Average Error: 17.8 → 0.1
Time: 5.7s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1630045945.92276692 \lor \neg \left(y \le 76447033.3308169693\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -1630045945.92276692 \lor \neg \left(y \le 76447033.3308169693\right):\\
\;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{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 r475355 = 1.0;
        double r475356 = x;
        double r475357 = y;
        double r475358 = r475356 - r475357;
        double r475359 = r475355 - r475357;
        double r475360 = r475358 / r475359;
        double r475361 = r475355 - r475360;
        double r475362 = log(r475361);
        double r475363 = r475355 - r475362;
        return r475363;
}
↓
double f(double x, double y) {
        double r475364 = y;
        double r475365 = -1630045945.922767;
        bool r475366 = r475364 <= r475365;
        double r475367 = 76447033.33081697;
        bool r475368 = r475364 <= r475367;
        double r475369 = !r475368;
        bool r475370 = r475366 || r475369;
        double r475371 = 1.0;
        double r475372 = x;
        double r475373 = 2.0;
        double r475374 = pow(r475364, r475373);
        double r475375 = r475372 / r475374;
        double r475376 = 1.0;
        double r475377 = r475376 / r475364;
        double r475378 = r475375 - r475377;
        double r475379 = r475371 * r475378;
        double r475380 = r475372 / r475364;
        double r475381 = r475379 + r475380;
        double r475382 = log(r475381);
        double r475383 = r475371 - r475382;
        double r475384 = r475372 - r475364;
        double r475385 = r475371 - r475364;
        double r475386 = r475376 / r475385;
        double r475387 = r475384 * r475386;
        double r475388 = r475371 - r475387;
        double r475389 = log(r475388);
        double r475390 = r475371 - r475389;
        double r475391 = r475370 ? r475383 : r475390;
        return r475391;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.8
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1630045945.922767 or 76447033.33081697 < y

    1. Initial program 46.1

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

    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

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

    if -1630045945.922767 < y < 76447033.33081697

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1630045945.92276692 \lor \neg \left(y \le 76447033.3308169693\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))))