982;
        bool r258013 = r258009 <= r258012;
        double r258014 = !r258013;
        bool r258015 = r258011 || r258014;
        double r258016 = 1.0;
        double r258017 = r258016 / r258009;
        double r258018 = 1.0;
        double r258019 = r258017 + r258018;
        double r258020 = x;
        double r258021 = r258020 / r258009;
        double r258022 = r258019 * r258021;
        double r258023 = r258022 - r258017;
        double r258024 = log(r258023);
        double r258025 = r258016 - r258024;
        double r258026 = r258020 - r258009;
        double r258027 = r258016 - r258009;
        double r258028 = r258018 / r258027;
        double r258029 = r258026 * r258028;
        double r258030 = r258016 - r258029;
        double r258031 = log(r258030);
        double r258032 = r258016 - r258031;
        double r258033 = r258015 ? r258025 : r258032;
        return r258033;
}

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. Using strategy rm

    3. Applied div-inv45.8

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

    4. Taylor expanded around inf 0.1

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

    5. Simplified0.1

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

    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

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(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))))))