Average Error: 18.5 → 0.2
Time: 7.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -79025762686.63672 \lor \neg \left(y \le 80304997.2964148968\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -79025762686.63672 \lor \neg \left(y \le 80304997.2964148968\right):\\
\;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r347985 = 1.0;
        double r347986 = x;
        double r347987 = y;
        double r347988 = r347986 - r347987;
        double r347989 = r347985 - r347987;
        double r347990 = r347988 / r347989;
        double r347991 = r347985 - r347990;
        double r347992 = log(r347991);
        double r347993 = r347985 - r347992;
        return r347993;
}
↓
double f(double x, double y) {
        double r347994 = y;
        double r347995 = -79025762686.63672;
        bool r347996 = r347994 <= r347995;
        double r347997 = 80304997.2964149;
        bool r347998 = r347994 <= r347997;
        double r347999 = !r347998;
        bool r348000 = r347996 || r347999;
        double r348001 = 1.0;
        double r348002 = x;
        double r348003 = r348002 / r347994;
        double r348004 = 2.0;
        double r348005 = pow(r347994, r348004);
        double r348006 = r348002 / r348005;
        double r348007 = r348001 * r348006;
        double r348008 = r348003 + r348007;
        double r348009 = 1.0;
        double r348010 = r348009 / r347994;
        double r348011 = r348001 * r348010;
        double r348012 = r348008 - r348011;
        double r348013 = log(r348012);
        double r348014 = r348001 - r348013;
        double r348015 = r348002 - r347994;
        double r348016 = r348001 - r347994;
        double r348017 = r348015 / r348016;
        double r348018 = r348001 - r348017;
        double r348019 = sqrt(r348018);
        double r348020 = log(r348019);
        double r348021 = r348020 + r348020;
        double r348022 = r348001 - r348021;
        double r348023 = r348000 ? r348014 : r348022;
        return r348023;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.5
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -79025762686.63672 or 80304997.2964149 < y

    1. Initial program 46.9

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

    3. Applied flip3--53.5

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

    4. Simplified53.5

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

    5. Taylor expanded around inf 0.1

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

    if -79025762686.63672 < y < 80304997.2964149

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied log-prod0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -79025762686.63672 \lor \neg \left(y \le 80304997.2964148968\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))))))