Average Error: 19.6 → 5.3
Time: 18.6s
Precision: 64
\[0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.364714454837564 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -3.2241936526231514 \cdot 10^{-170}:\\ \;\;\;\;\sqrt[3]{\left(\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}} \cdot \frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}}\\ \mathbf{elif}\;y \le -1.0663270652980943 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 6.518473908454177 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.364714454837564 \cdot 10^{+154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -3.2241936526231514 \cdot 10^{-170}:\\
\;\;\;\;\sqrt[3]{\left(\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}} \cdot \frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}}\\

\mathbf{elif}\;y \le -1.0663270652980943 \cdot 10^{-198}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le 6.518473908454177 \cdot 10^{-162}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}\\

\end{array}
double f(double x, double y) {
        double r2162631 = x;
        double r2162632 = y;
        double r2162633 = r2162631 - r2162632;
        double r2162634 = r2162631 + r2162632;
        double r2162635 = r2162633 * r2162634;
        double r2162636 = r2162631 * r2162631;
        double r2162637 = r2162632 * r2162632;
        double r2162638 = r2162636 + r2162637;
        double r2162639 = r2162635 / r2162638;
        return r2162639;
}
↓
double f(double x, double y) {
        double r2162640 = y;
        double r2162641 = -1.364714454837564e+154;
        bool r2162642 = r2162640 <= r2162641;
        double r2162643 = -1.0;
        double r2162644 = -3.2241936526231514e-170;
        bool r2162645 = r2162640 <= r2162644;
        double r2162646 = 1.0;
        double r2162647 = x;
        double r2162648 = r2162647 * r2162647;
        double r2162649 = fma(r2162640, r2162640, r2162648);
        double r2162650 = r2162647 - r2162640;
        double r2162651 = r2162640 + r2162647;
        double r2162652 = r2162650 * r2162651;
        double r2162653 = r2162649 / r2162652;
        double r2162654 = r2162646 / r2162653;
        double r2162655 = r2162654 * r2162654;
        double r2162656 = r2162655 * r2162654;
        double r2162657 = cbrt(r2162656);
        double r2162658 = -1.0663270652980943e-198;
        bool r2162659 = r2162640 <= r2162658;
        double r2162660 = 6.518473908454177e-162;
        bool r2162661 = r2162640 <= r2162660;
        double r2162662 = r2162661 ? r2162646 : r2162654;
        double r2162663 = r2162659 ? r2162643 : r2162662;
        double r2162664 = r2162645 ? r2162657 : r2162663;
        double r2162665 = r2162642 ? r2162643 : r2162664;
        return r2162665;
}

Error

Bits error versus x

Bits error versus y

Target

Original19.6
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 4 regimes

  2. if y < -1.364714454837564e+154 or -3.2241936526231514e-170 < y < -1.0663270652980943e-198

    1. Initial program 58.3

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

    2. Simplified58.3

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

    4. Applied clear-num58.3

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube58.3

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}} \cdot \frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}}}\]

    7. Taylor expanded around inf 6.2

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

    if -1.364714454837564e+154 < y < -3.2241936526231514e-170

    1. Initial program 0.6

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

    2. Simplified0.6

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

    4. Applied clear-num0.6

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube0.6

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}} \cdot \frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}}}\]

    if -1.0663270652980943e-198 < y < 6.518473908454177e-162

    1. Initial program 28.6

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

    2. Simplified28.6

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

    3. Taylor expanded around inf 13.5

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

    if 6.518473908454177e-162 < y

    1. Initial program 0.0

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

    2. Simplified0.0

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

    4. Applied clear-num0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.364714454837564 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -3.2241936526231514 \cdot 10^{-170}:\\ \;\;\;\;\sqrt[3]{\left(\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}} \cdot \frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}}\\ \mathbf{elif}\;y \le -1.0663270652980943 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 6.518473908454177 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(y, y, x \cdot x\right)}{\left(x - y\right) \cdot \left(y + x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 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))))