Average Error: 21.9 → 7.5
Time: 3.7s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -9.10910013050573315 \cdot 10^{36} \lor \neg \left(y \le 3.0451068833477929 \cdot 10^{29}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{\sqrt[3]{y + 1} \cdot \frac{\sqrt[3]{y \cdot y - 1 \cdot 1}}{\sqrt[3]{y - 1}}}}{\sqrt[3]{y + 1}}, x - 1, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -9.10910013050573315 \cdot 10^{36} \lor \neg \left(y \le 3.0451068833477929 \cdot 10^{29}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{\sqrt[3]{y + 1} \cdot \frac{\sqrt[3]{y \cdot y - 1 \cdot 1}}{\sqrt[3]{y - 1}}}}{\sqrt[3]{y + 1}}, x - 1, 1\right)\\

\end{array}
double f(double x, double y) {
        double r632617 = 1.0;
        double r632618 = x;
        double r632619 = r632617 - r632618;
        double r632620 = y;
        double r632621 = r632619 * r632620;
        double r632622 = r632620 + r632617;
        double r632623 = r632621 / r632622;
        double r632624 = r632617 - r632623;
        return r632624;
}
↓
double f(double x, double y) {
        double r632625 = y;
        double r632626 = -9.109100130505733e+36;
        bool r632627 = r632625 <= r632626;
        double r632628 = 3.045106883347793e+29;
        bool r632629 = r632625 <= r632628;
        double r632630 = !r632629;
        bool r632631 = r632627 || r632630;
        double r632632 = x;
        double r632633 = r632632 / r632625;
        double r632634 = 1.0;
        double r632635 = r632634 / r632625;
        double r632636 = r632635 - r632634;
        double r632637 = fma(r632633, r632636, r632632);
        double r632638 = r632625 + r632634;
        double r632639 = cbrt(r632638);
        double r632640 = r632625 * r632625;
        double r632641 = r632634 * r632634;
        double r632642 = r632640 - r632641;
        double r632643 = cbrt(r632642);
        double r632644 = r632625 - r632634;
        double r632645 = cbrt(r632644);
        double r632646 = r632643 / r632645;
        double r632647 = r632639 * r632646;
        double r632648 = r632625 / r632647;
        double r632649 = r632648 / r632639;
        double r632650 = r632632 - r632634;
        double r632651 = fma(r632649, r632650, r632634);
        double r632652 = r632631 ? r632637 : r632651;
        return r632652;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.9
Target0.2
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -9.109100130505733e+36 or 3.045106883347793e+29 < y

    1. Initial program 46.8

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

    2. Simplified29.4

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

    3. Taylor expanded around inf 14.2

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

    4. Simplified14.2

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

    if -9.109100130505733e+36 < y < 3.045106883347793e+29

    1. Initial program 2.4

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

    2. Simplified2.2

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

    4. Applied add-cube-cbrt2.3

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

    5. Applied associate-/r*2.3

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

    7. Applied flip-+2.3

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

    8. Applied cbrt-div2.3

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.10910013050573315 \cdot 10^{36} \lor \neg \left(y \le 3.0451068833477929 \cdot 10^{29}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{\sqrt[3]{y + 1} \cdot \frac{\sqrt[3]{y \cdot y - 1 \cdot 1}}{\sqrt[3]{y - 1}}}}{\sqrt[3]{y + 1}}, x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))))