Average Error: 18.4 → 0.1
Time: 6.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -270508391.61573327 \lor \neg \left(y \le 148253897.024537414\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -270508391.61573327 \lor \neg \left(y \le 148253897.024537414\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 r419823 = 1.0;
        double r419824 = x;
        double r419825 = y;
        double r419826 = r419824 - r419825;
        double r419827 = r419823 - r419825;
        double r419828 = r419826 / r419827;
        double r419829 = r419823 - r419828;
        double r419830 = log(r419829);
        double r419831 = r419823 - r419830;
        return r419831;
}
↓
double f(double x, double y) {
        double r419832 = y;
        double r419833 = -270508391.61573327;
        bool r419834 = r419832 <= r419833;
        double r419835 = 148253897.0245374;
        bool r419836 = r419832 <= r419835;
        double r419837 = !r419836;
        bool r419838 = r419834 || r419837;
        double r419839 = 1.0;
        double r419840 = exp(r419839);
        double r419841 = x;
        double r419842 = 2.0;
        double r419843 = pow(r419832, r419842);
        double r419844 = r419841 / r419843;
        double r419845 = 1.0;
        double r419846 = r419845 / r419832;
        double r419847 = r419844 - r419846;
        double r419848 = r419839 * r419847;
        double r419849 = r419841 / r419832;
        double r419850 = r419848 + r419849;
        double r419851 = r419840 / r419850;
        double r419852 = log(r419851);
        double r419853 = r419841 - r419832;
        double r419854 = r419839 - r419832;
        double r419855 = r419853 / r419854;
        double r419856 = r419839 - r419855;
        double r419857 = r419840 / r419856;
        double r419858 = log(r419857);
        double r419859 = r419838 ? r419852 : r419858;
        return r419859;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.4
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 < -270508391.61573327 or 148253897.0245374 < y

    1. Initial program 47.3

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

    3. Applied add-log-exp47.3

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

    4. Applied diff-log47.3

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

    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 -270508391.61573327 < y < 148253897.0245374

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -270508391.61573327 \lor \neg \left(y \le 148253897.024537414\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))))))