Average Error: 22.9 → 0.2
Time: 2.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -378501471949656.688 \lor \neg \left(y \le 325972206.362031639\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -378501471949656.688 \lor \neg \left(y \le 325972206.362031639\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 r772937 = 1.0;
        double r772938 = x;
        double r772939 = r772937 - r772938;
        double r772940 = y;
        double r772941 = r772939 * r772940;
        double r772942 = r772940 + r772937;
        double r772943 = r772941 / r772942;
        double r772944 = r772937 - r772943;
        return r772944;
}
↓
double f(double x, double y) {
        double r772945 = y;
        double r772946 = -378501471949656.7;
        bool r772947 = r772945 <= r772946;
        double r772948 = 325972206.36203164;
        bool r772949 = r772945 <= r772948;
        double r772950 = !r772949;
        bool r772951 = r772947 || r772950;
        double r772952 = 1.0;
        double r772953 = 1.0;
        double r772954 = r772953 / r772945;
        double r772955 = x;
        double r772956 = r772955 / r772945;
        double r772957 = r772954 - r772956;
        double r772958 = r772952 * r772957;
        double r772959 = r772958 + r772955;
        double r772960 = r772952 - r772955;
        double r772961 = r772945 + r772952;
        double r772962 = r772945 / r772961;
        double r772963 = r772960 * r772962;
        double r772964 = r772952 - r772963;
        double r772965 = r772951 ? r772959 : r772964;
        return r772965;
}

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
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -378501471949656.7 or 325972206.36203164 < y

    1. Initial program 46.2

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

    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 -378501471949656.7 < y < 325972206.36203164

    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-frac0.3

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

    5. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -378501471949656.688 \lor \neg \left(y \le 325972206.362031639\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))