Average Error: 22.8 → 0.1
Time: 14.7s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2868269604.961124897003173828125 \lor \neg \left(y \le 181304849.2823911607265472412109375\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(1 - x\right), \frac{y}{y + 1}, {\left(\sqrt[3]{1}\right)}^{3}\right) + \frac{y}{y + 1} \cdot \left(\left(-\left(1 - x\right)\right) + \left(1 - x\right)\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2868269604.961124897003173828125 \lor \neg \left(y \le 181304849.2823911607265472412109375\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r452351 = 1.0;
        double r452352 = x;
        double r452353 = r452351 - r452352;
        double r452354 = y;
        double r452355 = r452353 * r452354;
        double r452356 = r452354 + r452351;
        double r452357 = r452355 / r452356;
        double r452358 = r452351 - r452357;
        return r452358;
}
↓
double f(double x, double y) {
        double r452359 = y;
        double r452360 = -2868269604.961125;
        bool r452361 = r452359 <= r452360;
        double r452362 = 181304849.28239116;
        bool r452363 = r452359 <= r452362;
        double r452364 = !r452363;
        bool r452365 = r452361 || r452364;
        double r452366 = 1.0;
        double r452367 = 1.0;
        double r452368 = r452367 / r452359;
        double r452369 = x;
        double r452370 = r452369 / r452359;
        double r452371 = r452368 - r452370;
        double r452372 = fma(r452366, r452371, r452369);
        double r452373 = r452366 - r452369;
        double r452374 = -r452373;
        double r452375 = r452359 + r452366;
        double r452376 = r452359 / r452375;
        double r452377 = cbrt(r452366);
        double r452378 = 3.0;
        double r452379 = pow(r452377, r452378);
        double r452380 = fma(r452374, r452376, r452379);
        double r452381 = r452374 + r452373;
        double r452382 = r452376 * r452381;
        double r452383 = r452380 + r452382;
        double r452384 = r452365 ? r452372 : r452383;
        return r452384;
}

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Split input into 2 regimes

  2. if y < -2868269604.961125 or 181304849.28239116 < y

    1. Initial program 46.9

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

    2. Simplified30.4

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

    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

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

    if -2868269604.961125 < y < 181304849.28239116

    1. Initial program 0.2

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

    3. Applied *-un-lft-identity0.2

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

    4. Applied times-frac0.2

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{1} \cdot \frac{y}{y + 1}}\]

    5. Applied add-cube-cbrt0.2

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

    6. Applied prod-diff0.2

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

    7. Simplified0.2

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

    8. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2868269604.961124897003173828125 \lor \neg \left(y \le 181304849.2823911607265472412109375\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(1 - x\right), \frac{y}{y + 1}, {\left(\sqrt[3]{1}\right)}^{3}\right) + \frac{y}{y + 1} \cdot \left(\left(-\left(1 - x\right)\right) + \left(1 - x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.84827882972468) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891003) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))