Given's Rotation SVD example

Average Error: 13.4 → 13.5

Time: 7.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.727158006251015 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{{\left(e^{\sqrt[3]{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}\right)}}\right) + \frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\\ \end{array}\]

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) {
	double VAR;
	if ((x <= -1.727158006251015e-119)) {
		VAR = ((double) sqrt(((double) (0.5 * ((double) (((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) pow(((double) exp(((double) (((double) cbrt(((double) log(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))) * ((double) cbrt(((double) log(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))))))), ((double) cbrt(((double) log(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))))))))))) + ((double) (((double) (x * x)) / ((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x))))))))))))));
	} else {
		VAR = ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (x / ((double) (((double) fabs(((double) cbrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))) * ((double) sqrt(((double) cbrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x))))))))))))))))));
	}
	return VAR;
}

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. Split input into 2 regimes

2. if x < -1.727158006251015e-119

   1. Initial program 28.4

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

   3. Applied flip3-+ 28.4

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

   4. Simplified 28.4

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

   6. Applied add-exp-log 28.4

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

   8. Applied add-cube-cbrt 28.4

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

   9. Applied exp-prod 28.4

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

   if -1.727158006251015e-119 < x

   1. Initial program 1.6

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

   3. Applied add-cube-cbrt 1.7

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

   4. Applied sqrt-prod 1.7

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

   5. Simplified 1.7

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.727158006251015 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{{\left(e^{\sqrt[3]{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}\right)}}\right) + \frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(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))))))))