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

↓

\[\begin{array}{l} \mathbf{if}\;y \le \frac{-64129817537929}{1048576} \lor \neg \left(y \le \frac{4535396767118165}{134217728}\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} - \left(\frac{1}{y} - 1 \cdot \frac{x}{{y}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le \frac{-64129817537929}{1048576} \lor \neg \left(y \le \frac{4535396767118165}{134217728}\right):\\
\;\;\;\;1 - \log \left(\frac{x}{y} - \left(\frac{1}{y} - 1 \cdot \frac{x}{{y}^{2}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\

\end{array}

double f(double x, double y) {
        double r326530 = 1.0;
        double r326531 = x;
        double r326532 = y;
        double r326533 = r326531 - r326532;
        double r326534 = r326530 - r326532;
        double r326535 = r326533 / r326534;
        double r326536 = r326530 - r326535;
        double r326537 = log(r326536);
        double r326538 = r326530 - r326537;
        return r326538;
}

↓

double f(double x, double y) {
        double r326539 = y;
        double r326540 = -64129817537929.0;
        double r326541 = 1048576.0;
        double r326542 = r326540 / r326541;
        bool r326543 = r326539 <= r326542;
        double r326544 = 4535396767118165.0;
        double r326545 = 134217728.0;
        double r326546 = r326544 / r326545;
        bool r326547 = r326539 <= r326546;
        double r326548 = !r326547;
        bool r326549 = r326543 || r326548;
        double r326550 = 1.0;
        double r326551 = x;
        double r326552 = r326551 / r326539;
        double r326553 = r326550 / r326539;
        double r326554 = 2.0;
        double r326555 = pow(r326539, r326554);
        double r326556 = r326551 / r326555;
        double r326557 = r326550 * r326556;
        double r326558 = r326553 - r326557;
        double r326559 = r326552 - r326558;
        double r326560 = log(r326559);
        double r326561 = r326550 - r326560;
        double r326562 = 1.0;
        double r326563 = r326550 - r326539;
        double r326564 = cbrt(r326563);
        double r326565 = r326564 * r326564;
        double r326566 = r326562 / r326565;
        double r326567 = r326551 - r326539;
        double r326568 = r326567 / r326564;
        double r326569 = r326566 * r326568;
        double r326570 = r326550 - r326569;
        double r326571 = log(r326570);
        double r326572 = r326550 - r326571;
        double r326573 = r326549 ? r326561 : r326572;
        return r326573;
}

Original	17.8
Target	0.1
Herbie	0.1

Derivation

Split input into 2 regimes
if y < -61158959.90174198 or 33791339.15244166 < y
1. Initial program 46.1
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied add-cube-cbrt42.3
  \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]
4. Applied *-un-lft-identity42.3
  \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{1 \cdot \left(x - y\right)}}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}\right)\]
5. Applied times-frac42.2
  \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}}\right)\]
6. Taylor expanded around inf 0.1
  \[\leadsto 1 - \log \color{blue}{\left(\left(1 \cdot \frac{x}{{y}^{2}} + \frac{x}{y}\right) - 1 \cdot \frac{1}{y}\right)}\]
7. Simplified0.1
  \[\leadsto 1 - \log \color{blue}{\left(\frac{x}{y} - \left(\frac{1}{y} - 1 \cdot \frac{x}{{y}^{2}}\right)\right)}\]
if -61158959.90174198 < y < 33791339.15244166
1. Initial program 0.1
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied add-cube-cbrt0.1
  \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]
4. Applied *-un-lft-identity0.1
  \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{1 \cdot \left(x - y\right)}}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}\right)\]
5. Applied times-frac0.1
  \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}}\right)\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le \frac{-64129817537929}{1048576} \lor \neg \left(y \le \frac{4535396767118165}{134217728}\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} - \left(\frac{1}{y} - 1 \cdot \frac{x}{{y}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.619472414) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))

Error

Try it out

Target

Derivation

`if y < -61158959.90174198 or 33791339.15244166 < y`

`if -61158959.90174198 < y < 33791339.15244166`

Reproduce

In
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