Average Error: 18.0 → 0.1
Time: 22.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -166149344.053797900676727294921875:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{elif}\;y \le 89236118.13017952442169189453125:\\ \;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(1 - \frac{x - y}{1 - y}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -166149344.053797900676727294921875:\\
\;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\

\mathbf{elif}\;y \le 89236118.13017952442169189453125:\\
\;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(1 - \frac{x - y}{1 - y}\right) \cdot \frac{1}{2}\\

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

\end{array}
double f(double x, double y) {
        double r21581839 = 1.0;
        double r21581840 = x;
        double r21581841 = y;
        double r21581842 = r21581840 - r21581841;
        double r21581843 = r21581839 - r21581841;
        double r21581844 = r21581842 / r21581843;
        double r21581845 = r21581839 - r21581844;
        double r21581846 = log(r21581845);
        double r21581847 = r21581839 - r21581846;
        return r21581847;
}
↓
double f(double x, double y) {
        double r21581848 = y;
        double r21581849 = -166149344.0537979;
        bool r21581850 = r21581848 <= r21581849;
        double r21581851 = 1.0;
        double r21581852 = 1.0;
        double r21581853 = r21581851 / r21581848;
        double r21581854 = r21581852 + r21581853;
        double r21581855 = x;
        double r21581856 = r21581855 / r21581848;
        double r21581857 = r21581854 * r21581856;
        double r21581858 = r21581857 - r21581853;
        double r21581859 = log(r21581858);
        double r21581860 = r21581851 - r21581859;
        double r21581861 = 89236118.13017952;
        bool r21581862 = r21581848 <= r21581861;
        double r21581863 = r21581855 - r21581848;
        double r21581864 = r21581851 - r21581848;
        double r21581865 = r21581863 / r21581864;
        double r21581866 = r21581851 - r21581865;
        double r21581867 = sqrt(r21581866);
        double r21581868 = log(r21581867);
        double r21581869 = r21581851 - r21581868;
        double r21581870 = log(r21581866);
        double r21581871 = 0.5;
        double r21581872 = r21581870 * r21581871;
        double r21581873 = r21581869 - r21581872;
        double r21581874 = r21581862 ? r21581873 : r21581860;
        double r21581875 = r21581850 ? r21581860 : r21581874;
        return r21581875;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.0
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 < -166149344.0537979 or 89236118.13017952 < y

    1. Initial program 46.3

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

    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

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

    if -166149344.0537979 < y < 89236118.13017952

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied log-prod0.1

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

    5. Applied associate--r+0.1

      \[\leadsto \color{blue}{\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)}\]
    6. Using strategy rm

    7. Applied pow1/20.1

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

    8. Applied log-pow0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -166149344.053797900676727294921875:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{elif}\;y \le 89236118.13017952442169189453125:\\ \;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(1 - \frac{x - y}{1 - y}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))))