Average Error: 20.1 → 5.4
Time: 11.0s
Precision: 64
\[0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.347493887093548 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.721956094789157 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y}{\sqrt{x \cdot x + y \cdot y}}\\ \mathbf{elif}\;y \le 3.455947184275326 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.347493887093548 \cdot 10^{+154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -2.721956094789157 \cdot 10^{-159}:\\
\;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y}{\sqrt{x \cdot x + y \cdot y}}\\

\mathbf{elif}\;y \le 3.455947184275326 \cdot 10^{-157}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y}{\sqrt{x \cdot x + y \cdot y}}\\

\end{array}
double f(double x, double y) {
        double r2179682 = x;
        double r2179683 = y;
        double r2179684 = r2179682 - r2179683;
        double r2179685 = r2179682 + r2179683;
        double r2179686 = r2179684 * r2179685;
        double r2179687 = r2179682 * r2179682;
        double r2179688 = r2179683 * r2179683;
        double r2179689 = r2179687 + r2179688;
        double r2179690 = r2179686 / r2179689;
        return r2179690;
}
↓
double f(double x, double y) {
        double r2179691 = y;
        double r2179692 = -1.347493887093548e+154;
        bool r2179693 = r2179691 <= r2179692;
        double r2179694 = -1.0;
        double r2179695 = -2.721956094789157e-159;
        bool r2179696 = r2179691 <= r2179695;
        double r2179697 = x;
        double r2179698 = r2179697 * r2179697;
        double r2179699 = r2179691 * r2179691;
        double r2179700 = r2179698 + r2179699;
        double r2179701 = r2179698 / r2179700;
        double r2179702 = sqrt(r2179700);
        double r2179703 = r2179691 / r2179702;
        double r2179704 = r2179703 * r2179703;
        double r2179705 = r2179701 - r2179704;
        double r2179706 = 3.455947184275326e-157;
        bool r2179707 = r2179691 <= r2179706;
        double r2179708 = 1.0;
        double r2179709 = r2179707 ? r2179708 : r2179705;
        double r2179710 = r2179696 ? r2179705 : r2179709;
        double r2179711 = r2179693 ? r2179694 : r2179710;
        return r2179711;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.0
Herbie5.4
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.347493887093548e+154

    1. Initial program 63.6

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

    2. Simplified63.6

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

    3. Taylor expanded around 0 0

      \[\leadsto \color{blue}{-1}\]

    if -1.347493887093548e+154 < y < -2.721956094789157e-159 or 3.455947184275326e-157 < y

    1. Initial program 0.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y \cdot y}{x \cdot x + y \cdot y}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.0

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

    5. Applied times-frac0.1

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

    if -2.721956094789157e-159 < y < 3.455947184275326e-157

    1. Initial program 29.2

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

    2. Simplified29.2

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

    3. Taylor expanded around inf 16.3

      \[\leadsto \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.347493887093548 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -2.721956094789157 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y}{\sqrt{x \cdot x + y \cdot y}}\\ \mathbf{elif}\;y \le 3.455947184275326 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

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