Average Error: 20.2 → 5.2
Time: 1.4s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.23956771186146647 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.8097602109050456 \cdot 10^{-149}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 4.6575319845407228 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.23956771186146647 \cdot 10^{151}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -4.8097602109050456 \cdot 10^{-149}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 4.6575319845407228 \cdot 10^{-169}:\\
\;\;\;\;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 r86511 = x;
        double r86512 = y;
        double r86513 = r86511 - r86512;
        double r86514 = r86511 + r86512;
        double r86515 = r86513 * r86514;
        double r86516 = r86511 * r86511;
        double r86517 = r86512 * r86512;
        double r86518 = r86516 + r86517;
        double r86519 = r86515 / r86518;
        return r86519;
}
↓
double f(double x, double y) {
        double r86520 = y;
        double r86521 = -1.2395677118614665e+151;
        bool r86522 = r86520 <= r86521;
        double r86523 = -1.0;
        double r86524 = -4.809760210905046e-149;
        bool r86525 = r86520 <= r86524;
        double r86526 = x;
        double r86527 = r86526 - r86520;
        double r86528 = r86526 + r86520;
        double r86529 = r86527 * r86528;
        double r86530 = r86526 * r86526;
        double r86531 = r86520 * r86520;
        double r86532 = r86530 + r86531;
        double r86533 = r86529 / r86532;
        double r86534 = 4.657531984540723e-169;
        bool r86535 = r86520 <= r86534;
        double r86536 = 1.0;
        double r86537 = r86535 ? r86536 : r86533;
        double r86538 = r86525 ? r86533 : r86537;
        double r86539 = r86522 ? r86523 : r86538;
        return r86539;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if y < -1.2395677118614665e+151

    1. Initial program 63.1

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

    2. Taylor expanded around 0 0

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

    if -1.2395677118614665e+151 < y < -4.809760210905046e-149 or 4.657531984540723e-169 < y

    1. Initial program 0.3

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

    if -4.809760210905046e-149 < y < 4.657531984540723e-169

    1. Initial program 29.8

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

    2. Taylor expanded around inf 16.2

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.23956771186146647 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.8097602109050456 \cdot 10^{-149}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 4.6575319845407228 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))))