Average Error: 17.6 → 0.1
Time: 20.0s
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 r19977508 = 1.0;
        double r19977509 = x;
        double r19977510 = y;
        double r19977511 = r19977509 - r19977510;
        double r19977512 = r19977508 - r19977510;
        double r19977513 = r19977511 / r19977512;
        double r19977514 = r19977508 - r19977513;
        double r19977515 = log(r19977514);
        double r19977516 = r19977508 - r19977515;
        return r19977516;
}
↓
double f(double x, double y) {
        double r19977517 = y;
        double r19977518 = -156542919.49421751;
        bool r19977519 = r19977517 <= r19977518;
        double r19977520 = 1.0;
        double r19977521 = x;
        double r19977522 = r19977521 / r19977517;
        double r19977523 = r19977520 / r19977517;
        double r19977524 = r19977522 * r19977523;
        double r19977525 = r19977524 - r19977523;
        double r19977526 = r19977522 + r19977525;
        double r19977527 = log(r19977526);
        double r19977528 = r19977520 - r19977527;
        double r19977529 = 41723711.509616;
        bool r19977530 = r19977517 <= r19977529;
        double r19977531 = r19977521 - r19977517;
        double r19977532 = 1.0;
        double r19977533 = r19977520 - r19977517;
        double r19977534 = r19977532 / r19977533;
        double r19977535 = r19977531 * r19977534;
        double r19977536 = r19977520 - r19977535;
        double r19977537 = log(r19977536);
        double r19977538 = r19977520 - r19977537;
        double r19977539 = r19977530 ? r19977538 : r19977528;
        double r19977540 = r19977519 ? r19977528 : r19977539;
        return r19977540;
}

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))))))