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

↓

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

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 r431555 = 1.0;
        double r431556 = x;
        double r431557 = r431555 - r431556;
        double r431558 = y;
        double r431559 = r431557 * r431558;
        double r431560 = r431558 + r431555;
        double r431561 = r431559 / r431560;
        double r431562 = r431555 - r431561;
        return r431562;
}

↓

double f(double x, double y) {
        double r431563 = y;
        double r431564 = -61215075097455.13;
        bool r431565 = r431563 <= r431564;
        double r431566 = 150550976.72319537;
        bool r431567 = r431563 <= r431566;
        double r431568 = !r431567;
        bool r431569 = r431565 || r431568;
        double r431570 = 1.0;
        double r431571 = 1.0;
        double r431572 = r431571 / r431563;
        double r431573 = x;
        double r431574 = r431573 / r431563;
        double r431575 = r431572 - r431574;
        double r431576 = fma(r431570, r431575, r431573);
        double r431577 = r431563 + r431570;
        double r431578 = r431573 / r431577;
        double r431579 = r431570 / r431577;
        double r431580 = r431578 - r431579;
        double r431581 = fma(r431580, r431563, r431570);
        double r431582 = r431569 ? r431576 : r431581;
        return r431582;
}

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