Average Error: 22.6 → 0.2
Time: 6.2s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3338147193416.8515625 \lor \neg \left(y \le 176122843.7413960397243499755859375\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3338147193416.8515625 \lor \neg \left(y \le 176122843.7413960397243499755859375\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 r485177 = 1.0;
        double r485178 = x;
        double r485179 = r485177 - r485178;
        double r485180 = y;
        double r485181 = r485179 * r485180;
        double r485182 = r485180 + r485177;
        double r485183 = r485181 / r485182;
        double r485184 = r485177 - r485183;
        return r485184;
}
↓
double f(double x, double y) {
        double r485185 = y;
        double r485186 = -3338147193416.8516;
        bool r485187 = r485185 <= r485186;
        double r485188 = 176122843.74139604;
        bool r485189 = r485185 <= r485188;
        double r485190 = !r485189;
        bool r485191 = r485187 || r485190;
        double r485192 = 1.0;
        double r485193 = 1.0;
        double r485194 = r485193 / r485185;
        double r485195 = x;
        double r485196 = r485195 / r485185;
        double r485197 = r485194 - r485196;
        double r485198 = r485192 * r485197;
        double r485199 = r485198 + r485195;
        double r485200 = r485192 - r485195;
        double r485201 = r485185 + r485192;
        double r485202 = r485185 / r485201;
        double r485203 = r485200 * r485202;
        double r485204 = r485192 - r485203;
        double r485205 = r485191 ? r485199 : r485204;
        return r485205;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.6
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 < -3338147193416.8516 or 176122843.74139604 < y

    1. Initial program 46.0

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

    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 -3338147193416.8516 < y < 176122843.74139604

    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-frac0.2

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

    5. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3338147193416.8515625 \lor \neg \left(y \le 176122843.7413960397243499755859375\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 1978988140 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.84827882972468) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891003) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))