Average Error: 22.5 → 0.2
Time: 16.8s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -17413228841055.01171875:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x \cdot 1}{y}\\ \mathbf{elif}\;y \le 256213875.7381432950496673583984375:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x \cdot 1}{y}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -17413228841055.01171875:\\
\;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x \cdot 1}{y}\\

\mathbf{elif}\;y \le 256213875.7381432950496673583984375:\\
\;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\

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

\end{array}
double f(double x, double y) {
        double r33948338 = 1.0;
        double r33948339 = x;
        double r33948340 = r33948338 - r33948339;
        double r33948341 = y;
        double r33948342 = r33948340 * r33948341;
        double r33948343 = r33948341 + r33948338;
        double r33948344 = r33948342 / r33948343;
        double r33948345 = r33948338 - r33948344;
        return r33948345;
}
↓
double f(double x, double y) {
        double r33948346 = y;
        double r33948347 = -17413228841055.012;
        bool r33948348 = r33948346 <= r33948347;
        double r33948349 = x;
        double r33948350 = 1.0;
        double r33948351 = r33948350 / r33948346;
        double r33948352 = r33948349 + r33948351;
        double r33948353 = r33948349 * r33948350;
        double r33948354 = r33948353 / r33948346;
        double r33948355 = r33948352 - r33948354;
        double r33948356 = 256213875.7381433;
        bool r33948357 = r33948346 <= r33948356;
        double r33948358 = r33948350 - r33948349;
        double r33948359 = r33948346 + r33948350;
        double r33948360 = r33948346 / r33948359;
        double r33948361 = r33948358 * r33948360;
        double r33948362 = r33948350 - r33948361;
        double r33948363 = r33948357 ? r33948362 : r33948355;
        double r33948364 = r33948348 ? r33948355 : r33948363;
        return r33948364;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.5
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 < -17413228841055.012 or 256213875.7381433 < y

    1. Initial program 46.6

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

    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

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

    if -17413228841055.012 < y < 256213875.7381433

    1. Initial program 0.3

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

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

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -17413228841055.01171875:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x \cdot 1}{y}\\ \mathbf{elif}\;y \le 256213875.7381432950496673583984375:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x \cdot 1}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))