Average Error: 13.0 → 14.0
Time: 5.1s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \frac{{1}^{3} - {\left(\frac{x}{e^{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}}\right)}^{3}}{\frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x} + \left(\frac{1 \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1 \cdot 1\right)} + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]
\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{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \frac{{1}^{3} - {\left(\frac{x}{e^{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}}\right)}^{3}}{\frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x} + \left(\frac{1 \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1 \cdot 1\right)} + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}
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) (((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) (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) (x / ((double) exp(((double) log(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))))), 3.0)))) / ((double) (((double) (((double) (x * x)) / ((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))) + ((double) (((double) (((double) (1.0 * x)) / ((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))) + ((double) (1.0 * 1.0)))))))))) + ((double) (((double) pow(x, 2.0)) / ((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x))))))))))))));
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.0
Target13.0
Herbie14.0
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

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 flip3-+13.0

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

  4. Simplified13.0

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{\color{blue}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\]
  5. Using strategy rm

  6. Applied add-exp-log14.0

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\color{blue}{e^{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]
  7. Using strategy rm

  8. Applied flip3--14.0

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \color{blue}{\frac{{1}^{3} - {\left(\frac{x}{e^{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{e^{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}} \cdot \frac{x}{e^{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}} + 1 \cdot \frac{x}{e^{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}}\right)}} + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]

  9. Simplified14.0

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \frac{{1}^{3} - {\left(\frac{x}{e^{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}}\right)}^{3}}{\color{blue}{\frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x} + \left(\frac{1 \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1 \cdot 1\right)}} + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]

  10. Final simplification14.0

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \frac{{1}^{3} - {\left(\frac{x}{e^{\log \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)}}\right)}^{3}}{\frac{x \cdot x}{\left(4 \cdot p\right) \cdot p + x \cdot x} + \left(\frac{1 \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1 \cdot 1\right)} + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\]

Reproduce

herbie shell --seed 2020113 
(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))))))))