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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.03024915003939501398355638173143233871087:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.03024915003939501398355638173143233871087:\\
\;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\

\end{array}

double f(double x, double y) {
        double r215317 = 1.0;
        double r215318 = x;
        double r215319 = y;
        double r215320 = r215318 - r215319;
        double r215321 = r215317 - r215319;
        double r215322 = r215320 / r215321;
        double r215323 = r215317 - r215322;
        double r215324 = log(r215323);
        double r215325 = r215317 - r215324;
        return r215325;
}

↓

double f(double x, double y) {
        double r215326 = x;
        double r215327 = y;
        double r215328 = r215326 - r215327;
        double r215329 = 1.0;
        double r215330 = r215329 - r215327;
        double r215331 = r215328 / r215330;
        double r215332 = 0.030249150039395014;
        bool r215333 = r215331 <= r215332;
        double r215334 = 1.0;
        double r215335 = r215334 / r215330;
        double r215336 = r215328 * r215335;
        double r215337 = r215329 - r215336;
        double r215338 = log(r215337);
        double r215339 = r215329 - r215338;
        double r215340 = exp(r215329);
        double r215341 = 2.0;
        double r215342 = pow(r215327, r215341);
        double r215343 = r215326 / r215342;
        double r215344 = r215326 / r215327;
        double r215345 = fma(r215329, r215343, r215344);
        double r215346 = r215329 / r215327;
        double r215347 = r215345 - r215346;
        double r215348 = r215340 / r215347;
        double r215349 = log(r215348);
        double r215350 = r215333 ? r215339 : r215349;
        return r215350;
}

Original	18.2
Target	0.1
Herbie	0.4

Derivation

Split input into 2 regimes
if (/ (- x y) (- 1.0 y)) < 0.030249150039395014
1. Initial program 0.0
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied div-inv0.0
  \[\leadsto 1 - \log \left(1 - \color{blue}{\left(x - y\right) \cdot \frac{1}{1 - y}}\right)\]
if 0.030249150039395014 < (/ (- x y) (- 1.0 y))
1. Initial program 60.5
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied add-cbrt-cube60.5
  \[\leadsto \color{blue}{\sqrt[3]{\left(\left(1 - \log \left(1 - \frac{x - y}{1 - y}\right)\right) \cdot \left(1 - \log \left(1 - \frac{x - y}{1 - y}\right)\right)\right) \cdot \left(1 - \log \left(1 - \frac{x - y}{1 - y}\right)\right)}}\]
4. Simplified60.5
  \[\leadsto \sqrt[3]{\color{blue}{{\left(1 - \log \left(1 - \frac{x - y}{1 - y}\right)\right)}^{3}}}\]
5. Using strategy rm
6. Applied add-log-exp60.5
  \[\leadsto \color{blue}{\log \left(e^{\sqrt[3]{{\left(1 - \log \left(1 - \frac{x - y}{1 - y}\right)\right)}^{3}}}\right)}\]
7. Simplified60.5
  \[\leadsto \log \color{blue}{\left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)}\]
8. Taylor expanded around inf 1.1
  \[\leadsto \log \left(\frac{e^{1}}{\color{blue}{\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}}}\right)\]
9. Simplified1.1
  \[\leadsto \log \left(\frac{e^{1}}{\color{blue}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}}\right)\]
Recombined 2 regimes into one program.
Final simplification0.4
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.03024915003939501398355638173143233871087:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(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

Target

Derivation

`if (/ (- x y) (- 1.0 y)) < 0.030249150039395014`

`if 0.030249150039395014 < (/ (- x y) (- 1.0 y))`

Reproduce