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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1123308855491800.2 \lor \neg \left(y \le 106063957.48696265\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -1123308855491800.2 \lor \neg \left(y \le 106063957.48696265\right):\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \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 r263755 = 1.0;
        double r263756 = x;
        double r263757 = y;
        double r263758 = r263756 - r263757;
        double r263759 = r263755 - r263757;
        double r263760 = r263758 / r263759;
        double r263761 = r263755 - r263760;
        double r263762 = log(r263761);
        double r263763 = r263755 - r263762;
        return r263763;
}

↓

double f(double x, double y) {
        double r263764 = y;
        double r263765 = -1123308855491800.2;
        bool r263766 = r263764 <= r263765;
        double r263767 = 106063957.48696265;
        bool r263768 = r263764 <= r263767;
        double r263769 = !r263768;
        bool r263770 = r263766 || r263769;
        double r263771 = 1.0;
        double r263772 = x;
        double r263773 = 2.0;
        double r263774 = pow(r263764, r263773);
        double r263775 = r263772 / r263774;
        double r263776 = r263772 / r263764;
        double r263777 = fma(r263771, r263775, r263776);
        double r263778 = r263771 / r263764;
        double r263779 = r263777 - r263778;
        double r263780 = log(r263779);
        double r263781 = r263771 - r263780;
        double r263782 = r263772 - r263764;
        double r263783 = 1.0;
        double r263784 = r263771 - r263764;
        double r263785 = r263783 / r263784;
        double r263786 = r263782 * r263785;
        double r263787 = r263771 - r263786;
        double r263788 = log(r263787);
        double r263789 = r263771 - r263788;
        double r263790 = r263770 ? r263781 : r263789;
        return r263790;
}

Original	17.9
Target	0.1
Herbie	0.2

Derivation

Split input into 2 regimes
if y < -1123308855491800.2 or 106063957.48696265 < y
1. Initial program 46.6
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Taylor expanded around inf 0.0
  \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)}\]
3. Simplified0.0
  \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right)}\]
if -1123308855491800.2 < y < 106063957.48696265
1. Initial program 0.3
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied div-inv0.3
  \[\leadsto 1 - \log \left(1 - \color{blue}{\left(x - y\right) \cdot \frac{1}{1 - y}}\right)\]
Recombined 2 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1123308855491800.2 \lor \neg \left(y \le 106063957.48696265\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(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))))))

Error

Target

Derivation

`if y < -1123308855491800.2 or 106063957.48696265 < y`

`if -1123308855491800.2 < y < 106063957.48696265`

Reproduce