Average Error: 21.1 → 0.3
Time: 14.5s
Precision: 64
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.477097239245187 \cdot 10^{+19}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\ \mathbf{elif}\;y \le 140606760.23202688:\\ \;\;\;\;1.0 - \left(y - 1.0\right) \cdot \left(\frac{y}{y + 1.0} \cdot \frac{1.0 - x}{y - 1.0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\ \end{array}\]
1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.477097239245187 \cdot 10^{+19}:\\
\;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\

\mathbf{elif}\;y \le 140606760.23202688:\\
\;\;\;\;1.0 - \left(y - 1.0\right) \cdot \left(\frac{y}{y + 1.0} \cdot \frac{1.0 - x}{y - 1.0}\right)\\

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

\end{array}
double f(double x, double y) {
        double r36032364 = 1.0;
        double r36032365 = x;
        double r36032366 = r36032364 - r36032365;
        double r36032367 = y;
        double r36032368 = r36032366 * r36032367;
        double r36032369 = r36032367 + r36032364;
        double r36032370 = r36032368 / r36032369;
        double r36032371 = r36032364 - r36032370;
        return r36032371;
}
↓
double f(double x, double y) {
        double r36032372 = y;
        double r36032373 = -3.477097239245187e+19;
        bool r36032374 = r36032372 <= r36032373;
        double r36032375 = x;
        double r36032376 = 1.0;
        double r36032377 = r36032376 / r36032372;
        double r36032378 = r36032375 + r36032377;
        double r36032379 = r36032375 / r36032372;
        double r36032380 = r36032379 * r36032376;
        double r36032381 = r36032378 - r36032380;
        double r36032382 = 140606760.23202688;
        bool r36032383 = r36032372 <= r36032382;
        double r36032384 = r36032372 - r36032376;
        double r36032385 = r36032372 + r36032376;
        double r36032386 = r36032372 / r36032385;
        double r36032387 = r36032376 - r36032375;
        double r36032388 = r36032387 / r36032384;
        double r36032389 = r36032386 * r36032388;
        double r36032390 = r36032384 * r36032389;
        double r36032391 = r36032376 - r36032390;
        double r36032392 = r36032383 ? r36032391 : r36032381;
        double r36032393 = r36032374 ? r36032381 : r36032392;
        return r36032393;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.1
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.477097239245187e+19 or 140606760.23202688 < y

    1. Initial program 44.8

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

    3. Applied *-un-lft-identity44.8

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

    4. Applied times-frac29.6

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

    5. Simplified29.6

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

    6. Taylor expanded around inf 0.1

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

    7. Simplified0.1

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

    if -3.477097239245187e+19 < y < 140606760.23202688

    1. Initial program 0.5

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

    3. Applied *-un-lft-identity0.5

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

    4. Applied times-frac0.5

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

    5. Simplified0.5

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

    7. Applied flip-+0.5

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

    8. Applied associate-/r/0.5

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

    9. Applied associate-*r*0.5

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

    10. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.477097239245187 \cdot 10^{+19}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\ \mathbf{elif}\;y \le 140606760.23202688:\\ \;\;\;\;1.0 - \left(y - 1.0\right) \cdot \left(\frac{y}{y + 1.0} \cdot \frac{1.0 - x}{y - 1.0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(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))))