Average Error: 22.1 → 0.1
Time: 4.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -108879375.674216911 \lor \neg \left(y \le 157030215.622216791\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -108879375.674216911 \lor \neg \left(y \le 157030215.622216791\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r630377 = 1.0;
        double r630378 = x;
        double r630379 = r630377 - r630378;
        double r630380 = y;
        double r630381 = r630379 * r630380;
        double r630382 = r630380 + r630377;
        double r630383 = r630381 / r630382;
        double r630384 = r630377 - r630383;
        return r630384;
}
↓
double f(double x, double y) {
        double r630385 = y;
        double r630386 = -108879375.67421691;
        bool r630387 = r630385 <= r630386;
        double r630388 = 157030215.6222168;
        bool r630389 = r630385 <= r630388;
        double r630390 = !r630389;
        bool r630391 = r630387 || r630390;
        double r630392 = 1.0;
        double r630393 = 1.0;
        double r630394 = r630393 / r630385;
        double r630395 = x;
        double r630396 = r630395 / r630385;
        double r630397 = r630394 - r630396;
        double r630398 = r630392 * r630397;
        double r630399 = r630398 + r630395;
        double r630400 = r630392 - r630395;
        double r630401 = r630385 + r630392;
        double r630402 = r630385 / r630401;
        double r630403 = r630400 * r630402;
        double r630404 = r630392 - r630403;
        double r630405 = r630391 ? r630399 : r630404;
        return r630405;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.1
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -108879375.67421691 or 157030215.6222168 < y

    1. Initial program 45.7

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -108879375.67421691 < y < 157030215.6222168

    1. Initial program 0.1

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

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

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -108879375.674216911 \lor \neg \left(y \le 157030215.622216791\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))))