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 r95130 = x;
        double r95131 = y;
        double r95132 = r95130 - r95131;
        double r95133 = r95130 + r95131;
        double r95134 = r95132 * r95133;
        double r95135 = r95130 * r95130;
        double r95136 = r95131 * r95131;
        double r95137 = r95135 + r95136;
        double r95138 = r95134 / r95137;
        return r95138;
}
↓
double f(double x, double y) {
        double r95139 = y;
        double r95140 = -1.3404988451868662e+154;
        bool r95141 = r95139 <= r95140;
        double r95142 = -1.0;
        double r95143 = -5.692859475155119e-160;
        bool r95144 = r95139 <= r95143;
        double r95145 = x;
        double r95146 = r95145 - r95139;
        double r95147 = r95145 + r95139;
        double r95148 = r95146 * r95147;
        double r95149 = r95145 * r95145;
        double r95150 = r95139 * r95139;
        double r95151 = r95149 + r95150;
        double r95152 = r95148 / r95151;
        double r95153 = 1.2490407105203886e-166;
        bool r95154 = r95139 <= r95153;
        double r95155 = 1.0;
        double r95156 = r95154 ? r95155 : r95152;
        double r95157 = r95144 ? r95152 : r95156;
        double r95158 = r95141 ? r95142 : r95157;
        return r95158;
}

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 
(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))))