Average Error: 22.7 → 0.2
Time: 2.6s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -64063762.177069880068302154541015625 \lor \neg \left(y \le 35775621.8817711770534515380859375\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -64063762.177069880068302154541015625 \lor \neg \left(y \le 35775621.8817711770534515380859375\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

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

\end{array}
double f(double x, double y) {
        double r742178 = 1.0;
        double r742179 = x;
        double r742180 = r742178 - r742179;
        double r742181 = y;
        double r742182 = r742180 * r742181;
        double r742183 = r742181 + r742178;
        double r742184 = r742182 / r742183;
        double r742185 = r742178 - r742184;
        return r742185;
}
↓
double f(double x, double y) {
        double r742186 = y;
        double r742187 = -64063762.17706988;
        bool r742188 = r742186 <= r742187;
        double r742189 = 35775621.88177118;
        bool r742190 = r742186 <= r742189;
        double r742191 = !r742190;
        bool r742192 = r742188 || r742191;
        double r742193 = 1.0;
        double r742194 = 1.0;
        double r742195 = r742194 / r742186;
        double r742196 = x;
        double r742197 = r742196 / r742186;
        double r742198 = r742195 - r742197;
        double r742199 = r742193 * r742198;
        double r742200 = r742199 + r742196;
        double r742201 = r742186 + r742193;
        double r742202 = r742194 / r742201;
        double r742203 = r742193 - r742196;
        double r742204 = r742203 * r742186;
        double r742205 = r742202 * r742204;
        double r742206 = r742193 - r742205;
        double r742207 = r742192 ? r742200 : r742206;
        return r742207;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.7
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 < -64063762.17706988 or 35775621.88177118 < y

    1. Initial program 45.6

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

    if -64063762.17706988 < y < 35775621.88177118

    1. Initial program 0.1

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

    3. Applied associate-/l*0.2

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\]
    4. Using strategy rm

    5. Applied div-inv0.2

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

    6. Applied *-un-lft-identity0.2

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

    7. Applied times-frac0.2

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

    8. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -64063762.177069880068302154541015625 \lor \neg \left(y \le 35775621.8817711770534515380859375\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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))))