Average Error: 22.5 → 7.2
Time: 4.1s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.0766327124897609 \cdot 10^{33} \lor \neg \left(y \le 5.3592131041254416 \cdot 10^{25}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right), x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.0766327124897609 \cdot 10^{33} \lor \neg \left(y \le 5.3592131041254416 \cdot 10^{25}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r652583 = 1.0;
        double r652584 = x;
        double r652585 = r652583 - r652584;
        double r652586 = y;
        double r652587 = r652585 * r652586;
        double r652588 = r652586 + r652583;
        double r652589 = r652587 / r652588;
        double r652590 = r652583 - r652589;
        return r652590;
}
↓
double f(double x, double y) {
        double r652591 = y;
        double r652592 = -2.076632712489761e+33;
        bool r652593 = r652591 <= r652592;
        double r652594 = 5.359213104125442e+25;
        bool r652595 = r652591 <= r652594;
        double r652596 = !r652595;
        bool r652597 = r652593 || r652596;
        double r652598 = x;
        double r652599 = r652598 / r652591;
        double r652600 = 1.0;
        double r652601 = r652600 / r652591;
        double r652602 = r652601 - r652600;
        double r652603 = fma(r652599, r652602, r652598);
        double r652604 = r652591 * r652591;
        double r652605 = r652600 * r652600;
        double r652606 = r652604 - r652605;
        double r652607 = r652591 / r652606;
        double r652608 = r652591 - r652600;
        double r652609 = r652607 * r652608;
        double r652610 = r652598 - r652600;
        double r652611 = fma(r652609, r652610, r652600);
        double r652612 = r652597 ? r652603 : r652611;
        return r652612;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.5
Target0.2
Herbie7.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.076632712489761e+33 or 5.359213104125442e+25 < y

    1. Initial program 47.2

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

    2. Simplified29.7

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

    3. Taylor expanded around inf 13.7

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{x}{y}}\]

    4. Simplified13.7

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

    if -2.076632712489761e+33 < y < 5.359213104125442e+25

    1. Initial program 1.8

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

    2. Simplified1.7

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

    4. Applied flip-+1.7

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

    5. Applied associate-/r/1.7

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.0766327124897609 \cdot 10^{33} \lor \neg \left(y \le 5.3592131041254416 \cdot 10^{25}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))))