Average Error: 22.0 → 0.2
Time: 17.9s
Precision: 64
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -142333186.8719295:\\ \;\;\;\;\left(x - \frac{1.0 \cdot x}{y}\right) + \frac{1.0}{y}\\ \mathbf{elif}\;y \le 181059310.44765908:\\ \;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{1.0 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{1.0 \cdot x}{y}\right) + \frac{1.0}{y}\\ \end{array}\]
1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -142333186.8719295:\\
\;\;\;\;\left(x - \frac{1.0 \cdot x}{y}\right) + \frac{1.0}{y}\\

\mathbf{elif}\;y \le 181059310.44765908:\\
\;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{1.0 + y}\\

\mathbf{else}:\\
\;\;\;\;\left(x - \frac{1.0 \cdot x}{y}\right) + \frac{1.0}{y}\\

\end{array}
double f(double x, double y) {
        double r32128348 = 1.0;
        double r32128349 = x;
        double r32128350 = r32128348 - r32128349;
        double r32128351 = y;
        double r32128352 = r32128350 * r32128351;
        double r32128353 = r32128351 + r32128348;
        double r32128354 = r32128352 / r32128353;
        double r32128355 = r32128348 - r32128354;
        return r32128355;
}
↓
double f(double x, double y) {
        double r32128356 = y;
        double r32128357 = -142333186.8719295;
        bool r32128358 = r32128356 <= r32128357;
        double r32128359 = x;
        double r32128360 = 1.0;
        double r32128361 = r32128360 * r32128359;
        double r32128362 = r32128361 / r32128356;
        double r32128363 = r32128359 - r32128362;
        double r32128364 = r32128360 / r32128356;
        double r32128365 = r32128363 + r32128364;
        double r32128366 = 181059310.44765908;
        bool r32128367 = r32128356 <= r32128366;
        double r32128368 = r32128360 - r32128359;
        double r32128369 = r32128360 + r32128356;
        double r32128370 = r32128356 / r32128369;
        double r32128371 = r32128368 * r32128370;
        double r32128372 = r32128360 - r32128371;
        double r32128373 = r32128367 ? r32128372 : r32128365;
        double r32128374 = r32128358 ? r32128365 : r32128373;
        return r32128374;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.0
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -142333186.8719295 or 181059310.44765908 < y

    1. Initial program 44.6

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]

    2. Taylor expanded around inf 0.2

      \[\leadsto \color{blue}{\left(x + 1.0 \cdot \frac{1}{y}\right) - 1.0 \cdot \frac{x}{y}}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\frac{1.0}{y} + \left(x - \frac{x \cdot 1.0}{y}\right)}\]

    if -142333186.8719295 < y < 181059310.44765908

    1. Initial program 0.1

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.1

      \[\leadsto 1.0 - \frac{\left(1.0 - x\right) \cdot y}{\color{blue}{1 \cdot \left(y + 1.0\right)}}\]

    4. Applied times-frac0.1

      \[\leadsto 1.0 - \color{blue}{\frac{1.0 - x}{1} \cdot \frac{y}{y + 1.0}}\]

    5. Simplified0.1

      \[\leadsto 1.0 - \color{blue}{\left(1.0 - x\right)} \cdot \frac{y}{y + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -142333186.8719295:\\ \;\;\;\;\left(x - \frac{1.0 \cdot x}{y}\right) + \frac{1.0}{y}\\ \mathbf{elif}\;y \le 181059310.44765908:\\ \;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{1.0 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{1.0 \cdot x}{y}\right) + \frac{1.0}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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))))