Average Error: 17.6 → 0.1
Time: 18.7s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -116643585.83872083:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 51693858.328214146:\\ \;\;\;\;1.0 - \log \left(1.0 - \left(x - y\right) \cdot \frac{1}{1.0 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]
1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -116643585.83872083:\\
\;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\

\mathbf{elif}\;y \le 51693858.328214146:\\
\;\;\;\;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} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r20535482 = 1.0;
        double r20535483 = x;
        double r20535484 = y;
        double r20535485 = r20535483 - r20535484;
        double r20535486 = r20535482 - r20535484;
        double r20535487 = r20535485 / r20535486;
        double r20535488 = r20535482 - r20535487;
        double r20535489 = log(r20535488);
        double r20535490 = r20535482 - r20535489;
        return r20535490;
}
↓
double f(double x, double y) {
        double r20535491 = y;
        double r20535492 = -116643585.83872083;
        bool r20535493 = r20535491 <= r20535492;
        double r20535494 = 1.0;
        double r20535495 = x;
        double r20535496 = r20535495 / r20535491;
        double r20535497 = r20535494 / r20535491;
        double r20535498 = r20535496 * r20535497;
        double r20535499 = r20535496 - r20535497;
        double r20535500 = r20535498 + r20535499;
        double r20535501 = log(r20535500);
        double r20535502 = r20535494 - r20535501;
        double r20535503 = 51693858.328214146;
        bool r20535504 = r20535491 <= r20535503;
        double r20535505 = r20535495 - r20535491;
        double r20535506 = 1.0;
        double r20535507 = r20535494 - r20535491;
        double r20535508 = r20535506 / r20535507;
        double r20535509 = r20535505 * r20535508;
        double r20535510 = r20535494 - r20535509;
        double r20535511 = log(r20535510);
        double r20535512 = r20535494 - r20535511;
        double r20535513 = r20535504 ? r20535512 : r20535502;
        double r20535514 = r20535493 ? r20535502 : r20535513;
        return r20535514;
}

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 < -116643585.83872083 or 51693858.328214146 < y

    1. Initial program 46.1

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

    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

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

    if -116643585.83872083 < y < 51693858.328214146

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -116643585.83872083:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 51693858.328214146:\\ \;\;\;\;1.0 - \log \left(1.0 - \left(x - y\right) \cdot \frac{1}{1.0 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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))))))