Given's Rotation SVD example

Average Error: 13.4 → 13.5

Time: 8.0s

Precision: binary64

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

Math

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

↓

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

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

Error

Bits error versus p

Bits error versus x

Target

Original	13.4
Target	13.4
Herbie	13.5

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

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 div-inv 13.5

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

5. Applied add-log-exp 13.5

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

7. Applied flip3-+ 13.5

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

8. Simplified 13.5

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

9. Final simplification 13.5

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

Reproduce

herbie shell --seed 2020147 
(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))))))))