Average Error: 18.2 → 0.3
Time: 53.6s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.09185737703913710028302830323809757828712:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{1 - y} \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + \frac{x}{y} \cdot \frac{1}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.09185737703913710028302830323809757828712:\\
\;\;\;\;1 - \log \left(1 - \frac{1}{1 - y} \cdot \left(x - y\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\left(\frac{x}{y} + \frac{x}{y} \cdot \frac{1}{y}\right) - \frac{1}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r19968542 = 1.0;
        double r19968543 = x;
        double r19968544 = y;
        double r19968545 = r19968543 - r19968544;
        double r19968546 = r19968542 - r19968544;
        double r19968547 = r19968545 / r19968546;
        double r19968548 = r19968542 - r19968547;
        double r19968549 = log(r19968548);
        double r19968550 = r19968542 - r19968549;
        return r19968550;
}
↓
double f(double x, double y) {
        double r19968551 = x;
        double r19968552 = y;
        double r19968553 = r19968551 - r19968552;
        double r19968554 = 1.0;
        double r19968555 = r19968554 - r19968552;
        double r19968556 = r19968553 / r19968555;
        double r19968557 = 0.0918573770391371;
        bool r19968558 = r19968556 <= r19968557;
        double r19968559 = 1.0;
        double r19968560 = r19968559 / r19968555;
        double r19968561 = r19968560 * r19968553;
        double r19968562 = r19968554 - r19968561;
        double r19968563 = log(r19968562);
        double r19968564 = r19968554 - r19968563;
        double r19968565 = r19968551 / r19968552;
        double r19968566 = r19968554 / r19968552;
        double r19968567 = r19968565 * r19968566;
        double r19968568 = r19968565 + r19968567;
        double r19968569 = r19968568 - r19968566;
        double r19968570 = log(r19968569);
        double r19968571 = r19968554 - r19968570;
        double r19968572 = r19968558 ? r19968564 : r19968571;
        return r19968572;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.2
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.0918573770391371

    1. Initial program 0.0

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    2. Using strategy rm

    3. Applied div-inv0.0

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

    if 0.0918573770391371 < (/ (- x y) (- 1.0 y))

    1. Initial program 60.7

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    2. Using strategy rm

    3. Applied div-inv59.2

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

    4. Taylor expanded around inf 1.0

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

    5. Simplified1.0

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.09185737703913710028302830323809757828712:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{1 - y} \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + \frac{x}{y} \cdot \frac{1}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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))))))