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

↓

\[\begin{array}{l} \mathbf{if}\;1 - \frac{x - y}{1 - y} \le 2.61378141 \cdot 10^{-10}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;1 - \frac{x - y}{1 - y} \le 2.61378141 \cdot 10^{-10}:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\

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

\end{array}

double f(double x, double y) {
        double r383529 = 1.0;
        double r383530 = x;
        double r383531 = y;
        double r383532 = r383530 - r383531;
        double r383533 = r383529 - r383531;
        double r383534 = r383532 / r383533;
        double r383535 = r383529 - r383534;
        double r383536 = log(r383535);
        double r383537 = r383529 - r383536;
        return r383537;
}

↓

double f(double x, double y) {
        double r383538 = 1.0;
        double r383539 = x;
        double r383540 = y;
        double r383541 = r383539 - r383540;
        double r383542 = r383538 - r383540;
        double r383543 = r383541 / r383542;
        double r383544 = r383538 - r383543;
        double r383545 = 2.6137814135296367e-10;
        bool r383546 = r383544 <= r383545;
        double r383547 = 2.0;
        double r383548 = pow(r383540, r383547);
        double r383549 = r383539 / r383548;
        double r383550 = 1.0;
        double r383551 = r383550 / r383540;
        double r383552 = r383549 - r383551;
        double r383553 = r383539 / r383540;
        double r383554 = fma(r383538, r383552, r383553);
        double r383555 = log(r383554);
        double r383556 = r383538 - r383555;
        double r383557 = cbrt(r383541);
        double r383558 = r383557 * r383557;
        double r383559 = r383557 / r383542;
        double r383560 = r383558 * r383559;
        double r383561 = r383538 - r383560;
        double r383562 = log(r383561);
        double r383563 = r383538 - r383562;
        double r383564 = r383546 ? r383556 : r383563;
        return r383564;
}

Original	18.3
Target	0.1
Herbie	0.2

Derivation

Split input into 2 regimes
if (- 1.0 (/ (- x y) (- 1.0 y))) < 2.6137814135296367e-10
1. Initial program 63.4
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
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
  \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)}\]
if 2.6137814135296367e-10 < (- 1.0 (/ (- x y) (- 1.0 y)))
1. Initial program 0.2
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied *-un-lft-identity0.2
  \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{1 \cdot \left(1 - y\right)}}\right)\]
4. Applied add-cube-cbrt0.2
  \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{\left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \sqrt[3]{x - y}}}{1 \cdot \left(1 - y\right)}\right)\]
5. Applied times-frac0.3
  \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{1} \cdot \frac{\sqrt[3]{x - y}}{1 - y}}\right)\]
6. Simplified0.3
  \[\leadsto 1 - \log \left(1 - \color{blue}{\left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right)} \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\]
Recombined 2 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{x - y}{1 - y} \le 2.61378141 \cdot 10^{-10}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 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

Target

Derivation

`if (- 1.0 (/ (- x y) (- 1.0 y))) < 2.6137814135296367e-10`

`if 2.6137814135296367e-10 < (- 1.0 (/ (- x y) (- 1.0 y)))`

Reproduce