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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -106501299.420671895 \lor \neg \left(y \le 73789365.95162582\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \left(\sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}} \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right) + \log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -106501299.420671895 \lor \neg \left(y \le 73789365.95162582\right):\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \left(\sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}} \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}, \frac{x}{y}\right)\right)\\

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

\end{array}

double f(double x, double y) {
        double r386635 = 1.0;
        double r386636 = x;
        double r386637 = y;
        double r386638 = r386636 - r386637;
        double r386639 = r386635 - r386637;
        double r386640 = r386638 / r386639;
        double r386641 = r386635 - r386640;
        double r386642 = log(r386641);
        double r386643 = r386635 - r386642;
        return r386643;
}

↓

double f(double x, double y) {
        double r386644 = y;
        double r386645 = -106501299.4206719;
        bool r386646 = r386644 <= r386645;
        double r386647 = 73789365.95162582;
        bool r386648 = r386644 <= r386647;
        double r386649 = !r386648;
        bool r386650 = r386646 || r386649;
        double r386651 = 1.0;
        double r386652 = x;
        double r386653 = 2.0;
        double r386654 = pow(r386644, r386653);
        double r386655 = r386652 / r386654;
        double r386656 = 1.0;
        double r386657 = r386656 / r386644;
        double r386658 = r386655 - r386657;
        double r386659 = cbrt(r386658);
        double r386660 = r386659 * r386659;
        double r386661 = r386660 * r386659;
        double r386662 = r386652 / r386644;
        double r386663 = fma(r386651, r386661, r386662);
        double r386664 = log(r386663);
        double r386665 = r386651 - r386664;
        double r386666 = r386652 - r386644;
        double r386667 = r386651 - r386644;
        double r386668 = r386656 / r386667;
        double r386669 = r386666 * r386668;
        double r386670 = r386651 - r386669;
        double r386671 = sqrt(r386670);
        double r386672 = log(r386671);
        double r386673 = r386672 + r386672;
        double r386674 = r386651 - r386673;
        double r386675 = r386650 ? r386665 : r386674;
        return r386675;
}

Original	18.4
Target	0.1
Herbie	0.1

Derivation

Split input into 2 regimes
if y < -106501299.4206719 or 73789365.95162582 < y
1. Initial program 46.8
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Taylor expanded around inf 0.2
  \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)}\]
3. Simplified0.2
  \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)}\]
4. Using strategy rm
5. Applied add-cube-cbrt0.2
  \[\leadsto 1 - \log \left(\mathsf{fma}\left(1, \color{blue}{\left(\sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}} \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}}, \frac{x}{y}\right)\right)\]
if -106501299.4206719 < y < 73789365.95162582
1. Initial program 0.1
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied div-inv0.1
  \[\leadsto 1 - \log \left(1 - \color{blue}{\left(x - y\right) \cdot \frac{1}{1 - y}}\right)\]
4. Using strategy rm
5. Applied add-sqr-sqrt0.1
  \[\leadsto 1 - \log \color{blue}{\left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}} \cdot \sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)}\]
6. Applied log-prod0.1
  \[\leadsto 1 - \color{blue}{\left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right) + \log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -106501299.420671895 \lor \neg \left(y \le 73789365.95162582\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \left(\sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}} \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}\right) \cdot \sqrt[3]{\frac{x}{{y}^{2}} - \frac{1}{y}}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right) + \log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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 y < -106501299.4206719 or 73789365.95162582 < y`

`if -106501299.4206719 < y < 73789365.95162582`

Reproduce