Given's Rotation SVD example

Percentage Accurate: 79.9% → 99.7%

Time: 12.8s

Alternatives: 8

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\\ \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{1 + {t_0}^{3}}{\mathsf{fma}\left(t_0, \frac{-1 + {t_0}^{2}}{1 + t_0}, 1\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ x (hypot x (* p 2.0)))))
   (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
     (/ (- p) x)
     (sqrt
      (*
       0.5
       (/
        (+ 1.0 (pow t_0 3.0))
        (fma t_0 (/ (+ -1.0 (pow t_0 2.0)) (+ 1.0 t_0)) 1.0)))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = x / hypot(x, (p * 2.0));
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * ((1.0 + pow(t_0, 3.0)) / fma(t_0, ((-1.0 + pow(t_0, 2.0)) / (1.0 + t_0)), 1.0))));
	}
	return tmp;
}

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(x / hypot(x, Float64(p * 2.0)))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(Float64(1.0 + (t_0 ^ 3.0)) / fma(t_0, Float64(Float64(-1.0 + (t_0 ^ 2.0)) / Float64(1.0 + t_0)), 1.0))));
	end
	return tmp
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[(1.0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(-1.0 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 10.2%

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

   2. Taylor expanded in x around -inf 55.7%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 55.7%

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

      2. associate-*r/ 55.7%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{2 \cdot {p}^{2}}{x \cdot x}}} \]

      3. times-frac 55.8%

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

      4. unpow2 55.8%

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

      5. associate-/l* 61.3%

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

   4. Simplified 61.3%

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

   5. Taylor expanded in x around -inf 42.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 42.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot p}{x}} \]

      2. mul-1-neg 42.8%

         \[\leadsto \frac{\color{blue}{-p}}{x} \]

   7. Simplified 42.8%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. flip3-+ 99.6%

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

   3. Applied egg-rr 99.7%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}} \]
   4. Step-by-step derivation

      1. flip-- 99.7%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \color{blue}{\frac{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1 \cdot 1}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 1}}, 1\right)}} \]

      2. unpow2 99.7%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{\color{blue}{{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{2}} - 1 \cdot 1}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 1}, 1\right)}} \]

      3. metadata-eval 99.7%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{2} - \color{blue}{1}}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 1}, 1\right)}} \]

      4. sub-neg 99.7%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{\color{blue}{{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{2} + \left(-1\right)}}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 1}, 1\right)}} \]

      5. metadata-eval 99.7%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{2} + \color{blue}{-1}}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 1}, 1\right)}} \]

   5. Applied egg-rr 99.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, \frac{-1 + {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}}{1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}, 1\right)}}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\\ \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{1 + {t_0}^{3}}{\mathsf{fma}\left(t_0, -1 + t_0, 1\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ x (hypot x (* p 2.0)))))
   (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
     (/ (- p) x)
     (sqrt (* 0.5 (/ (+ 1.0 (pow t_0 3.0)) (fma t_0 (+ -1.0 t_0) 1.0)))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = x / hypot(x, (p * 2.0));
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * ((1.0 + pow(t_0, 3.0)) / fma(t_0, (-1.0 + t_0), 1.0))));
	}
	return tmp;
}

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(x / hypot(x, Float64(p * 2.0)))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(Float64(1.0 + (t_0 ^ 3.0)) / fma(t_0, Float64(-1.0 + t_0), 1.0))));
	end
	return tmp
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[(1.0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(-1.0 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 10.2%

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

   2. Taylor expanded in x around -inf 55.7%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 55.7%

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

      2. associate-*r/ 55.7%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{2 \cdot {p}^{2}}{x \cdot x}}} \]

      3. times-frac 55.8%

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

      4. unpow2 55.8%

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

      5. associate-/l* 61.3%

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

   4. Simplified 61.3%

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

   5. Taylor expanded in x around -inf 42.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 42.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot p}{x}} \]

      2. mul-1-neg 42.8%

         \[\leadsto \frac{\color{blue}{-p}}{x} \]

   7. Simplified 42.8%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. flip3-+ 99.6%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, -1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 1\right)}}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
   (/ (- p) x)
   (sqrt (* 0.5 (exp (log1p (/ x (hypot x (* p 2.0)))))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * exp(log1p((x / hypot(x, (p * 2.0)))))));
	}
	return tmp;
}

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt((0.5 * Math.exp(Math.log1p((x / Math.hypot(x, (p * 2.0)))))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0:
		tmp = -p / x
	else:
		tmp = math.sqrt((0.5 * math.exp(math.log1p((x / math.hypot(x, (p * 2.0)))))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * exp(log1p(Float64(x / hypot(x, Float64(p * 2.0)))))));
	end
	return tmp
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[Exp[N[Log[1 + N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 10.2%

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

   2. Taylor expanded in x around -inf 55.7%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 55.7%

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

      2. associate-*r/ 55.7%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{2 \cdot {p}^{2}}{x \cdot x}}} \]

      3. times-frac 55.8%

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

      4. unpow2 55.8%

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

      5. associate-/l* 61.3%

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

   4. Simplified 61.3%

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

   5. Taylor expanded in x around -inf 42.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 42.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot p}{x}} \]

      2. mul-1-neg 42.8%

         \[\leadsto \frac{\color{blue}{-p}}{x} \]

   7. Simplified 42.8%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. add-exp-log 99.6%

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

      2. log1p-udef 99.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. +-commutative 99.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      4. add-sqr-sqrt 99.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      5. hypot-def 99.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      6. associate-*l* 99.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      7. sqrt-prod 99.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      8. metadata-eval 99.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

      9. sqrt-unprod 45.7%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

      10. add-sqr-sqrt 99.7%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
   (/ (- p) x)
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	return tmp;
}

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0:
		tmp = -p / x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0)
		tmp = -p / x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 10.2%

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

   2. Taylor expanded in x around -inf 55.7%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 55.7%

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

      2. associate-*r/ 55.7%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{2 \cdot {p}^{2}}{x \cdot x}}} \]

      3. times-frac 55.8%

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

      4. unpow2 55.8%

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

      5. associate-/l* 61.3%

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

   4. Simplified 61.3%

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

   5. Taylor expanded in x around -inf 42.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 42.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot p}{x}} \]

      2. mul-1-neg 42.8%

         \[\leadsto \frac{\color{blue}{-p}}{x} \]

   7. Simplified 42.8%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 99.6%

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

      2. hypot-def 99.6%

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

      3. associate-*l* 99.6%

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

      4. sqrt-prod 99.6%

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

      5. metadata-eval 99.6%

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

      6. sqrt-unprod 45.7%

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

      7. add-sqr-sqrt 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 5: 80.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + 2 \cdot \frac{p \cdot p}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= x -1.8e+50)
     t_0
     (if (<= x -7.5e-12)
       (sqrt 0.5)
       (if (<= x -2.6e-28)
         t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (+ x (* 2.0 (/ (* p p) x))))))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (x <= -1.8e+50) {
		tmp = t_0;
	} else if (x <= -7.5e-12) {
		tmp = sqrt(0.5);
	} else if (x <= -2.6e-28) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p / x
    if (x <= (-1.8d+50)) then
        tmp = t_0
    else if (x <= (-7.5d-12)) then
        tmp = sqrt(0.5d0)
    else if (x <= (-2.6d-28)) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (x + (2.0d0 * ((p * p) / x)))))))
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (x <= -1.8e+50) {
		tmp = t_0;
	} else if (x <= -7.5e-12) {
		tmp = Math.sqrt(0.5);
	} else if (x <= -2.6e-28) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -p / x
	tmp = 0
	if x <= -1.8e+50:
		tmp = t_0
	elif x <= -7.5e-12:
		tmp = math.sqrt(0.5)
	elif x <= -2.6e-28:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (x <= -1.8e+50)
		tmp = t_0;
	elseif (x <= -7.5e-12)
		tmp = sqrt(0.5);
	elseif (x <= -2.6e-28)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(x + Float64(2.0 * Float64(Float64(p * p) / x)))))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -p / x;
	tmp = 0.0;
	if (x <= -1.8e+50)
		tmp = t_0;
	elseif (x <= -7.5e-12)
		tmp = sqrt(0.5);
	elseif (x <= -2.6e-28)
		tmp = t_0;
	else
		tmp = sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[((-p) / x), $MachinePrecision]}, If[LessEqual[x, -1.8e+50], t$95$0, If[LessEqual[x, -7.5e-12], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[x, -2.6e-28], t$95$0, N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(x + N[(2.0 * N[(N[(p * p), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999993e50 or -7.5e-12 < x < -2.6e-28

   1. Initial program 47.7%

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

   2. Taylor expanded in x around -inf 35.1%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 35.1%

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

      2. associate-*r/ 35.1%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{2 \cdot {p}^{2}}{x \cdot x}}} \]

      3. times-frac 35.1%

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

      4. unpow2 35.1%

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

      5. associate-/l* 37.1%

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

   4. Simplified 37.1%

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

   5. Taylor expanded in x around -inf 30.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 30.2%

         \[\leadsto \color{blue}{\frac{-1 \cdot p}{x}} \]

      2. mul-1-neg 30.2%

         \[\leadsto \frac{\color{blue}{-p}}{x} \]

   7. Simplified 30.2%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if -1.79999999999999993e50 < x < -7.5e-12

   1. Initial program 71.3%

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

   2. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]

   if -2.6e-28 < x

   1. Initial program 93.1%

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

   2. Taylor expanded in p around 0 91.2%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{x + 2 \cdot \frac{{p}^{2}}{x}}}\right)} \]
   3. Step-by-step derivation

      1. unpow2 91.2%

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

   4. Simplified 91.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+50}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + 2 \cdot \frac{p \cdot p}{x}}\right)}\\ \end{array} \]

Alternative 6: 67.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 4.1 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= p 4.1e-172) 1.0 (if (<= p 1.2e-17) (/ (- p) x) (sqrt 0.5))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 4.1e-172) {
		tmp = 1.0;
	} else if (p <= 1.2e-17) {
		tmp = -p / x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 4.1d-172) then
        tmp = 1.0d0
    else if (p <= 1.2d-17) then
        tmp = -p / x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (p <= 4.1e-172) {
		tmp = 1.0;
	} else if (p <= 1.2e-17) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 4.1e-172:
		tmp = 1.0
	elif p <= 1.2e-17:
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 4.1e-172)
		tmp = 1.0;
	elseif (p <= 1.2e-17)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 4.1e-172)
		tmp = 1.0;
	elseif (p <= 1.2e-17)
		tmp = -p / x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[p, 4.1e-172], 1.0, If[LessEqual[p, 1.2e-17], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 4.1e-172

   1. Initial program 79.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 79.0%

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

      2. hypot-def 79.0%

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

      3. associate-*l* 79.0%

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

      4. sqrt-prod 79.0%

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

      5. metadata-eval 79.0%

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

      6. sqrt-unprod 9.4%

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

      7. add-sqr-sqrt 79.1%

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

   3. Applied egg-rr 79.1%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(2 \cdot p, x\right)}}\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 79.1%

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

      2. distribute-lft-in 79.1%

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

      3. metadata-eval 79.1%

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

   5. Applied egg-rr 79.1%

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

   6. Taylor expanded in x around inf 34.7%

      \[\leadsto \color{blue}{1} \]

   if 4.1e-172 < p < 1.19999999999999993e-17

   1. Initial program 56.0%

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

   2. Taylor expanded in x around -inf 30.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 30.8%

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

      2. associate-*r/ 30.8%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{2 \cdot {p}^{2}}{x \cdot x}}} \]

      3. times-frac 30.9%

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

      4. unpow2 30.9%

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

      5. associate-/l* 32.6%

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

   4. Simplified 32.6%

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

   5. Taylor expanded in x around -inf 49.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 49.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot p}{x}} \]

      2. mul-1-neg 49.0%

         \[\leadsto \frac{\color{blue}{-p}}{x} \]

   7. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if 1.19999999999999993e-17 < p

   1. Initial program 98.2%

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

   2. Taylor expanded in x around 0 95.0%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.1 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 52.7% accurate, 35.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -4e-78) (/ (- p) x) 1.0))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -4e-78) {
		tmp = -p / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4d-78)) then
        tmp = -p / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -4e-78) {
		tmp = -p / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -4e-78:
		tmp = -p / x
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -4e-78)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -4e-78)
		tmp = -p / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -4e-78], N[((-p) / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4e-78

   1. Initial program 56.4%

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

   2. Taylor expanded in x around -inf 32.2%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. unpow2 32.2%

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

      2. associate-*r/ 32.2%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{2 \cdot {p}^{2}}{x \cdot x}}} \]

      3. times-frac 32.2%

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

      4. unpow2 32.2%

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

      5. associate-/l* 35.4%

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

   4. Simplified 35.4%

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

   5. Taylor expanded in x around -inf 23.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ 23.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot p}{x}} \]

      2. mul-1-neg 23.1%

         \[\leadsto \frac{\color{blue}{-p}}{x} \]

   7. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if -4e-78 < x

   1. Initial program 97.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 97.6%

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

      2. hypot-def 97.6%

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

      3. associate-*l* 97.6%

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

      4. sqrt-prod 97.6%

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

      5. metadata-eval 97.6%

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

      6. sqrt-unprod 46.8%

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

      7. add-sqr-sqrt 97.6%

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

   3. Applied egg-rr 97.6%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(2 \cdot p, x\right)}}\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 97.5%

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

      2. distribute-lft-in 97.5%

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

      3. metadata-eval 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in x around inf 42.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 36.0% accurate, 215.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ 1 \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 1.0)

C

p = abs(p);
double code(double p, double x) {
	return 1.0;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	return 1.0;
}

Python

p = abs(p)
def code(p, x):
	return 1.0

Julia

p = abs(p)
function code(p, x)
	return 1.0
end

MATLAB

p = abs(p)
function tmp = code(p, x)
	tmp = 1.0;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := 1.0

Derivation

1. Initial program 81.8%

   \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
2. Step-by-step derivation

   1. add-sqr-sqrt 81.8%

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

   2. hypot-def 81.8%

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

   3. associate-*l* 81.8%

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

   4. sqrt-prod 81.8%

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

   5. metadata-eval 81.8%

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

   6. sqrt-unprod 37.7%

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

   7. add-sqr-sqrt 81.8%

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

3. Applied egg-rr 81.8%

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(2 \cdot p, x\right)}}\right)} \]
4. Step-by-step derivation

   1. add-log-exp 81.8%

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

   2. distribute-lft-in 81.8%

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

   3. metadata-eval 81.8%

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

5. Applied egg-rr 81.8%

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

6. Taylor expanded in x around inf 30.9%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 30.9%

   \[\leadsto 1 \]

Developer target: 79.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x))))))

C

double code(double p, double x) {
	return sqrt((0.5 + (copysign(0.5, x) / hypot(1.0, ((2.0 * p) / x)))));
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 + (Math.copySign(0.5, x) / Math.hypot(1.0, ((2.0 * p) / x)))));
}

Python

def code(p, x):
	return math.sqrt((0.5 + (math.copysign(0.5, x) / math.hypot(1.0, ((2.0 * p) / x)))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 + Float64(copysign(0.5, x) / hypot(1.0, Float64(Float64(2.0 * p) / x)))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 + ((sign(x) * abs(0.5)) / hypot(1.0, ((2.0 * p) / x)))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 + N[(N[With[{TMP1 = Abs[0.5], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * p), $MachinePrecision] / x), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2023279 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (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))))))))