Given's Rotation SVD example

Average Error: 13.2 → 13.2

Time: 4.5s

Precision: 64

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

Math

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

↓

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

C

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

↓

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

Error

Bits error versus p

Bits error versus x

Target

Original	13.2
Target	13.2
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 13.2

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

3. Applied add-sqr-sqrt 13.2

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

5. Applied add-log-exp 13.2

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

6. Applied add-log-exp 13.2

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

7. Applied sum-log 13.2

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

8. Simplified 13.2

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

10. Applied add-sqr-sqrt 13.2

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

11. Applied log-prod 13.2

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

12. Final simplification 13.2

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

Reproduce

herbie shell --seed 2020092 
(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 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))