Average Error: 20.2 → 5.0
Time: 1.1s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.3404988451868662 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.69285947515511926 \cdot 10^{-160}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 1.2490407105203886 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.3404988451868662 \cdot 10^{154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -5.69285947515511926 \cdot 10^{-160}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 1.2490407105203886 \cdot 10^{-166}:\\
\;\;\;\;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 r83926 = x;
        double r83927 = y;
        double r83928 = r83926 - r83927;
        double r83929 = r83926 + r83927;
        double r83930 = r83928 * r83929;
        double r83931 = r83926 * r83926;
        double r83932 = r83927 * r83927;
        double r83933 = r83931 + r83932;
        double r83934 = r83930 / r83933;
        return r83934;
}
↓
double f(double x, double y) {
        double r83935 = y;
        double r83936 = -1.3404988451868662e+154;
        bool r83937 = r83935 <= r83936;
        double r83938 = -1.0;
        double r83939 = -5.692859475155119e-160;
        bool r83940 = r83935 <= r83939;
        double r83941 = x;
        double r83942 = r83941 - r83935;
        double r83943 = r83941 + r83935;
        double r83944 = r83942 * r83943;
        double r83945 = r83941 * r83941;
        double r83946 = r83935 * r83935;
        double r83947 = r83945 + r83946;
        double r83948 = r83944 / r83947;
        double r83949 = 1.2490407105203886e-166;
        bool r83950 = r83935 <= r83949;
        double r83951 = 1.0;
        double r83952 = r83950 ? r83951 : r83948;
        double r83953 = r83940 ? r83948 : r83952;
        double r83954 = r83937 ? r83938 : r83953;
        return r83954;
}

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

Derivation

  1. Split input into 3 regimes

  2. if y < -1.3404988451868662e+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 -1.3404988451868662e+154 < y < -5.692859475155119e-160 or 1.2490407105203886e-166 < y

    1. Initial program 0.2

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

    if -5.692859475155119e-160 < y < 1.2490407105203886e-166

    1. Initial program 30.1

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

    2. Taylor expanded around inf 15.2

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

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3404988451868662 \cdot 10^{154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.69285947515511926 \cdot 10^{-160}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 1.2490407105203886 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(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))))