Average Error: 20.3 → 4.9
Time: 14.6s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -6.684928343955356131371166176678318827411 \cdot 10^{152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.83212255427073711513933578216523599775 \cdot 10^{-161} \lor \neg \left(y \le 8.828733525662879851097652584775790113009 \cdot 10^{-187}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -6.684928343955356131371166176678318827411 \cdot 10^{152}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -5.83212255427073711513933578216523599775 \cdot 10^{-161} \lor \neg \left(y \le 8.828733525662879851097652584775790113009 \cdot 10^{-187}\right):\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\

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

\end{array}
double f(double x, double y) {
        double r67225 = x;
        double r67226 = y;
        double r67227 = r67225 - r67226;
        double r67228 = r67225 + r67226;
        double r67229 = r67227 * r67228;
        double r67230 = r67225 * r67225;
        double r67231 = r67226 * r67226;
        double r67232 = r67230 + r67231;
        double r67233 = r67229 / r67232;
        return r67233;
}
↓
double f(double x, double y) {
        double r67234 = y;
        double r67235 = -6.684928343955356e+152;
        bool r67236 = r67234 <= r67235;
        double r67237 = -1.0;
        double r67238 = -5.832122554270737e-161;
        bool r67239 = r67234 <= r67238;
        double r67240 = 8.82873352566288e-187;
        bool r67241 = r67234 <= r67240;
        double r67242 = !r67241;
        bool r67243 = r67239 || r67242;
        double r67244 = x;
        double r67245 = r67244 - r67234;
        double r67246 = r67234 + r67244;
        double r67247 = r67245 * r67246;
        double r67248 = r67234 * r67234;
        double r67249 = r67244 * r67244;
        double r67250 = r67248 + r67249;
        double r67251 = r67247 / r67250;
        double r67252 = 1.0;
        double r67253 = r67243 ? r67251 : r67252;
        double r67254 = r67236 ? r67237 : r67253;
        return r67254;
}

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
Herbie4.9
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -6.684928343955356e+152

    1. Initial program 63.4

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

    2. Taylor expanded around 0 0

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

    if -6.684928343955356e+152 < y < -5.832122554270737e-161 or 8.82873352566288e-187 < y

    1. Initial program 1.6

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

    if -5.832122554270737e-161 < y < 8.82873352566288e-187

    1. Initial program 29.3

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

    2. Taylor expanded around inf 13.7

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.684928343955356131371166176678318827411 \cdot 10^{152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.83212255427073711513933578216523599775 \cdot 10^{-161} \lor \neg \left(y \le 8.828733525662879851097652584775790113009 \cdot 10^{-187}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 0.0 x 1.0) (< y 1.0))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2.0) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))