Average Error: 20.0 → 5.5
Time: 10.1s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.279730354411318047085995485446303040422 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.503517045334009834542053854668863535885 \cdot 10^{-159} \lor \neg \left(y \le 4.121579147791191678316023927451779242359 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.279730354411318047085995485446303040422 \cdot 10^{153}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.503517045334009834542053854668863535885 \cdot 10^{-159} \lor \neg \left(y \le 4.121579147791191678316023927451779242359 \cdot 10^{-161}\right):\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
double f(double x, double y) {
        double r87485 = x;
        double r87486 = y;
        double r87487 = r87485 - r87486;
        double r87488 = r87485 + r87486;
        double r87489 = r87487 * r87488;
        double r87490 = r87485 * r87485;
        double r87491 = r87486 * r87486;
        double r87492 = r87490 + r87491;
        double r87493 = r87489 / r87492;
        return r87493;
}
↓
double f(double x, double y) {
        double r87494 = y;
        double r87495 = -2.279730354411318e+153;
        bool r87496 = r87494 <= r87495;
        double r87497 = -1.0;
        double r87498 = -1.5035170453340098e-159;
        bool r87499 = r87494 <= r87498;
        double r87500 = 4.121579147791192e-161;
        bool r87501 = r87494 <= r87500;
        double r87502 = !r87501;
        bool r87503 = r87499 || r87502;
        double r87504 = x;
        double r87505 = r87504 - r87494;
        double r87506 = r87504 + r87494;
        double r87507 = r87505 * r87506;
        double r87508 = r87504 * r87504;
        double r87509 = r87494 * r87494;
        double r87510 = r87508 + r87509;
        double r87511 = r87507 / r87510;
        double r87512 = 1.0;
        double r87513 = r87503 ? r87511 : r87512;
        double r87514 = r87496 ? r87497 : r87513;
        return r87514;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.1
Herbie5.5
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.279730354411318e+153

    1. Initial program 63.8

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

    2. Taylor expanded around 0 0

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

    if -2.279730354411318e+153 < y < -1.5035170453340098e-159 or 4.121579147791192e-161 < y

    1. Initial program 0.0

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

    if -1.5035170453340098e-159 < y < 4.121579147791192e-161

    1. Initial program 30.5

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

    2. Taylor expanded around inf 16.9

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.279730354411318047085995485446303040422 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.503517045334009834542053854668863535885 \cdot 10^{-159} \lor \neg \left(y \le 4.121579147791191678316023927451779242359 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))))