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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -61215075097455.1328125 \lor \neg \left(y \le 150550976.723195374011993408203125\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y + 1} - \frac{1}{y + 1}, y, 1\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -61215075097455.1328125 \lor \neg \left(y \le 150550976.723195374011993408203125\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y + 1} - \frac{1}{y + 1}, y, 1\right)\\

\end{array}

double f(double x, double y) {
        double r463812 = 1.0;
        double r463813 = x;
        double r463814 = r463812 - r463813;
        double r463815 = y;
        double r463816 = r463814 * r463815;
        double r463817 = r463815 + r463812;
        double r463818 = r463816 / r463817;
        double r463819 = r463812 - r463818;
        return r463819;
}

↓

double f(double x, double y) {
        double r463820 = y;
        double r463821 = -61215075097455.13;
        bool r463822 = r463820 <= r463821;
        double r463823 = 150550976.72319537;
        bool r463824 = r463820 <= r463823;
        double r463825 = !r463824;
        bool r463826 = r463822 || r463825;
        double r463827 = 1.0;
        double r463828 = 1.0;
        double r463829 = r463828 / r463820;
        double r463830 = x;
        double r463831 = r463830 / r463820;
        double r463832 = r463829 - r463831;
        double r463833 = fma(r463827, r463832, r463830);
        double r463834 = r463820 + r463827;
        double r463835 = r463830 / r463834;
        double r463836 = r463827 / r463834;
        double r463837 = r463835 - r463836;
        double r463838 = fma(r463837, r463820, r463827);
        double r463839 = r463826 ? r463833 : r463838;
        return r463839;
}

Original	22.8
Target	0.2
Herbie	0.2

Derivation

Split input into 2 regimes
if y < -61215075097455.13 or 150550976.72319537 < y
1. Initial program 46.7
  \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
2. Simplified30.0
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)}\]
3. Using strategy rm
4. Applied div-sub30.0
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y + 1} - \frac{1}{y + 1}}, y, 1\right)\]
5. Taylor expanded around inf 0.1
  \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}}\]
6. Simplified0.1
  \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)}\]
if -61215075097455.13 < y < 150550976.72319537
1. Initial program 0.3
  \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
2. Simplified0.3
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)}\]
3. Using strategy rm
4. Applied div-sub0.3
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y + 1} - \frac{1}{y + 1}}, y, 1\right)\]
Recombined 2 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -61215075097455.1328125 \lor \neg \left(y \le 150550976.723195374011993408203125\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y + 1} - \frac{1}{y + 1}, y, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))

Error

Target

Derivation

`if y < -61215075097455.13 or 150550976.72319537 < y`

`if -61215075097455.13 < y < 150550976.72319537`

Reproduce