Average Error: 22.9 → 0.2
Time: 16.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -147449421.0190339386463165283203125 \lor \neg \left(y \le 261047506.93679630756378173828125\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -147449421.0190339386463165283203125 \lor \neg \left(y \le 261047506.93679630756378173828125\right):\\
\;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\

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

\end{array}
double f(double x, double y) {
        double r496084 = 1.0;
        double r496085 = x;
        double r496086 = r496084 - r496085;
        double r496087 = y;
        double r496088 = r496086 * r496087;
        double r496089 = r496087 + r496084;
        double r496090 = r496088 / r496089;
        double r496091 = r496084 - r496090;
        return r496091;
}
↓
double f(double x, double y) {
        double r496092 = y;
        double r496093 = -147449421.01903394;
        bool r496094 = r496092 <= r496093;
        double r496095 = 261047506.9367963;
        bool r496096 = r496092 <= r496095;
        double r496097 = !r496096;
        bool r496098 = r496094 || r496097;
        double r496099 = x;
        double r496100 = 1.0;
        double r496101 = r496100 / r496092;
        double r496102 = r496099 + r496101;
        double r496103 = r496099 / r496092;
        double r496104 = r496100 * r496103;
        double r496105 = r496102 - r496104;
        double r496106 = r496100 - r496099;
        double r496107 = r496092 + r496100;
        double r496108 = r496092 / r496107;
        double r496109 = r496106 * r496108;
        double r496110 = r496100 - r496109;
        double r496111 = r496098 ? r496105 : r496110;
        return r496111;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.9
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 < -147449421.01903394 or 261047506.9367963 < y

    1. Initial program 46.3

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

    3. Applied *-un-lft-identity46.3

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

    4. Applied times-frac30.0

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

    5. Simplified30.0

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

    6. Taylor expanded around inf 0.1

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

    7. Simplified0.1

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

    if -147449421.01903394 < y < 261047506.9367963

    1. Initial program 0.2

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

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

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -147449421.0190339386463165283203125 \lor \neg \left(y \le 261047506.93679630756378173828125\right):\\ \;\;\;\;\left(x + \frac{1}{y}\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))