Average Error: 20.5 → 5.3
Time: 4.2s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.2350588066957499 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -7.33106158512129338 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 1.3645633303893502 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.2350588066957499 \cdot 10^{153}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -7.33106158512129338 \cdot 10^{-158}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 1.3645633303893502 \cdot 10^{-189}:\\
\;\;\;\;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 r96837 = x;
        double r96838 = y;
        double r96839 = r96837 - r96838;
        double r96840 = r96837 + r96838;
        double r96841 = r96839 * r96840;
        double r96842 = r96837 * r96837;
        double r96843 = r96838 * r96838;
        double r96844 = r96842 + r96843;
        double r96845 = r96841 / r96844;
        return r96845;
}
↓
double f(double x, double y) {
        double r96846 = y;
        double r96847 = -3.23505880669575e+153;
        bool r96848 = r96846 <= r96847;
        double r96849 = -1.0;
        double r96850 = -7.331061585121293e-158;
        bool r96851 = r96846 <= r96850;
        double r96852 = x;
        double r96853 = r96852 - r96846;
        double r96854 = r96852 + r96846;
        double r96855 = r96853 * r96854;
        double r96856 = r96852 * r96852;
        double r96857 = r96846 * r96846;
        double r96858 = r96856 + r96857;
        double r96859 = r96855 / r96858;
        double r96860 = 1.3645633303893502e-189;
        bool r96861 = r96846 <= r96860;
        double r96862 = 1.0;
        double r96863 = r96861 ? r96862 : r96859;
        double r96864 = r96851 ? r96859 : r96863;
        double r96865 = r96848 ? r96849 : r96864;
        return r96865;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.5
Target0.1
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -3.23505880669575e+153

    1. Initial program 63.5

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

    2. Taylor expanded around 0 0

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

    if -3.23505880669575e+153 < y < -7.331061585121293e-158 or 1.3645633303893502e-189 < y

    1. Initial program 1.7

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

    if -7.331061585121293e-158 < y < 1.3645633303893502e-189

    1. Initial program 30.4

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

    2. Taylor expanded around inf 14.9

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.2350588066957499 \cdot 10^{153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -7.33106158512129338 \cdot 10^{-158}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 1.3645633303893502 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))))