Kahan p9 Example

Average Error: 19.9 → 0.1

Time: 18.4s

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 r71444 = x;
        double r71445 = y;
        double r71446 = r71444 - r71445;
        double r71447 = r71444 + r71445;
        double r71448 = r71446 * r71447;
        double r71449 = r71444 * r71444;
        double r71450 = r71445 * r71445;
        double r71451 = r71449 + r71450;
        double r71452 = r71448 / r71451;
        return r71452;
}

↓

double f(double x, double y) {
        double r71453 = x;
        double r71454 = y;
        double r71455 = r71453 - r71454;
        double r71456 = hypot(r71453, r71454);
        double r71457 = r71455 / r71456;
        double r71458 = r71453 + r71454;
        double r71459 = r71458 / r71456;
        double r71460 = r71457 * r71459;
        double r71461 = cbrt(r71460);
        double r71462 = r71461 * r71461;
        double r71463 = r71462 * r71461;
        return r71463;
}

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))))