Average Error: 22.5 → 0.9
Time: 13.2s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -9.186493886184318 \cdot 10^{36} \lor \neg \left(y \le 165453570.87693185\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -9.186493886184318 \cdot 10^{36} \lor \neg \left(y \le 165453570.87693185\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

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

\end{array}
double f(double x, double y) {
        double r451289 = 1.0;
        double r451290 = x;
        double r451291 = r451289 - r451290;
        double r451292 = y;
        double r451293 = r451291 * r451292;
        double r451294 = r451292 + r451289;
        double r451295 = r451293 / r451294;
        double r451296 = r451289 - r451295;
        return r451296;
}
↓
double f(double x, double y) {
        double r451297 = y;
        double r451298 = -9.186493886184318e+36;
        bool r451299 = r451297 <= r451298;
        double r451300 = 165453570.87693185;
        bool r451301 = r451297 <= r451300;
        double r451302 = !r451301;
        bool r451303 = r451299 || r451302;
        double r451304 = 1.0;
        double r451305 = 1.0;
        double r451306 = r451305 / r451297;
        double r451307 = x;
        double r451308 = r451307 / r451297;
        double r451309 = r451306 - r451308;
        double r451310 = fma(r451304, r451309, r451307);
        double r451311 = r451307 - r451304;
        double r451312 = r451297 + r451304;
        double r451313 = r451311 / r451312;
        double r451314 = fma(r451313, r451297, r451304);
        double r451315 = r451303 ? r451310 : r451314;
        return r451315;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.5
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -9.186493886184318e+36 or 165453570.87693185 < y

    1. Initial program 46.6

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

    2. Simplified29.5

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

    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 -9.186493886184318e+36 < y < 165453570.87693185

    1. Initial program 1.7

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

    2. Simplified1.6

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.186493886184318 \cdot 10^{36} \lor \neg \left(y \le 165453570.87693185\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - 1}{y + 1}, y, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))