Average Error: 20.3 → 5.7
Time: 2.5s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.6952782580207462 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.5683661590480565 \cdot 10^{-150}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 3.8646556168450438 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.6952782580207462 \cdot 10^{153}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.5683661590480565 \cdot 10^{-150}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 3.8646556168450438 \cdot 10^{-160}:\\
\;\;\;\;1\\

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

\end{array}
double f(double x, double y) {
        double r64609 = x;
        double r64610 = y;
        double r64611 = r64609 - r64610;
        double r64612 = r64609 + r64610;
        double r64613 = r64611 * r64612;
        double r64614 = r64609 * r64609;
        double r64615 = r64610 * r64610;
        double r64616 = r64614 + r64615;
        double r64617 = r64613 / r64616;
        return r64617;
}
↓
double f(double x, double y) {
        double r64618 = y;
        double r64619 = -1.695278258020746e+153;
        bool r64620 = r64618 <= r64619;
        double r64621 = -1.0;
        double r64622 = -1.5683661590480565e-150;
        bool r64623 = r64618 <= r64622;
        double r64624 = x;
        double r64625 = r64624 - r64618;
        double r64626 = r64624 + r64618;
        double r64627 = r64625 * r64626;
        double r64628 = r64624 * r64624;
        double r64629 = r64618 * r64618;
        double r64630 = r64628 + r64629;
        double r64631 = r64627 / r64630;
        double r64632 = 3.864655616845044e-160;
        bool r64633 = r64618 <= r64632;
        double r64634 = 1.0;
        double r64635 = r64633 ? r64634 : r64631;
        double r64636 = r64623 ? r64631 : r64635;
        double r64637 = r64620 ? r64621 : r64636;
        return r64637;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.3
Target0.0
Herbie5.7
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.695278258020746e+153

    1. Initial program 63.5

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

    2. Taylor expanded around 0 0

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

    if -1.695278258020746e+153 < y < -1.5683661590480565e-150 or 3.864655616845044e-160 < y

    1. Initial program 0.0

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

    if -1.5683661590480565e-150 < y < 3.864655616845044e-160

    1. Initial program 29.1

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

    2. Taylor expanded around inf 16.8

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.6952782580207462 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.5683661590480565 \cdot 10^{-150}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 3.8646556168450438 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 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))))