Average Error: 20.3 → 5.7
Time: 1.8s
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 r113274 = x;
        double r113275 = y;
        double r113276 = r113274 - r113275;
        double r113277 = r113274 + r113275;
        double r113278 = r113276 * r113277;
        double r113279 = r113274 * r113274;
        double r113280 = r113275 * r113275;
        double r113281 = r113279 + r113280;
        double r113282 = r113278 / r113281;
        return r113282;
}
↓
double f(double x, double y) {
        double r113283 = y;
        double r113284 = -1.695278258020746e+153;
        bool r113285 = r113283 <= r113284;
        double r113286 = -1.0;
        double r113287 = -1.5683661590480565e-150;
        bool r113288 = r113283 <= r113287;
        double r113289 = x;
        double r113290 = r113289 - r113283;
        double r113291 = r113289 + r113283;
        double r113292 = r113290 * r113291;
        double r113293 = r113289 * r113289;
        double r113294 = r113283 * r113283;
        double r113295 = r113293 + r113294;
        double r113296 = r113292 / r113295;
        double r113297 = 3.864655616845044e-160;
        bool r113298 = r113283 <= r113297;
        double r113299 = 1.0;
        double r113300 = r113298 ? r113299 : r113296;
        double r113301 = r113288 ? r113296 : r113300;
        double r113302 = r113285 ? r113286 : r113301;
        return r113302;
}

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))))