Average Error: 17.6 → 0.1
Time: 19.7s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -156542919.49421751:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1.0}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 41723711.509616:\\ \;\;\;\;1.0 - \log \left(1.0 - \left(x - y\right) \cdot \frac{1}{1.0 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1.0}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]
1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -156542919.49421751:\\
\;\;\;\;1.0 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1.0}{y} - \frac{1.0}{y}\right)\right)\\

\mathbf{elif}\;y \le 41723711.509616:\\
\;\;\;\;1.0 - \log \left(1.0 - \left(x - y\right) \cdot \frac{1}{1.0 - y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r16490361 = 1.0;
        double r16490362 = x;
        double r16490363 = y;
        double r16490364 = r16490362 - r16490363;
        double r16490365 = r16490361 - r16490363;
        double r16490366 = r16490364 / r16490365;
        double r16490367 = r16490361 - r16490366;
        double r16490368 = log(r16490367);
        double r16490369 = r16490361 - r16490368;
        return r16490369;
}
↓
double f(double x, double y) {
        double r16490370 = y;
        double r16490371 = -156542919.49421751;
        bool r16490372 = r16490370 <= r16490371;
        double r16490373 = 1.0;
        double r16490374 = x;
        double r16490375 = r16490374 / r16490370;
        double r16490376 = r16490373 / r16490370;
        double r16490377 = r16490375 * r16490376;
        double r16490378 = r16490377 - r16490376;
        double r16490379 = r16490375 + r16490378;
        double r16490380 = log(r16490379);
        double r16490381 = r16490373 - r16490380;
        double r16490382 = 41723711.509616;
        bool r16490383 = r16490370 <= r16490382;
        double r16490384 = r16490374 - r16490370;
        double r16490385 = 1.0;
        double r16490386 = r16490373 - r16490370;
        double r16490387 = r16490385 / r16490386;
        double r16490388 = r16490384 * r16490387;
        double r16490389 = r16490373 - r16490388;
        double r16490390 = log(r16490389);
        double r16490391 = r16490373 - r16490390;
        double r16490392 = r16490383 ? r16490391 : r16490381;
        double r16490393 = r16490372 ? r16490381 : r16490392;
        return r16490393;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.6
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 y < -156542919.49421751 or 41723711.509616 < y

    1. Initial program 45.2

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

    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

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

    if -156542919.49421751 < y < 41723711.509616

    1. Initial program 0.1

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

    3. Applied div-inv0.1

      \[\leadsto 1.0 - \log \left(1.0 - \color{blue}{\left(x - y\right) \cdot \frac{1}{1.0 - y}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -156542919.49421751:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1.0}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 41723711.509616:\\ \;\;\;\;1.0 - \log \left(1.0 - \left(x - y\right) \cdot \frac{1}{1.0 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1.0}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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))))))