Kahan p9 Example

Average Error: 19.9 → 0.1

Time: 19.4s

Precision: 64

\[0.0 \lt x \lt 1 \land y \lt 1\]

Math

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

↓

\[\left(\sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \cdot \sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}}\right) \cdot \sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}}\]

C

double f(double x, double y) {
        double r68317 = x;
        double r68318 = y;
        double r68319 = r68317 - r68318;
        double r68320 = r68317 + r68318;
        double r68321 = r68319 * r68320;
        double r68322 = r68317 * r68317;
        double r68323 = r68318 * r68318;
        double r68324 = r68322 + r68323;
        double r68325 = r68321 / r68324;
        return r68325;
}

↓

double f(double x, double y) {
        double r68326 = x;
        double r68327 = y;
        double r68328 = r68326 - r68327;
        double r68329 = hypot(r68326, r68327);
        double r68330 = r68328 / r68329;
        double r68331 = r68326 + r68327;
        double r68332 = r68331 / r68329;
        double r68333 = r68330 * r68332;
        double r68334 = cbrt(r68333);
        double r68335 = r68334 * r68334;
        double r68336 = r68335 * r68334;
        return r68336;
}

Error

Bits error versus x

Bits error versus y

Target

Original	19.9
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 19.9

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

3. Applied add-sqr-sqrt 19.9

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

4. Applied times-frac 20.0

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

5. Simplified 20.0

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

6. Simplified 0.0

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

8. Applied add-cube-cbrt 0.1

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

9. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019323 +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))))