Average Error: 13.6 → 11.7

Time: 15.3s

Precision: binary64

Cost: 26369

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

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -3.9489225518133:\\ \;\;\;\;\sqrt{\left|\frac{p}{x}\right|} \cdot \sqrt{\left|\frac{p}{x}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}}\right)}\\ \end{array}\]

\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}\;x \leq -3.9489225518133:\\
\;\;\;\;\sqrt{\left|\frac{p}{x}\right|} \cdot \sqrt{\left|\frac{p}{x}\right|}\\

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

\end{array}

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

↓

(FPCore (p x)
 :precision binary64
 (if (<= x -3.9489225518133)
   (* (sqrt (fabs (/ p x))) (sqrt (fabs (/ p x))))
   (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* p (* p 4.0)) (* x x)))))))))

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) {
	double tmp;
	if (x <= -3.9489225518133) {
		tmp = sqrt(fabs(p / x)) * sqrt(fabs(p / x));
	} else {
		tmp = sqrt(0.5 * (1.0 + (x / sqrt((p * (p * 4.0)) + (x * x)))));
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `p`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	32.4
Cost	26560

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

Alternative 2
Error	31.7
Cost	6976

\[\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot -2}\right)}\]

Alternative 3
Error	46.6
Cost	19392

\[e^{\log \left(\left|\frac{p}{x}\right|\right)}\]

Alternative 4
Error	28.5
Cost	7488

\[\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2 + 0.25 \cdot \frac{x \cdot x}{p}}\right)}\]

Alternative 5
Error	31.7
Cost	6976

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

Alternative 6
Error	51.7
Cost	7232

\[\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{\frac{1}{\frac{p}{x \cdot x}}}\right)}\]

Alternative 7
Error	55.8
Cost	13248

\[\frac{\sqrt{p}}{\sqrt{\frac{x \cdot x}{p}}}\]

Alternative 8
Error	14.7
Cost	33600

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

Alternative 9
Error	51.7
Cost	7104

\[\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{\frac{x}{\frac{p}{x}}}\right)}\]

Alternative 10
Error	51.7
Cost	7104

\[\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{x \cdot \frac{x}{p}}\right)}\]

Alternative 11
Error	45.7
Cost	39104

\[\left(\sqrt[3]{\left|\frac{p}{x}\right|} \cdot \sqrt[3]{\left|\frac{p}{x}\right|}\right) \cdot \sqrt[3]{\left|\frac{p}{x}\right|}\]

Alternative 12
Error	16.2
Cost	7360

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

Alternative 13
Error	41.2
Cost	6848

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

Alternative 14
Error	15.1
Cost	34240

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

Alternative 15
Error	28.8
Cost	6464

\[\sqrt{0.5}\]

Alternative 16
Error	54.7
Cost	19456

\[\sqrt[3]{{\left(\left|\frac{p}{x}\right|\right)}^{3}}\]

Alternative 17
Error	53.3
Cost	256

\[\frac{-p}{x}\]

Alternative 18
Error	54.1
Cost	33152

\[3 \cdot \left(\frac{\sqrt{0.5}}{\sqrt{2}} \cdot {\left(\frac{p}{x}\right)}^{3}\right) - \frac{\sqrt{0.5} \cdot \sqrt{2}}{\frac{x}{p}}\]

Alternative 19
Error	15.5
Cost	47040

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

Alternative 20
Error	51.7
Cost	7104

\[\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{\frac{x \cdot x}{p}}\right)}\]

Alternative 21
Error	41.2
Cost	6592

\[\sqrt{0.5 \cdot 2}\]

Alternative 22
Error	28.8
Cost	6592

\[\sqrt{0.5 \cdot 1}\]

Alternative 23
Error	59.9
Cost	6912

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

Alternative 24
Error	51.7
Cost	13504

\[\sqrt{0.5} \cdot \sqrt{2 \cdot \frac{p}{\frac{x \cdot x}{p}}}\]

Alternative 25
Error	13.6
Cost	13760

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

Alternative 26
Error	26.4
Cost	26624

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

Alternative 27
Error	28.5
Cost	7488

\[\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot -2 - 0.25 \cdot \frac{x \cdot x}{p}}\right)}\]

Alternative 28
Error	19.9
Cost	26624

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

Alternative 29
Error	51.7
Cost	7104

\[\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\]

Alternative 30
Error	53.1
Cost	192

\[\frac{p}{x}\]

Alternative 31
Error	16.5
Cost	26560

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

Alternative 32
Error	29.0
Cost	14784

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

Alternative 33
Error	16.5
Cost	26560

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

Alternative 34
Error	58.1
Cost	19904

\[\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{e^{\log \left(\frac{x \cdot x}{p}\right)}}\right)}\]

Alternative 35
Error	45.5
Cost	26048

\[\sqrt{\left|\frac{p}{x}\right|} \cdot \sqrt{\left|\frac{p}{x}\right|}\]

Alternative 36
Error	15.0
Cost	33600

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

Alternative 37
Error	53.5
Cost	13312

\[-\frac{\sqrt{0.5} \cdot \sqrt{2}}{\frac{x}{p}}\]

Alternative 38
Error	58.4
Cost	26112

\[1 - \log \left(e^{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right)\]

Alternative 39
Error	58.4
Cost	58624

\[1 - {\left(\sqrt[3]{\sqrt{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)}^{5} \cdot \sqrt[3]{\sqrt{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\]

Alternative 40
Error	58.4
Cost	45696

\[1 - \sqrt{0.5 + \frac{\frac{0.5}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 41
Error	58.4
Cost	39360

\[1 - \left|\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right| \cdot \sqrt{\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 42
Error	58.4
Cost	26112

\[e^{\log \left(1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}\]

Alternative 43
Error	58.4
Cost	45824

\[1 - \sqrt{0.5 + \frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{0.5}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 44
Error	58.4
Cost	39360

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

Alternative 45
Error	58.4
Cost	32960

\[1 - \sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 46
Error	58.4
Cost	84672

\[1 - \left(\sqrt[3]{\sqrt{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \cdot \sqrt[3]{\sqrt{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right) \cdot {\left(\sqrt[3]{\sqrt{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)}^{4}\]

Alternative 47
Error	58.4
Cost	32832

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

Alternative 48
Error	58.4
Cost	45952

\[1 - \sqrt{0.5 + \left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 49
Error	58.4
Cost	26112

\[1 - \sqrt{0.5 + \log \left(e^{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}\]

Alternative 50
Error	59.4
Cost	65600

\[1 - \frac{\sqrt{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \sqrt{\sqrt{0.25 - \frac{0.25}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}}}{\sqrt{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\]

Alternative 51
Error	58.4
Cost	72064

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

Alternative 52
Error	58.4
Cost	26176

\[1 - \sqrt{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}\]

Alternative 53
Error	58.4
Cost	32960

\[\log \left(e^{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)\]

Alternative 54
Error	58.4
Cost	39424

\[1 + \left(-\left|\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right|\right) \cdot \sqrt{\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 55
Error	58.4
Cost	26112

\[\log \left(e^{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right)\]

Alternative 56
Error	58.4
Cost	52224

\[1 - {\left(\sqrt[3]{\sqrt{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)}^{4} \cdot \sqrt[3]{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 57
Error	58.4
Cost	39488

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

Alternative 58
Error	58.4
Cost	65088

\[\log \left(\sqrt{e^{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right) + \log \left(\sqrt{e^{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)\]

Alternative 59
Error	58.4
Cost	20160

\[\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 60
Error	58.4
Cost	39872

\[\frac{1 - {\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}\]

Alternative 61
Error	58.4
Cost	59264

\[\left(\sqrt[3]{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \sqrt[3]{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right) \cdot \sqrt[3]{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 62
Error	58.4
Cost	26176

\[\sqrt[3]{{\left(1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}}\]

Alternative 63
Error	58.4
Cost	65216

\[\log \left(\sqrt[3]{e^{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right) \cdot 2 + \log \left(\sqrt[3]{e^{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)\]

Alternative 64
Error	58.4
Cost	58624

\[1 - \sqrt{\sqrt{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \cdot {\left(\sqrt{\sqrt{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)}^{3}\]

Alternative 65
Error	59.5
Cost	33152

\[1 - \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 66
Error	58.4
Cost	39744

\[1 - \frac{\sqrt{{\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} + 0.125}}{\sqrt{0.25 + \frac{\frac{0.25}{\mathsf{hypot}\left(1, x\right)} + -0.25}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 67
Error	59.5
Cost	45952

\[\log \left(e^{1 - \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)\]

Alternative 68
Error	58.4
Cost	38976

\[\log \left(e^{1 - \sqrt[3]{{\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}}}\right)\]

Alternative 69
Error	58.4
Cost	26176

\[1 - \sqrt[3]{{\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}}\]

Alternative 70
Error	59.5
Cost	26752

\[1 - \sqrt{\frac{0.25 - \frac{0.25}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 71
Error	58.4
Cost	33344

\[1 - \sqrt{\frac{{\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} + 0.125}{0.25 + \frac{\frac{0.25}{\mathsf{hypot}\left(1, x\right)} + -0.25}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 72
Error	58.4
Cost	45824

\[1 - \sqrt{{\left(\sqrt[3]{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right)}^{4} \cdot \sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\]

Alternative 73
Error	58.4
Cost	26176

\[1 - {\left(\sqrt{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right)}^{2}\]

Alternative 74
Error	58.4
Cost	13312

\[1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\]

Alternative 75
Error	58.4
Cost	13440

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

Error

Derivation

Split input into 2 regimes
if x < -3.94892255181329999
1. Initial program 30.5
  \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
2. Taylor expanded around -inf 38.6
  \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}}\]
3. Simplified38.2
  \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{p}{\frac{x \cdot x}{p}}\right)}}\]
4. Using strategy rm
5. Applied add-sqr-sqrt_binary64_10038.3
  \[\leadsto \color{blue}{\sqrt{\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{\frac{x \cdot x}{p}}\right)}} \cdot \sqrt{\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{\frac{x \cdot x}{p}}\right)}}}\]
6. Simplified38.2
  \[\leadsto \color{blue}{\sqrt{\left|\frac{p}{x}\right|}} \cdot \sqrt{\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{\frac{x \cdot x}{p}}\right)}}\]
7. Simplified23.2
  \[\leadsto \sqrt{\left|\frac{p}{x}\right|} \cdot \color{blue}{\sqrt{\left|\frac{p}{x}\right|}}\]
if -3.94892255181329999 < x
1. Initial program 7.7
  \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
Recombined 2 regimes into one program.
Final simplification11.7
\[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9489225518133:\\ \;\;\;\;\sqrt{\left|\frac{p}{x}\right|} \cdot \sqrt{\left|\frac{p}{x}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1e-150 (fabs x) 1e+150)
  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

Error

Try it out

Alternatives

Error

Derivation

`if x < -3.94892255181329999`

`if -3.94892255181329999 < x`

Reproduce