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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -93297022.8796742707 \lor \neg \left(y \le 106211441.56073992\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -93297022.8796742707 \lor \neg \left(y \le 106211441.56073992\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

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

\end{array}

double f(double x, double y) {
        double r796794 = 1.0;
        double r796795 = x;
        double r796796 = r796794 - r796795;
        double r796797 = y;
        double r796798 = r796796 * r796797;
        double r796799 = r796797 + r796794;
        double r796800 = r796798 / r796799;
        double r796801 = r796794 - r796800;
        return r796801;
}

↓

double f(double x, double y) {
        double r796802 = y;
        double r796803 = -93297022.87967427;
        bool r796804 = r796802 <= r796803;
        double r796805 = 106211441.56073992;
        bool r796806 = r796802 <= r796805;
        double r796807 = !r796806;
        bool r796808 = r796804 || r796807;
        double r796809 = 1.0;
        double r796810 = 1.0;
        double r796811 = r796810 / r796802;
        double r796812 = x;
        double r796813 = r796812 / r796802;
        double r796814 = r796811 - r796813;
        double r796815 = r796809 * r796814;
        double r796816 = r796815 + r796812;
        double r796817 = r796809 - r796812;
        double r796818 = r796817 * r796802;
        double r796819 = r796802 * r796802;
        double r796820 = r796809 * r796809;
        double r796821 = r796819 - r796820;
        double r796822 = r796818 / r796821;
        double r796823 = r796802 - r796809;
        double r796824 = r796822 * r796823;
        double r796825 = r796809 - r796824;
        double r796826 = r796808 ? r796816 : r796825;
        return r796826;
}

Original	22.2
Target	0.2
Herbie	0.2

Derivation

Split input into 2 regimes
if y < -93297022.87967427 or 106211441.56073992 < y
1. Initial program 45.7
  \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
2. Taylor expanded around inf 0.1
  \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}}\]
3. Simplified0.1
  \[\leadsto \color{blue}{1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x}\]
if -93297022.87967427 < y < 106211441.56073992
1. Initial program 0.2
  \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
2. Using strategy rm
3. Applied flip-+0.2
  \[\leadsto 1 - \frac{\left(1 - x\right) \cdot y}{\color{blue}{\frac{y \cdot y - 1 \cdot 1}{y - 1}}}\]
4. Applied associate-/r/0.2
  \[\leadsto 1 - \color{blue}{\frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)}\]
Recombined 2 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -93297022.8796742707 \lor \neg \left(y \le 106211441.56073992\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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

Try it out

Target

Derivation

`if y < -93297022.87967427 or 106211441.56073992 < y`

`if -93297022.87967427 < y < 106211441.56073992`

Reproduce

In
Out