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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -119070188.93960175 \lor \neg \left(y \le 12232505.825313114\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -119070188.93960175 \lor \neg \left(y \le 12232505.825313114\right):\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\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 r449420 = 1.0;
        double r449421 = x;
        double r449422 = y;
        double r449423 = r449421 - r449422;
        double r449424 = r449420 - r449422;
        double r449425 = r449423 / r449424;
        double r449426 = r449420 - r449425;
        double r449427 = log(r449426);
        double r449428 = r449420 - r449427;
        return r449428;
}

↓

double f(double x, double y) {
        double r449429 = y;
        double r449430 = -119070188.93960175;
        bool r449431 = r449429 <= r449430;
        double r449432 = 12232505.825313114;
        bool r449433 = r449429 <= r449432;
        double r449434 = !r449433;
        bool r449435 = r449431 || r449434;
        double r449436 = 1.0;
        double r449437 = x;
        double r449438 = 2.0;
        double r449439 = pow(r449429, r449438);
        double r449440 = r449437 / r449439;
        double r449441 = 1.0;
        double r449442 = r449441 / r449429;
        double r449443 = r449440 - r449442;
        double r449444 = r449437 / r449429;
        double r449445 = fma(r449436, r449443, r449444);
        double r449446 = log(r449445);
        double r449447 = r449436 - r449446;
        double r449448 = r449437 - r449429;
        double r449449 = r449436 - r449429;
        double r449450 = r449441 / r449449;
        double r449451 = r449448 * r449450;
        double r449452 = r449436 - r449451;
        double r449453 = log(r449452);
        double r449454 = r449436 - r449453;
        double r449455 = r449435 ? r449447 : r449454;
        return r449455;
}

Original	18.1
Target	0.1
Herbie	0.1

Derivation

Split input into 2 regimes
if y < -119070188.93960175 or 12232505.825313114 < y
1. Initial program 46.8
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Taylor expanded around inf 0.1
  \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)}\]
3. Simplified0.1
  \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)}\]
if -119070188.93960175 < y < 12232505.825313114
1. Initial program 0.1
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied div-inv0.1
  \[\leadsto 1 - \log \left(1 - \color{blue}{\left(x - y\right) \cdot \frac{1}{1 - y}}\right)\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -119070188.93960175 \lor \neg \left(y \le 12232505.825313114\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))))))

Error

Target

Derivation

`if y < -119070188.93960175 or 12232505.825313114 < y`

`if -119070188.93960175 < y < 12232505.825313114`

Reproduce