Given's Rotation SVD example

Average Error: 13.4 → 13.4

Time: 6.3s

Precision: binary64

\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]

Math

\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]

↓

\[\sqrt{0.5 \cdot \frac{\log \left(\sqrt{e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}\right) + \log \left(\sqrt{e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]

C

double code(double p, double x) {
	return ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (x / ((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x))))))))))))));
}

↓

double code(double p, double x) {
	return ((double) sqrt(((double) (0.5 * ((double) (((double) (((double) log(((double) sqrt(((double) exp(((double) (((double) pow(1.0, 3.0)) + ((double) pow(((double) (x / ((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))), 3.0)))))))))) + ((double) log(((double) sqrt(((double) exp(((double) (((double) pow(1.0, 3.0)) + ((double) pow(((double) (x / ((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))), 3.0)))))))))))) / ((double) (((double) (1.0 * ((double) (1.0 - ((double) (x / ((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))))) + ((double) (((double) pow(x, 2.0)) / ((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x))))))))))))));
}

Error

Bits error versus p

Bits error versus x

Target

Original	13.4
Target	13.4
Herbie	13.4

\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

1. Initial program 13.4

   \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
2. Using strategy rm

3. Applied flip3-+ 13.4

   \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}\]

4. Simplified 13.4

   \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{\color{blue}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\]
5. Using strategy rm

6. Applied add-log-exp 13.4

   \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + \color{blue}{\log \left(e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]

7. Applied add-log-exp 13.4

   \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{\log \left(e^{{1}^{3}}\right)} + \log \left(e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]

8. Applied sum-log 13.4

   \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{\log \left(e^{{1}^{3}} \cdot e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]

9. Simplified 13.4

   \[\leadsto \sqrt{0.5 \cdot \frac{\log \color{blue}{\left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]
10. Using strategy rm

11. Applied add-sqr-sqrt 13.4

    \[\leadsto \sqrt{0.5 \cdot \frac{\log \color{blue}{\left(\sqrt{e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}} \cdot \sqrt{e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]

12. Applied log-prod 13.4

    \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{\log \left(\sqrt{e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}\right) + \log \left(\sqrt{e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]

13. Final simplification 13.4

    \[\leadsto \sqrt{0.5 \cdot \frac{\log \left(\sqrt{e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}\right) + \log \left(\sqrt{e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]

Reproduce

herbie shell --seed 2020149 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))