Average Error: 21.8 → 0.1
Time: 8.3s
Precision: 64
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -119130230.14897136:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\ \mathbf{elif}\;y \le 166873327.66733637:\\ \;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\ \end{array}\]
1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -119130230.14897136:\\
\;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\

\mathbf{elif}\;y \le 166873327.66733637:\\
\;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{y + 1.0}\\

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

\end{array}
double f(double x, double y) {
        double r12152269 = 1.0;
        double r12152270 = x;
        double r12152271 = r12152269 - r12152270;
        double r12152272 = y;
        double r12152273 = r12152271 * r12152272;
        double r12152274 = r12152272 + r12152269;
        double r12152275 = r12152273 / r12152274;
        double r12152276 = r12152269 - r12152275;
        return r12152276;
}
↓
double f(double x, double y) {
        double r12152277 = y;
        double r12152278 = -119130230.14897136;
        bool r12152279 = r12152277 <= r12152278;
        double r12152280 = x;
        double r12152281 = 1.0;
        double r12152282 = r12152281 / r12152277;
        double r12152283 = r12152280 + r12152282;
        double r12152284 = r12152277 / r12152281;
        double r12152285 = r12152280 / r12152284;
        double r12152286 = r12152283 - r12152285;
        double r12152287 = 166873327.66733637;
        bool r12152288 = r12152277 <= r12152287;
        double r12152289 = r12152281 - r12152280;
        double r12152290 = r12152277 + r12152281;
        double r12152291 = r12152277 / r12152290;
        double r12152292 = r12152289 * r12152291;
        double r12152293 = r12152281 - r12152292;
        double r12152294 = r12152288 ? r12152293 : r12152286;
        double r12152295 = r12152279 ? r12152286 : r12152294;
        return r12152295;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.8
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -119130230.14897136 or 166873327.66733637 < y

    1. Initial program 45.1

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

    3. Applied *-un-lft-identity45.1

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

    4. Applied times-frac29.8

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

    5. Simplified29.8

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

    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

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

    if -119130230.14897136 < y < 166873327.66733637

    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 simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -119130230.14897136:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\ \mathbf{elif}\;y \le 166873327.66733637:\\ \;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{\frac{y}{1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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))))