Average Error: 19.9 → 5.5
Time: 2.7s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.9550884380039752 \cdot 10^{149}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -3.2249699642398948 \cdot 10^{-151}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}\\ \mathbf{elif}\;y \le 1.981429660423225 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.9550884380039752 \cdot 10^{149}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -3.2249699642398948 \cdot 10^{-151}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}\\

\mathbf{elif}\;y \le 1.981429660423225 \cdot 10^{-178}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}\\

\end{array}
double f(double x, double y) {
        double r64760 = x;
        double r64761 = y;
        double r64762 = r64760 - r64761;
        double r64763 = r64760 + r64761;
        double r64764 = r64762 * r64763;
        double r64765 = r64760 * r64760;
        double r64766 = r64761 * r64761;
        double r64767 = r64765 + r64766;
        double r64768 = r64764 / r64767;
        return r64768;
}
↓
double f(double x, double y) {
        double r64769 = y;
        double r64770 = -1.9550884380039752e+149;
        bool r64771 = r64769 <= r64770;
        double r64772 = -1.0;
        double r64773 = -3.224969964239895e-151;
        bool r64774 = r64769 <= r64773;
        double r64775 = x;
        double r64776 = r64775 - r64769;
        double r64777 = r64775 + r64769;
        double r64778 = r64776 * r64777;
        double r64779 = r64775 * r64775;
        double r64780 = r64769 * r64769;
        double r64781 = r64779 + r64780;
        double r64782 = r64778 / r64781;
        double r64783 = 3.0;
        double r64784 = pow(r64782, r64783);
        double r64785 = cbrt(r64784);
        double r64786 = 1.981429660423225e-178;
        bool r64787 = r64769 <= r64786;
        double r64788 = 1.0;
        double r64789 = r64787 ? r64788 : r64785;
        double r64790 = r64774 ? r64785 : r64789;
        double r64791 = r64771 ? r64772 : r64790;
        return r64791;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.0
Herbie5.5
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.9550884380039752e+149

    1. Initial program 62.0

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

    2. Taylor expanded around 0 0

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

    if -1.9550884380039752e+149 < y < -3.224969964239895e-151 or 1.981429660423225e-178 < y

    1. Initial program 1.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube37.4

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

    4. Applied add-cbrt-cube37.6

      \[\leadsto \frac{\left(x - y\right) \cdot \color{blue}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

    5. Applied add-cbrt-cube37.8

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}} \cdot \sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

    6. Applied cbrt-unprod37.4

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

    7. Applied cbrt-undiv37.3

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

    8. Simplified1.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}}\]

    if -3.224969964239895e-151 < y < 1.981429660423225e-178

    1. Initial program 29.2

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

    2. Taylor expanded around inf 16.2

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.9550884380039752 \cdot 10^{149}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -3.2249699642398948 \cdot 10^{-151}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}\\ \mathbf{elif}\;y \le 1.981429660423225 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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))))