Average Error: 20.3 → 5.1
Time: 3.5s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.785532773814616831863731014933445423349 \cdot 10^{140}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.558923983411081976150022496277695268694 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot y}{\left(x - y\right) \cdot \left(x + y\right)}}\\ \mathbf{elif}\;y \le 3.047723682413627022763368644019132647617 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.785532773814616831863731014933445423349 \cdot 10^{140}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.558923983411081976150022496277695268694 \cdot 10^{-162}:\\
\;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot y}{\left(x - y\right) \cdot \left(x + y\right)}}\\

\mathbf{elif}\;y \le 3.047723682413627022763368644019132647617 \cdot 10^{-169}:\\
\;\;\;\;1\\

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

\end{array}
double f(double x, double y) {
        double r98562 = x;
        double r98563 = y;
        double r98564 = r98562 - r98563;
        double r98565 = r98562 + r98563;
        double r98566 = r98564 * r98565;
        double r98567 = r98562 * r98562;
        double r98568 = r98563 * r98563;
        double r98569 = r98567 + r98568;
        double r98570 = r98566 / r98569;
        return r98570;
}
↓
double f(double x, double y) {
        double r98571 = y;
        double r98572 = -1.7855327738146168e+140;
        bool r98573 = r98571 <= r98572;
        double r98574 = -1.0;
        double r98575 = -1.558923983411082e-162;
        bool r98576 = r98571 <= r98575;
        double r98577 = 1.0;
        double r98578 = x;
        double r98579 = r98578 * r98578;
        double r98580 = r98571 * r98571;
        double r98581 = r98579 + r98580;
        double r98582 = r98578 - r98571;
        double r98583 = r98578 + r98571;
        double r98584 = r98582 * r98583;
        double r98585 = r98581 / r98584;
        double r98586 = r98577 / r98585;
        double r98587 = 3.047723682413627e-169;
        bool r98588 = r98571 <= r98587;
        double r98589 = sqrt(r98581);
        double r98590 = r98582 / r98589;
        double r98591 = r98583 / r98589;
        double r98592 = r98590 * r98591;
        double r98593 = r98588 ? r98577 : r98592;
        double r98594 = r98576 ? r98586 : r98593;
        double r98595 = r98573 ? r98574 : r98594;
        return r98595;
}

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

Derivation

  1. Split input into 4 regimes

  2. if y < -1.7855327738146168e+140

    1. Initial program 58.1

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

    2. Taylor expanded around 0 0

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

    if -1.7855327738146168e+140 < y < -1.558923983411082e-162

    1. Initial program 0.0

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

    3. Applied clear-num0.0

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

    if -1.558923983411082e-162 < y < 3.047723682413627e-169

    1. Initial program 30.6

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

    3. Applied clear-num30.6

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

    4. Taylor expanded around inf 15.9

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

    if 3.047723682413627e-169 < y

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.1

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

    4. Applied times-frac1.6

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.785532773814616831863731014933445423349 \cdot 10^{140}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.558923983411081976150022496277695268694 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot y}{\left(x - y\right) \cdot \left(x + y\right)}}\\ \mathbf{elif}\;y \le 3.047723682413627022763368644019132647617 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))