Average Error: 22.5 → 0.3
Time: 13.5s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.020049338560968577027665560308378189802 \lor \neg \left(y \le 70729809.48992495238780975341796875\right):\\ \;\;\;\;\left(x - \frac{1 \cdot x}{y}\right) + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{\frac{y}{\sqrt{1 + y}}}{\sqrt{1 + y}}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.020049338560968577027665560308378189802 \lor \neg \left(y \le 70729809.48992495238780975341796875\right):\\
\;\;\;\;\left(x - \frac{1 \cdot x}{y}\right) + \frac{1}{y}\\

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

\end{array}
double f(double x, double y) {
        double r560821 = 1.0;
        double r560822 = x;
        double r560823 = r560821 - r560822;
        double r560824 = y;
        double r560825 = r560823 * r560824;
        double r560826 = r560824 + r560821;
        double r560827 = r560825 / r560826;
        double r560828 = r560821 - r560827;
        return r560828;
}
↓
double f(double x, double y) {
        double r560829 = y;
        double r560830 = -1.0200493385609686;
        bool r560831 = r560829 <= r560830;
        double r560832 = 70729809.48992495;
        bool r560833 = r560829 <= r560832;
        double r560834 = !r560833;
        bool r560835 = r560831 || r560834;
        double r560836 = x;
        double r560837 = 1.0;
        double r560838 = r560837 * r560836;
        double r560839 = r560838 / r560829;
        double r560840 = r560836 - r560839;
        double r560841 = r560837 / r560829;
        double r560842 = r560840 + r560841;
        double r560843 = r560837 - r560836;
        double r560844 = r560837 + r560829;
        double r560845 = sqrt(r560844);
        double r560846 = r560829 / r560845;
        double r560847 = r560846 / r560845;
        double r560848 = r560843 * r560847;
        double r560849 = r560837 - r560848;
        double r560850 = r560835 ? r560842 : r560849;
        return r560850;
}

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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.0200493385609686 or 70729809.48992495 < y

    1. Initial program 45.7

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

    2. Simplified29.1

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

    4. Applied add-sqr-sqrt46.9

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

    5. Applied associate-/r*46.9

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

    6. Taylor expanded around inf 0.5

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

    7. Simplified0.5

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

    if -1.0200493385609686 < y < 70729809.48992495

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied associate-/r*0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.020049338560968577027665560308378189802 \lor \neg \left(y \le 70729809.48992495238780975341796875\right):\\ \;\;\;\;\left(x - \frac{1 \cdot x}{y}\right) + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{\frac{y}{\sqrt{1 + y}}}{\sqrt{1 + y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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))))