Average Error: 22.3 → 0.2
Time: 29.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -295491748.0547482967376708984375 \lor \neg \left(y \le 237947981.839270412921905517578125\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y}, 1 - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -295491748.0547482967376708984375 \lor \neg \left(y \le 237947981.839270412921905517578125\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{y}, 1 - x, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r467601 = 1.0;
        double r467602 = x;
        double r467603 = r467601 - r467602;
        double r467604 = y;
        double r467605 = r467603 * r467604;
        double r467606 = r467604 + r467601;
        double r467607 = r467605 / r467606;
        double r467608 = r467601 - r467607;
        return r467608;
}
↓
double f(double x, double y) {
        double r467609 = y;
        double r467610 = -295491748.0547483;
        bool r467611 = r467609 <= r467610;
        double r467612 = 237947981.8392704;
        bool r467613 = r467609 <= r467612;
        double r467614 = !r467613;
        bool r467615 = r467611 || r467614;
        double r467616 = 1.0;
        double r467617 = r467616 / r467609;
        double r467618 = 1.0;
        double r467619 = x;
        double r467620 = r467618 - r467619;
        double r467621 = fma(r467617, r467620, r467619);
        double r467622 = r467619 - r467616;
        double r467623 = r467609 + r467616;
        double r467624 = r467622 / r467623;
        double r467625 = fma(r467624, r467609, r467616);
        double r467626 = r467615 ? r467621 : r467625;
        return r467626;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.3
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -295491748.0547483 or 237947981.8392704 < y

    1. Initial program 45.4

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

    2. Simplified29.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)}\]

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y}, 1 - x, x\right)}\]

    if -295491748.0547483 < y < 237947981.8392704

    1. Initial program 0.3

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

    2. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -295491748.0547482967376708984375 \lor \neg \left(y \le 237947981.839270412921905517578125\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y}, 1 - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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))))