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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1864702967354.43042 \lor \neg \left(y \le 28223629.6847505309\right):\\ \;\;\;\;1 - \log \left(\left(1 \cdot \frac{x}{{y}^{2}} + \frac{x}{y}\right) - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -1864702967354.43042 \lor \neg \left(y \le 28223629.6847505309\right):\\
\;\;\;\;1 - \log \left(\left(1 \cdot \frac{x}{{y}^{2}} + \frac{x}{y}\right) - \frac{1}{y}\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 r407480 = 1.0;
        double r407481 = x;
        double r407482 = y;
        double r407483 = r407481 - r407482;
        double r407484 = r407480 - r407482;
        double r407485 = r407483 / r407484;
        double r407486 = r407480 - r407485;
        double r407487 = log(r407486);
        double r407488 = r407480 - r407487;
        return r407488;
}

↓

double f(double x, double y) {
        double r407489 = y;
        double r407490 = -1864702967354.4304;
        bool r407491 = r407489 <= r407490;
        double r407492 = 28223629.68475053;
        bool r407493 = r407489 <= r407492;
        double r407494 = !r407493;
        bool r407495 = r407491 || r407494;
        double r407496 = 1.0;
        double r407497 = x;
        double r407498 = 2.0;
        double r407499 = pow(r407489, r407498);
        double r407500 = r407497 / r407499;
        double r407501 = r407496 * r407500;
        double r407502 = r407497 / r407489;
        double r407503 = r407501 + r407502;
        double r407504 = r407496 / r407489;
        double r407505 = r407503 - r407504;
        double r407506 = log(r407505);
        double r407507 = r407496 - r407506;
        double r407508 = 1.0;
        double r407509 = r407496 - r407489;
        double r407510 = cbrt(r407509);
        double r407511 = r407510 * r407510;
        double r407512 = r407508 / r407511;
        double r407513 = r407497 - r407489;
        double r407514 = r407513 / r407510;
        double r407515 = r407512 * r407514;
        double r407516 = r407496 - r407515;
        double r407517 = log(r407516);
        double r407518 = r407496 - r407517;
        double r407519 = r407495 ? r407507 : r407518;
        return r407519;
}

Original	18.3
Target	0.1
Herbie	0.1

Derivation

Split input into 2 regimes
if y < -1864702967354.4304 or 28223629.68475053 < y
1. Initial program 46.6
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied add-cube-cbrt42.9
  \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]
4. Applied *-un-lft-identity42.9
  \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{1 \cdot \left(x - y\right)}}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}\right)\]
5. Applied times-frac42.8
  \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}}\right)\]
6. Taylor expanded around inf 0.0
  \[\leadsto 1 - \log \color{blue}{\left(\left(1 \cdot \frac{x}{{y}^{2}} + \frac{x}{y}\right) - 1 \cdot \frac{1}{y}\right)}\]
7. Simplified0.0
  \[\leadsto 1 - \log \color{blue}{\left(\left(1 \cdot \frac{x}{{y}^{2}} + \frac{x}{y}\right) - \frac{1}{y}\right)}\]
if -1864702967354.4304 < y < 28223629.68475053
1. Initial program 0.2
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied add-cube-cbrt0.2
  \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]
4. Applied *-un-lft-identity0.2
  \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{1 \cdot \left(x - y\right)}}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}\right)\]
5. Applied times-frac0.2
  \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}}\right)\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1864702967354.43042 \lor \neg \left(y \le 28223629.6847505309\right):\\ \;\;\;\;1 - \log \left(\left(1 \cdot \frac{x}{{y}^{2}} + \frac{x}{y}\right) - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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))))))

Error

Try it out

Target

Derivation

`if y < -1864702967354.4304 or 28223629.68475053 < y`

`if -1864702967354.4304 < y < 28223629.68475053`

Reproduce

In
Out