Average Error: 22.5 → 0.2
Time: 4.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -33944091157892768 \lor \neg \left(y \le 139832134.2860770523548126220703125\right):\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -33944091157892768 \lor \neg \left(y \le 139832134.2860770523548126220703125\right):\\
\;\;\;\;1 \cdot \left(1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\right)\\

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

\end{array}
double f(double x, double y) {
        double r804892 = 1.0;
        double r804893 = x;
        double r804894 = r804892 - r804893;
        double r804895 = y;
        double r804896 = r804894 * r804895;
        double r804897 = r804895 + r804892;
        double r804898 = r804896 / r804897;
        double r804899 = r804892 - r804898;
        return r804899;
}
↓
double f(double x, double y) {
        double r804900 = y;
        double r804901 = -3.394409115789277e+16;
        bool r804902 = r804900 <= r804901;
        double r804903 = 139832134.28607705;
        bool r804904 = r804900 <= r804903;
        double r804905 = !r804904;
        bool r804906 = r804902 || r804905;
        double r804907 = 1.0;
        double r804908 = 1.0;
        double r804909 = r804907 / r804900;
        double r804910 = x;
        double r804911 = r804910 / r804900;
        double r804912 = r804909 - r804911;
        double r804913 = r804908 * r804912;
        double r804914 = r804913 + r804910;
        double r804915 = r804907 * r804914;
        double r804916 = r804908 - r804910;
        double r804917 = r804916 * r804900;
        double r804918 = r804900 + r804908;
        double r804919 = r804917 / r804918;
        double r804920 = r804908 - r804919;
        double r804921 = r804907 * r804920;
        double r804922 = r804906 ? r804915 : r804921;
        return r804922;
}

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 < -3.394409115789277e+16 or 139832134.28607705 < y

    1. Initial program 46.5

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

    3. Applied flip--50.0

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

    5. Applied flip-+51.3

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

    6. Applied associate-/r/51.3

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

    7. Simplified46.5

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

    8. Taylor expanded around inf 0.1

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

    9. Simplified0.1

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

    if -3.394409115789277e+16 < y < 139832134.28607705

    1. Initial program 0.3

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

    3. Applied flip--5.1

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

    5. Applied flip-+5.1

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

    6. Applied associate-/r/5.1

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

    7. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -33944091157892768 \lor \neg \left(y \le 139832134.2860770523548126220703125\right):\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(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))))