Average Error: 20.4 → 0.0
Time: 12.6s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\frac{\left(x - y\right) \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\frac{\left(x - y\right) \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}
double f(double x, double y) {
        double r93328 = x;
        double r93329 = y;
        double r93330 = r93328 - r93329;
        double r93331 = r93328 + r93329;
        double r93332 = r93330 * r93331;
        double r93333 = r93328 * r93328;
        double r93334 = r93329 * r93329;
        double r93335 = r93333 + r93334;
        double r93336 = r93332 / r93335;
        return r93336;
}
↓
double f(double x, double y) {
        double r93337 = x;
        double r93338 = y;
        double r93339 = r93337 - r93338;
        double r93340 = 1.0;
        double r93341 = hypot(r93337, r93338);
        double r93342 = r93337 + r93338;
        double r93343 = r93341 / r93342;
        double r93344 = r93340 / r93343;
        double r93345 = r93339 * r93344;
        double r93346 = r93345 / r93341;
        return r93346;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.4
Target0.0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Initial program 20.4

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

  3. Applied add-sqr-sqrt20.4

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

  4. Applied times-frac20.4

    \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]

  5. Simplified20.4

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

  6. Simplified0.0

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

  8. Applied associate-*r/0.0

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

  9. Simplified0.0

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

  11. Applied clear-num0.0

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

  12. Final simplification0.0

    \[\leadsto \frac{\left(x - y\right) \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\mathsf{hypot}\left(x, y\right)}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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))))