Average Error: 17.3 → 0.3
Time: 22.5s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.475504821605494 \cdot 10^{+19}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 98095783.98664801:\\ \;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]
1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.475504821605494 \cdot 10^{+19}:\\
\;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\

\mathbf{elif}\;y \le 98095783.98664801:\\
\;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r23636749 = 1.0;
        double r23636750 = x;
        double r23636751 = y;
        double r23636752 = r23636750 - r23636751;
        double r23636753 = r23636749 - r23636751;
        double r23636754 = r23636752 / r23636753;
        double r23636755 = r23636749 - r23636754;
        double r23636756 = log(r23636755);
        double r23636757 = r23636749 - r23636756;
        return r23636757;
}
↓
double f(double x, double y) {
        double r23636758 = y;
        double r23636759 = -3.475504821605494e+19;
        bool r23636760 = r23636758 <= r23636759;
        double r23636761 = 1.0;
        double r23636762 = x;
        double r23636763 = r23636762 / r23636758;
        double r23636764 = r23636761 / r23636758;
        double r23636765 = r23636763 * r23636764;
        double r23636766 = r23636763 - r23636764;
        double r23636767 = r23636765 + r23636766;
        double r23636768 = log(r23636767);
        double r23636769 = r23636761 - r23636768;
        double r23636770 = 98095783.98664801;
        bool r23636771 = r23636758 <= r23636770;
        double r23636772 = r23636762 - r23636758;
        double r23636773 = r23636761 - r23636758;
        double r23636774 = r23636772 / r23636773;
        double r23636775 = r23636761 - r23636774;
        double r23636776 = sqrt(r23636775);
        double r23636777 = log(r23636776);
        double r23636778 = r23636777 + r23636777;
        double r23636779 = r23636761 - r23636778;
        double r23636780 = r23636771 ? r23636779 : r23636769;
        double r23636781 = r23636760 ? r23636769 : r23636780;
        return r23636781;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.3
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.61947241:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.475504821605494e+19 or 98095783.98664801 < y

    1. Initial program 46.1

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]

    2. Taylor expanded around inf 0.0

      \[\leadsto 1.0 - \log \color{blue}{\left(\left(\frac{x}{y} + 1.0 \cdot \frac{x}{{y}^{2}}\right) - 1.0 \cdot \frac{1}{y}\right)}\]

    3. Simplified0.0

      \[\leadsto 1.0 - \log \color{blue}{\left(\frac{1.0}{y} \cdot \frac{x}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)}\]

    if -3.475504821605494e+19 < y < 98095783.98664801

    1. Initial program 0.5

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.5

      \[\leadsto 1.0 - \log \color{blue}{\left(\sqrt{1.0 - \frac{x - y}{1.0 - y}} \cdot \sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)}\]

    4. Applied log-prod0.5

      \[\leadsto 1.0 - \color{blue}{\left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.475504821605494 \cdot 10^{+19}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 98095783.98664801:\\ \;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))