Average Error: 22.5 → 7.3
Time: 3.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.3585420879223014 \cdot 10^{28} \lor \neg \left(y \le 111702921412086.141\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot y}{y + 1} + 1\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.3585420879223014 \cdot 10^{28} \lor \neg \left(y \le 111702921412086.141\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r704272 = 1.0;
        double r704273 = x;
        double r704274 = r704272 - r704273;
        double r704275 = y;
        double r704276 = r704274 * r704275;
        double r704277 = r704275 + r704272;
        double r704278 = r704276 / r704277;
        double r704279 = r704272 - r704278;
        return r704279;
}
↓
double f(double x, double y) {
        double r704280 = y;
        double r704281 = -1.3585420879223014e+28;
        bool r704282 = r704280 <= r704281;
        double r704283 = 111702921412086.14;
        bool r704284 = r704280 <= r704283;
        double r704285 = !r704284;
        bool r704286 = r704282 || r704285;
        double r704287 = x;
        double r704288 = r704287 / r704280;
        double r704289 = 1.0;
        double r704290 = r704289 / r704280;
        double r704291 = r704290 - r704289;
        double r704292 = fma(r704288, r704291, r704287);
        double r704293 = r704287 - r704289;
        double r704294 = r704293 * r704280;
        double r704295 = r704280 + r704289;
        double r704296 = r704294 / r704295;
        double r704297 = r704296 + r704289;
        double r704298 = r704286 ? r704292 : r704297;
        return r704298;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.5
Target0.2
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.3585420879223014e+28 or 111702921412086.14 < y

    1. Initial program 47.1

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

    2. Simplified29.8

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

    3. Taylor expanded around inf 14.6

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

    4. Simplified14.6

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

    if -1.3585420879223014e+28 < y < 111702921412086.14

    1. Initial program 1.0

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

    2. Simplified0.9

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

    4. Applied add-cube-cbrt1.0

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

    5. Applied associate-/r*1.0

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

    7. Applied fma-udef1.0

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

    8. Simplified1.0

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3585420879223014 \cdot 10^{28} \lor \neg \left(y \le 111702921412086.141\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y} - 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot y}{y + 1} + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))))