Average Error: 22.2 → 0.2
Time: 13.4s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 0.5438029697763243630603824385616462677717:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 1.00000021180969533851623509690398350358:\\ \;\;\;\;x + \left(\frac{1}{y} - \frac{1}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 0.5438029697763243630603824385616462677717:\\
\;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\

\mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 1.00000021180969533851623509690398350358:\\
\;\;\;\;x + \left(\frac{1}{y} - \frac{1}{\frac{y}{x}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r29722926 = 1.0;
        double r29722927 = x;
        double r29722928 = r29722926 - r29722927;
        double r29722929 = y;
        double r29722930 = r29722928 * r29722929;
        double r29722931 = r29722929 + r29722926;
        double r29722932 = r29722930 / r29722931;
        double r29722933 = r29722926 - r29722932;
        return r29722933;
}
↓
double f(double x, double y) {
        double r29722934 = 1.0;
        double r29722935 = x;
        double r29722936 = r29722934 - r29722935;
        double r29722937 = y;
        double r29722938 = r29722936 * r29722937;
        double r29722939 = r29722937 + r29722934;
        double r29722940 = r29722938 / r29722939;
        double r29722941 = 0.5438029697763244;
        bool r29722942 = r29722940 <= r29722941;
        double r29722943 = r29722937 / r29722939;
        double r29722944 = r29722936 * r29722943;
        double r29722945 = r29722934 - r29722944;
        double r29722946 = 1.0000002118096953;
        bool r29722947 = r29722940 <= r29722946;
        double r29722948 = r29722934 / r29722937;
        double r29722949 = r29722937 / r29722935;
        double r29722950 = r29722934 / r29722949;
        double r29722951 = r29722948 - r29722950;
        double r29722952 = r29722935 + r29722951;
        double r29722953 = r29722947 ? r29722952 : r29722945;
        double r29722954 = r29722942 ? r29722945 : r29722953;
        return r29722954;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.2
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 (/ (* (- 1.0 x) y) (+ y 1.0)) < 0.5438029697763244 or 1.0000002118096953 < (/ (* (- 1.0 x) y) (+ y 1.0))

    1. Initial program 10.5

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

    3. Applied *-un-lft-identity10.5

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

    if 0.5438029697763244 < (/ (* (- 1.0 x) y) (+ y 1.0)) < 1.0000002118096953

    1. Initial program 58.9

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

    3. Applied *-un-lft-identity58.9

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

    4. Applied times-frac58.8

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

    5. Simplified58.8

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

    6. Taylor expanded around inf 0.7

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

    7. Simplified0.7

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 0.5438029697763243630603824385616462677717:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y + 1} \le 1.00000021180969533851623509690398350358:\\ \;\;\;\;x + \left(\frac{1}{y} - \frac{1}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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))))