Given's Rotation SVD example

Average Error: 13.0 → 13.2

Time: 13.3s

Precision: binary64

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

Math

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

↓

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

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) (1.0 + ((double) (((double) log(((double) sqrt(((double) exp(((double) (x / ((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))))))) + ((double) (((double) (((double) (x / ((double) sqrt(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))) / 2.0)) * ((double) (1.0 / ((double) sqrt(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x))))))))))))))))))));
}

Error

Bits error versus p

Bits error versus x

Target

Original	13.0
Target	13.0
Herbie	13.2

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

Derivation

1. Initial program 13.0

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

3. Applied add-log-exp 13.0

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

5. Applied add-sqr-sqrt 13.0

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

6. Applied log-prod 13.0

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

8. Applied add-sqr-sqrt 13.0

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

9. Applied sqrt-prod 13.6

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

10. Applied *-un-lft-identity 13.6

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

11. Applied times-frac 13.5

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

12. Applied exp-prod 32.4

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

13. Applied sqrt-pow1 32.4

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

14. Applied log-pow 32.4

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

15. Simplified 13.2

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

16. Final simplification 13.2

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

Reproduce

herbie shell --seed 2020163 
(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))))))))