Average Error: 20.2 → 5.2
Time: 6.1s
Precision: 64
\[0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.3483719131786158 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -6.3662280950353905 \cdot 10^{-155}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y \cdot y}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 3.1003292371794966 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y \cdot y}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.3483719131786158 \cdot 10^{+154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -6.3662280950353905 \cdot 10^{-155}:\\
\;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y \cdot y}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 3.1003292371794966 \cdot 10^{-168}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y \cdot y}{x \cdot x + y \cdot y}\\

\end{array}
double f(double x, double y) {
        double r1452138 = x;
        double r1452139 = y;
        double r1452140 = r1452138 - r1452139;
        double r1452141 = r1452138 + r1452139;
        double r1452142 = r1452140 * r1452141;
        double r1452143 = r1452138 * r1452138;
        double r1452144 = r1452139 * r1452139;
        double r1452145 = r1452143 + r1452144;
        double r1452146 = r1452142 / r1452145;
        return r1452146;
}
↓
double f(double x, double y) {
        double r1452147 = y;
        double r1452148 = -1.3483719131786158e+154;
        bool r1452149 = r1452147 <= r1452148;
        double r1452150 = -1.0;
        double r1452151 = -6.3662280950353905e-155;
        bool r1452152 = r1452147 <= r1452151;
        double r1452153 = x;
        double r1452154 = r1452153 * r1452153;
        double r1452155 = r1452147 * r1452147;
        double r1452156 = r1452154 + r1452155;
        double r1452157 = r1452154 / r1452156;
        double r1452158 = r1452155 / r1452156;
        double r1452159 = r1452157 - r1452158;
        double r1452160 = 3.1003292371794966e-168;
        bool r1452161 = r1452147 <= r1452160;
        double r1452162 = 1.0;
        double r1452163 = r1452161 ? r1452162 : r1452159;
        double r1452164 = r1452152 ? r1452159 : r1452163;
        double r1452165 = r1452149 ? r1452150 : r1452164;
        return r1452165;
}

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
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 < -1.3483719131786158e+154

    1. Initial program 63.6

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

    2. Simplified63.6

      \[\leadsto \color{blue}{\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y \cdot y}{x \cdot x + y \cdot y}}\]

    3. Taylor expanded around 0 0

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

    if -1.3483719131786158e+154 < y < -6.3662280950353905e-155 or 3.1003292371794966e-168 < y

    1. Initial program 0.3

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

    2. Simplified0.3

      \[\leadsto \color{blue}{\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y \cdot y}{x \cdot x + y \cdot y}}\]

    if -6.3662280950353905e-155 < y < 3.1003292371794966e-168

    1. Initial program 28.9

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

    2. Simplified28.9

      \[\leadsto \color{blue}{\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y \cdot y}{x \cdot x + y \cdot y}}\]

    3. Taylor expanded around inf 16.0

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3483719131786158 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -6.3662280950353905 \cdot 10^{-155}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y \cdot y}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 3.1003292371794966 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot y} - \frac{y \cdot y}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 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))))