Given's Rotation SVD example

Average Error: 13.1 → 13.1

Time: 5.6s

Precision: binary64

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

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

↓

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

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

↓

(FPCore (p x)
 :precision binary64
 (sqrt
  (*
   0.5
   (/
    (+
     (log
      (sqrt (exp (+ 1.0 (pow (/ x (sqrt (+ (* p (* p 4.0)) (* x x)))) 3.0)))))
     (log
      (sqrt (exp (+ 1.0 (pow (/ x (sqrt (+ (* p (* p 4.0)) (* x x)))) 3.0))))))
    (+
     1.0
     (-
      (/ (* x x) (+ (* p (* p 4.0)) (* x x)))
      (/ x (sqrt (+ (* p (* p 4.0)) (* x x))))))))))

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

Error

Bits error versus p

Bits error versus x

Target

Original	13.1
Target	13.1
Herbie	13.1

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

Derivation

1. Initial program 13.1

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

3. Applied flip3-+_binary64_2809 13.1

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

4. Simplified 13.1

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

5. Simplified 13.1

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

7. Applied add-log-exp_binary64_2845 13.1

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

8. Applied add-log-exp_binary64_2845 13.1

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

9. Applied sum-log_binary64_2897 13.1

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

10. Simplified 13.1

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

12. Applied add-sqr-sqrt_binary64_2828 13.1

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

13. Applied log-prod_binary64_2892 13.1

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

14. Final simplification 13.1

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

Reproduce

herbie shell --seed 2020342 
(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))))))))