Average Error: 17.9 → 0.1
Time: 5.1s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -201288582.2730185985565185546875 \lor \neg \left(y \le 116648832255.3203277587890625\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -201288582.2730185985565185546875 \lor \neg \left(y \le 116648832255.3203277587890625\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r535596 = 1.0;
        double r535597 = x;
        double r535598 = y;
        double r535599 = r535597 - r535598;
        double r535600 = r535596 - r535598;
        double r535601 = r535599 / r535600;
        double r535602 = r535596 - r535601;
        double r535603 = log(r535602);
        double r535604 = r535596 - r535603;
        return r535604;
}
↓
double f(double x, double y) {
        double r535605 = y;
        double r535606 = -201288582.2730186;
        bool r535607 = r535605 <= r535606;
        double r535608 = 116648832255.32033;
        bool r535609 = r535605 <= r535608;
        double r535610 = !r535609;
        bool r535611 = r535607 || r535610;
        double r535612 = 1.0;
        double r535613 = exp(r535612);
        double r535614 = x;
        double r535615 = 2.0;
        double r535616 = pow(r535605, r535615);
        double r535617 = r535614 / r535616;
        double r535618 = 1.0;
        double r535619 = r535618 / r535605;
        double r535620 = r535617 - r535619;
        double r535621 = r535612 * r535620;
        double r535622 = r535614 / r535605;
        double r535623 = r535621 + r535622;
        double r535624 = r535613 / r535623;
        double r535625 = log(r535624);
        double r535626 = r535614 - r535605;
        double r535627 = r535612 - r535605;
        double r535628 = r535626 / r535627;
        double r535629 = r535612 - r535628;
        double r535630 = r535613 / r535629;
        double r535631 = log(r535630);
        double r535632 = r535611 ? r535625 : r535631;
        return r535632;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.9
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -201288582.2730186 or 116648832255.32033 < y

    1. Initial program 46.3

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

    3. Applied add-log-exp46.3

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

    4. Applied diff-log46.3

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

    if -201288582.2730186 < y < 116648832255.32033

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied diff-log0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -201288582.2730185985565185546875 \lor \neg \left(y \le 116648832255.3203277587890625\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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))))))