Average Error: 22.8 → 0.3
Time: 23.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -61215075097455.1328125 \lor \neg \left(y \le 150550976.723195374011993408203125\right):\\ \;\;\;\;x + \frac{\sqrt{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{y}} \cdot \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y + 1} - \frac{1}{y + 1}, y, 1\right)\\ \end{array}\]
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):\\
\;\;\;\;x + \frac{\sqrt{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{y}} \cdot \left(1 - x\right)\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 r443447 = 1.0;
        double r443448 = x;
        double r443449 = r443447 - r443448;
        double r443450 = y;
        double r443451 = r443449 * r443450;
        double r443452 = r443450 + r443447;
        double r443453 = r443451 / r443452;
        double r443454 = r443447 - r443453;
        return r443454;
}
↓
double f(double x, double y) {
        double r443455 = y;
        double r443456 = -61215075097455.13;
        bool r443457 = r443455 <= r443456;
        double r443458 = 150550976.72319537;
        bool r443459 = r443455 <= r443458;
        double r443460 = !r443459;
        bool r443461 = r443457 || r443460;
        double r443462 = x;
        double r443463 = 1.0;
        double r443464 = sqrt(r443463);
        double r443465 = cbrt(r443455);
        double r443466 = r443465 * r443465;
        double r443467 = r443464 / r443466;
        double r443468 = r443464 / r443465;
        double r443469 = 1.0;
        double r443470 = r443469 - r443462;
        double r443471 = r443468 * r443470;
        double r443472 = r443467 * r443471;
        double r443473 = r443462 + r443472;
        double r443474 = r443455 + r443463;
        double r443475 = r443462 / r443474;
        double r443476 = r443463 / r443474;
        double r443477 = r443475 - r443476;
        double r443478 = fma(r443477, r443455, r443463);
        double r443479 = r443461 ? r443473 : r443478;
        return r443479;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.8
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -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. 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

      \[\leadsto \color{blue}{x + \frac{1}{y} \cdot \left(1 - x\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.3

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

    7. Applied add-sqr-sqrt0.3

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

    8. Applied times-frac0.4

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

    9. Applied associate-*l*0.4

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -61215075097455.1328125 \lor \neg \left(y \le 150550976.723195374011993408203125\right):\\ \;\;\;\;x + \frac{\sqrt{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{y}} \cdot \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y + 1} - \frac{1}{y + 1}, y, 1\right)\\ \end{array}\]

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))))