Average Error: 22.0 → 0.2
Time: 18.8s
Precision: 64
\[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -142333186.8719295:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\ \mathbf{elif}\;y \le 181059310.44765908:\\ \;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\ \end{array}\]
1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;y \le -142333186.8719295:\\
\;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\

\mathbf{elif}\;y \le 181059310.44765908:\\
\;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{y + 1.0}\\

\mathbf{else}:\\
\;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\

\end{array}
double f(double x, double y) {
        double r34240944 = 1.0;
        double r34240945 = x;
        double r34240946 = r34240944 - r34240945;
        double r34240947 = y;
        double r34240948 = r34240946 * r34240947;
        double r34240949 = r34240947 + r34240944;
        double r34240950 = r34240948 / r34240949;
        double r34240951 = r34240944 - r34240950;
        return r34240951;
}
↓
double f(double x, double y) {
        double r34240952 = y;
        double r34240953 = -142333186.8719295;
        bool r34240954 = r34240952 <= r34240953;
        double r34240955 = x;
        double r34240956 = 1.0;
        double r34240957 = r34240956 / r34240952;
        double r34240958 = r34240955 + r34240957;
        double r34240959 = r34240955 / r34240952;
        double r34240960 = r34240959 * r34240956;
        double r34240961 = r34240958 - r34240960;
        double r34240962 = 181059310.44765908;
        bool r34240963 = r34240952 <= r34240962;
        double r34240964 = r34240956 - r34240955;
        double r34240965 = r34240952 + r34240956;
        double r34240966 = r34240952 / r34240965;
        double r34240967 = r34240964 * r34240966;
        double r34240968 = r34240956 - r34240967;
        double r34240969 = r34240963 ? r34240968 : r34240961;
        double r34240970 = r34240954 ? r34240961 : r34240969;
        return r34240970;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.0
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.8482788297247:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891:\\ \;\;\;\;1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.0}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -142333186.8719295 or 181059310.44765908 < y

    1. Initial program 44.6

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity44.6

      \[\leadsto 1.0 - \frac{\left(1.0 - x\right) \cdot y}{\color{blue}{1 \cdot \left(y + 1.0\right)}}\]

    4. Applied times-frac29.2

      \[\leadsto 1.0 - \color{blue}{\frac{1.0 - x}{1} \cdot \frac{y}{y + 1.0}}\]

    5. Simplified29.2

      \[\leadsto 1.0 - \color{blue}{\left(1.0 - x\right)} \cdot \frac{y}{y + 1.0}\]

    6. Taylor expanded around inf 0.2

      \[\leadsto \color{blue}{\left(x + 1.0 \cdot \frac{1}{y}\right) - 1.0 \cdot \frac{x}{y}}\]

    7. Simplified0.2

      \[\leadsto \color{blue}{\left(x + \frac{1.0}{y}\right) - 1.0 \cdot \frac{x}{y}}\]

    if -142333186.8719295 < y < 181059310.44765908

    1. Initial program 0.1

      \[1.0 - \frac{\left(1.0 - x\right) \cdot y}{y + 1.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.1

      \[\leadsto 1.0 - \frac{\left(1.0 - x\right) \cdot y}{\color{blue}{1 \cdot \left(y + 1.0\right)}}\]

    4. Applied times-frac0.1

      \[\leadsto 1.0 - \color{blue}{\frac{1.0 - x}{1} \cdot \frac{y}{y + 1.0}}\]

    5. Simplified0.1

      \[\leadsto 1.0 - \color{blue}{\left(1.0 - x\right)} \cdot \frac{y}{y + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -142333186.8719295:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\ \mathbf{elif}\;y \le 181059310.44765908:\\ \;\;\;\;1.0 - \left(1.0 - x\right) \cdot \frac{y}{y + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1.0}{y}\right) - \frac{x}{y} \cdot 1.0\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))