Average Error: 18.3 → 0.1
Time: 19.1s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9999999625628327:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\left(\frac{1.0}{y} \cdot \frac{x}{y} - \frac{1.0}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]
1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9999999625628327:\\
\;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r20202711 = 1.0;
        double r20202712 = x;
        double r20202713 = y;
        double r20202714 = r20202712 - r20202713;
        double r20202715 = r20202711 - r20202713;
        double r20202716 = r20202714 / r20202715;
        double r20202717 = r20202711 - r20202716;
        double r20202718 = log(r20202717);
        double r20202719 = r20202711 - r20202718;
        return r20202719;
}
↓
double f(double x, double y) {
        double r20202720 = x;
        double r20202721 = y;
        double r20202722 = r20202720 - r20202721;
        double r20202723 = 1.0;
        double r20202724 = r20202723 - r20202721;
        double r20202725 = r20202722 / r20202724;
        double r20202726 = 0.9999999625628327;
        bool r20202727 = r20202725 <= r20202726;
        double r20202728 = exp(r20202723);
        double r20202729 = r20202723 - r20202725;
        double r20202730 = r20202728 / r20202729;
        double r20202731 = log(r20202730);
        double r20202732 = r20202723 / r20202721;
        double r20202733 = r20202720 / r20202721;
        double r20202734 = r20202732 * r20202733;
        double r20202735 = r20202734 - r20202732;
        double r20202736 = r20202735 + r20202733;
        double r20202737 = log(r20202736);
        double r20202738 = r20202723 - r20202737;
        double r20202739 = r20202727 ? r20202731 : r20202738;
        return r20202739;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.3
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.61947241:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9999999625628327

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied diff-log0.1

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

    if 0.9999999625628327 < (/ (- x y) (- 1.0 y))

    1. Initial program 60.9

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1.0 - y} \le 0.9999999625628327:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\left(\frac{1.0}{y} \cdot \frac{x}{y} - \frac{1.0}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))))))