Average Error: 13.1 → 11.0
Time: 6.1s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -0.99999999999999978:\\ \;\;\;\;\sqrt{0.5 \cdot e^{2 \cdot \left(\log \left(\frac{-1}{x}\right) - \log \left(\frac{-1}{p}\right)\right) + \log 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\\ \end{array}\]
\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}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -0.99999999999999978:\\
\;\;\;\;\sqrt{0.5 \cdot e^{2 \cdot \left(\log \left(\frac{-1}{x}\right) - \log \left(\frac{-1}{p}\right)\right) + \log 2}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\\

\end{array}
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 ((((double) (x / ((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))) <= -0.9999999999999998)) {
		VAR = ((double) sqrt(((double) (0.5 * ((double) exp(((double) (((double) (2.0 * ((double) (((double) log(((double) (-1.0 / x)))) - ((double) log(((double) (-1.0 / p)))))))) + ((double) log(2.0))))))))));
	} else {
		VAR = ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (((double) (1.0 / ((double) sqrt(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))) * ((double) (x / ((double) sqrt(((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x))))))))))))))))));
	}
	return VAR;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.1
Target13.1
Herbie11.0
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ x (sqrt (+ (* (* 4.0 p) p) (* x x)))) < -0.9999999999999998

    1. Initial program 53.9

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

    3. Applied add-exp-log53.9

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

    4. Taylor expanded around -inf 45.1

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

    5. Simplified45.1

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

    if -0.9999999999999998 < (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied sqrt-prod0.2

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

    5. Applied *-un-lft-identity0.2

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

    6. Applied times-frac0.3

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \le -0.99999999999999978:\\ \;\;\;\;\sqrt{0.5 \cdot e^{2 \cdot \left(\log \left(\frac{-1}{x}\right) - \log \left(\frac{-1}{p}\right)\right) + \log 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020129 
(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))))))))