Average Error: 20.1 → 4.9
Time: 17.9s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.797856225877881060769082412965027708037 \cdot 10^{150}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.22186147190904949709756605256725936551 \cdot 10^{-158}:\\ \;\;\;\;\frac{x \cdot \left(x - y\right) + y \cdot \left(x - y\right)}{\mathsf{fma}\left(x, x, y \cdot y\right)}\\ \mathbf{elif}\;y \le 7.961862811311691246218405838467989119993 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - y\right) + y \cdot \left(x - y\right)}{\mathsf{fma}\left(x, x, y \cdot y\right)}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.797856225877881060769082412965027708037 \cdot 10^{150}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.22186147190904949709756605256725936551 \cdot 10^{-158}:\\
\;\;\;\;\frac{x \cdot \left(x - y\right) + y \cdot \left(x - y\right)}{\mathsf{fma}\left(x, x, y \cdot y\right)}\\

\mathbf{elif}\;y \le 7.961862811311691246218405838467989119993 \cdot 10^{-164}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot \left(x - y\right) + y \cdot \left(x - y\right)}{\mathsf{fma}\left(x, x, y \cdot y\right)}\\

\end{array}
double f(double x, double y) {
        double r6927851 = x;
        double r6927852 = y;
        double r6927853 = r6927851 - r6927852;
        double r6927854 = r6927851 + r6927852;
        double r6927855 = r6927853 * r6927854;
        double r6927856 = r6927851 * r6927851;
        double r6927857 = r6927852 * r6927852;
        double r6927858 = r6927856 + r6927857;
        double r6927859 = r6927855 / r6927858;
        return r6927859;
}
↓
double f(double x, double y) {
        double r6927860 = y;
        double r6927861 = -5.797856225877881e+150;
        bool r6927862 = r6927860 <= r6927861;
        double r6927863 = -1.0;
        double r6927864 = -1.2218614719090495e-158;
        bool r6927865 = r6927860 <= r6927864;
        double r6927866 = x;
        double r6927867 = r6927866 - r6927860;
        double r6927868 = r6927866 * r6927867;
        double r6927869 = r6927860 * r6927867;
        double r6927870 = r6927868 + r6927869;
        double r6927871 = r6927860 * r6927860;
        double r6927872 = fma(r6927866, r6927866, r6927871);
        double r6927873 = r6927870 / r6927872;
        double r6927874 = 7.961862811311691e-164;
        bool r6927875 = r6927860 <= r6927874;
        double r6927876 = 1.0;
        double r6927877 = r6927875 ? r6927876 : r6927873;
        double r6927878 = r6927865 ? r6927873 : r6927877;
        double r6927879 = r6927862 ? r6927863 : r6927878;
        return r6927879;
}

Error

Bits error versus x

Bits error versus y

Target

Original20.1
Target0.0
Herbie4.9
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -5.797856225877881e+150

    1. Initial program 62.3

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

    2. Simplified62.3

      \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot \left(x + y\right)}{\mathsf{fma}\left(x, x, y \cdot y\right)}}\]

    3. Taylor expanded around 0 0

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

    if -5.797856225877881e+150 < y < -1.2218614719090495e-158 or 7.961862811311691e-164 < y

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot \left(x + y\right)}{\mathsf{fma}\left(x, x, y \cdot y\right)}}\]
    3. Using strategy rm

    4. Applied distribute-rgt-in0.1

      \[\leadsto \frac{\color{blue}{x \cdot \left(x - y\right) + y \cdot \left(x - y\right)}}{\mathsf{fma}\left(x, x, y \cdot y\right)}\]

    if -1.2218614719090495e-158 < y < 7.961862811311691e-164

    1. Initial program 29.2

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

    2. Simplified29.2

      \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot \left(x + y\right)}{\mathsf{fma}\left(x, x, y \cdot y\right)}}\]

    3. Taylor expanded around inf 15.0

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

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.797856225877881060769082412965027708037 \cdot 10^{150}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.22186147190904949709756605256725936551 \cdot 10^{-158}:\\ \;\;\;\;\frac{x \cdot \left(x - y\right) + y \cdot \left(x - y\right)}{\mathsf{fma}\left(x, x, y \cdot y\right)}\\ \mathbf{elif}\;y \le 7.961862811311691246218405838467989119993 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - y\right) + y \cdot \left(x - y\right)}{\mathsf{fma}\left(x, x, y \cdot y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 0.0 x 1.0) (< y 1.0))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2.0) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))