Average Error: 20.2 → 4.9
Time: 13.5s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.0142476237434665 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.6194904278371335 \cdot 10^{-161} \lor \neg \left(y \le 4.425126226680901 \cdot 10^{-167}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.0142476237434665 \cdot 10^{154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.6194904278371335 \cdot 10^{-161} \lor \neg \left(y \le 4.425126226680901 \cdot 10^{-167}\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 r70827 = x;
        double r70828 = y;
        double r70829 = r70827 - r70828;
        double r70830 = r70827 + r70828;
        double r70831 = r70829 * r70830;
        double r70832 = r70827 * r70827;
        double r70833 = r70828 * r70828;
        double r70834 = r70832 + r70833;
        double r70835 = r70831 / r70834;
        return r70835;
}
↓
double f(double x, double y) {
        double r70836 = y;
        double r70837 = -1.0142476237434665e+154;
        bool r70838 = r70836 <= r70837;
        double r70839 = -1.0;
        double r70840 = -1.6194904278371335e-161;
        bool r70841 = r70836 <= r70840;
        double r70842 = 4.425126226680901e-167;
        bool r70843 = r70836 <= r70842;
        double r70844 = !r70843;
        bool r70845 = r70841 || r70844;
        double r70846 = x;
        double r70847 = r70846 - r70836;
        double r70848 = r70846 + r70836;
        double r70849 = r70847 * r70848;
        double r70850 = r70846 * r70846;
        double r70851 = r70836 * r70836;
        double r70852 = r70850 + r70851;
        double r70853 = r70849 / r70852;
        double r70854 = 1.0;
        double r70855 = r70845 ? r70853 : r70854;
        double r70856 = r70838 ? r70839 : r70855;
        return r70856;
}

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
Target0.1
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 < -1.0142476237434665e+154

    1. Initial program 64.0

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

    2. Taylor expanded around 0 0

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

    if -1.0142476237434665e+154 < y < -1.6194904278371335e-161 or 4.425126226680901e-167 < y

    1. Initial program 0.3

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

    if -1.6194904278371335e-161 < y < 4.425126226680901e-167

    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 15.4

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.0142476237434665 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.6194904278371335 \cdot 10^{-161} \lor \neg \left(y \le 4.425126226680901 \cdot 10^{-167}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(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))))