Average Error: 22.6 → 0.2
Time: 20.2s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -140476787.3433772623538970947265625:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 74766025.9185152947902679443359375:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -140476787.3433772623538970947265625:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\

\mathbf{elif}\;y \le 74766025.9185152947902679443359375:\\
\;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\

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

\end{array}
double f(double x, double y) {
        double r25284307 = 1.0;
        double r25284308 = x;
        double r25284309 = r25284307 - r25284308;
        double r25284310 = y;
        double r25284311 = r25284309 * r25284310;
        double r25284312 = r25284310 + r25284307;
        double r25284313 = r25284311 / r25284312;
        double r25284314 = r25284307 - r25284313;
        return r25284314;
}
↓
double f(double x, double y) {
        double r25284315 = y;
        double r25284316 = -140476787.34337726;
        bool r25284317 = r25284315 <= r25284316;
        double r25284318 = 1.0;
        double r25284319 = 1.0;
        double r25284320 = r25284319 / r25284315;
        double r25284321 = x;
        double r25284322 = r25284321 / r25284315;
        double r25284323 = r25284320 - r25284322;
        double r25284324 = fma(r25284318, r25284323, r25284321);
        double r25284325 = 74766025.9185153;
        bool r25284326 = r25284315 <= r25284325;
        double r25284327 = r25284318 - r25284321;
        double r25284328 = r25284327 * r25284315;
        double r25284329 = r25284318 + r25284315;
        double r25284330 = r25284328 / r25284329;
        double r25284331 = r25284318 - r25284330;
        double r25284332 = r25284326 ? r25284331 : r25284324;
        double r25284333 = r25284317 ? r25284324 : r25284332;
        return r25284333;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.6
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -140476787.34337726 or 74766025.9185153 < y

    1. Initial program 45.5

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

    2. Simplified29.1

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

    if -140476787.34337726 < y < 74766025.9185153

    1. Initial program 0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -140476787.3433772623538970947265625:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \mathbf{elif}\;y \le 74766025.9185152947902679443359375:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{y} - \frac{x}{y}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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))))