Average Error: 22.3 → 7.4
Time: 5.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -13972044976917.6426 \lor \neg \left(y \le 10613960846537648\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot y}{y + 1} + 1\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -13972044976917.6426 \lor \neg \left(y \le 10613960846537648\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r615572 = 1.0;
        double r615573 = x;
        double r615574 = r615572 - r615573;
        double r615575 = y;
        double r615576 = r615574 * r615575;
        double r615577 = r615575 + r615572;
        double r615578 = r615576 / r615577;
        double r615579 = r615572 - r615578;
        return r615579;
}
↓
double f(double x, double y) {
        double r615580 = y;
        double r615581 = -13972044976917.643;
        bool r615582 = r615580 <= r615581;
        double r615583 = 10613960846537648.0;
        bool r615584 = r615580 <= r615583;
        double r615585 = !r615584;
        bool r615586 = r615582 || r615585;
        double r615587 = 1.0;
        double r615588 = x;
        double r615589 = 2.0;
        double r615590 = pow(r615580, r615589);
        double r615591 = r615588 / r615590;
        double r615592 = r615588 / r615580;
        double r615593 = r615591 - r615592;
        double r615594 = fma(r615587, r615593, r615588);
        double r615595 = r615588 - r615587;
        double r615596 = r615595 * r615580;
        double r615597 = r615580 + r615587;
        double r615598 = r615596 / r615597;
        double r615599 = r615598 + r615587;
        double r615600 = r615586 ? r615594 : r615599;
        return r615600;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.3
Target0.2
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -13972044976917.643 or 10613960846537648.0 < y

    1. Initial program 46.7

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

    2. Simplified29.5

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

    4. Applied add-cube-cbrt30.3

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{\left(\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}\right) \cdot \sqrt[3]{y + 1}}}, x - 1, 1\right)\]

    5. Applied associate-/r*30.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{y}{\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}}}{\sqrt[3]{y + 1}}}, x - 1, 1\right)\]

    6. Taylor expanded around inf 15.1

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

    7. Simplified15.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)}\]

    if -13972044976917.643 < y < 10613960846537648.0

    1. Initial program 0.5

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

    2. Simplified0.5

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

    4. Applied add-cube-cbrt0.6

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{\left(\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}\right) \cdot \sqrt[3]{y + 1}}}, x - 1, 1\right)\]

    5. Applied associate-/r*0.6

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

    7. Applied fma-udef0.6

      \[\leadsto \color{blue}{\frac{\frac{y}{\sqrt[3]{y + 1} \cdot \sqrt[3]{y + 1}}}{\sqrt[3]{y + 1}} \cdot \left(x - 1\right) + 1}\]

    8. Simplified0.5

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -13972044976917.6426 \lor \neg \left(y \le 10613960846537648\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot y}{y + 1} + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))))