Given's Rotation SVD example

Percentage Accurate: 79.6% → 99.8%

Time: 7.6s

Alternatives: 6

Speedup: 0.7×

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: 79.6% 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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ (- p_m) x)
   (sqrt (+ 0.5 (* 0.5 (/ x (hypot x (* p_m 2.0))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 * N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 68.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p_m}{x}\\ t_1 := 1 + -0.5 \cdot \left(p_m \cdot \left(x \cdot \left(x \cdot p_m\right)\right)\right)\\ \mathbf{if}\;p_m \leq 2.05 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;p_m \leq 3.6 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 2.8 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;p_m \leq 3.4 \cdot 10^{-131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;1 + -0.5 \cdot \frac{x \cdot p_m}{\frac{x}{p_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)) (t_1 (+ 1.0 (* -0.5 (* p_m (* x (* x p_m)))))))
   (if (<= p_m 2.05e-241)
     t_1
     (if (<= p_m 3.6e-217)
       t_0
       (if (<= p_m 2.8e-194)
         t_1
         (if (<= p_m 3.4e-131)
           t_0
           (if (<= p_m 2.5e-38)
             (+ 1.0 (* -0.5 (/ (* x p_m) (/ x p_m))))
             (sqrt 0.5))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double t_1 = 1.0 + (-0.5 * (p_m * (x * (x * p_m))));
	double tmp;
	if (p_m <= 2.05e-241) {
		tmp = t_1;
	} else if (p_m <= 3.6e-217) {
		tmp = t_0;
	} else if (p_m <= 2.8e-194) {
		tmp = t_1;
	} else if (p_m <= 3.4e-131) {
		tmp = t_0;
	} else if (p_m <= 2.5e-38) {
		tmp = 1.0 + (-0.5 * ((x * p_m) / (x / p_m)));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -p_m / x
    t_1 = 1.0d0 + ((-0.5d0) * (p_m * (x * (x * p_m))))
    if (p_m <= 2.05d-241) then
        tmp = t_1
    else if (p_m <= 3.6d-217) then
        tmp = t_0
    else if (p_m <= 2.8d-194) then
        tmp = t_1
    else if (p_m <= 3.4d-131) then
        tmp = t_0
    else if (p_m <= 2.5d-38) then
        tmp = 1.0d0 + ((-0.5d0) * ((x * p_m) / (x / p_m)))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double t_1 = 1.0 + (-0.5 * (p_m * (x * (x * p_m))));
	double tmp;
	if (p_m <= 2.05e-241) {
		tmp = t_1;
	} else if (p_m <= 3.6e-217) {
		tmp = t_0;
	} else if (p_m <= 2.8e-194) {
		tmp = t_1;
	} else if (p_m <= 3.4e-131) {
		tmp = t_0;
	} else if (p_m <= 2.5e-38) {
		tmp = 1.0 + (-0.5 * ((x * p_m) / (x / p_m)));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	t_1 = 1.0 + (-0.5 * (p_m * (x * (x * p_m))))
	tmp = 0
	if p_m <= 2.05e-241:
		tmp = t_1
	elif p_m <= 3.6e-217:
		tmp = t_0
	elif p_m <= 2.8e-194:
		tmp = t_1
	elif p_m <= 3.4e-131:
		tmp = t_0
	elif p_m <= 2.5e-38:
		tmp = 1.0 + (-0.5 * ((x * p_m) / (x / p_m)))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	t_1 = Float64(1.0 + Float64(-0.5 * Float64(p_m * Float64(x * Float64(x * p_m)))))
	tmp = 0.0
	if (p_m <= 2.05e-241)
		tmp = t_1;
	elseif (p_m <= 3.6e-217)
		tmp = t_0;
	elseif (p_m <= 2.8e-194)
		tmp = t_1;
	elseif (p_m <= 3.4e-131)
		tmp = t_0;
	elseif (p_m <= 2.5e-38)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(x * p_m) / Float64(x / p_m))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	t_1 = 1.0 + (-0.5 * (p_m * (x * (x * p_m))));
	tmp = 0.0;
	if (p_m <= 2.05e-241)
		tmp = t_1;
	elseif (p_m <= 3.6e-217)
		tmp = t_0;
	elseif (p_m <= 2.8e-194)
		tmp = t_1;
	elseif (p_m <= 3.4e-131)
		tmp = t_0;
	elseif (p_m <= 2.5e-38)
		tmp = 1.0 + (-0.5 * ((x * p_m) / (x / p_m)));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[(p$95$m * N[(x * N[(x * p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p$95$m, 2.05e-241], t$95$1, If[LessEqual[p$95$m, 3.6e-217], t$95$0, If[LessEqual[p$95$m, 2.8e-194], t$95$1, If[LessEqual[p$95$m, 3.4e-131], t$95$0, If[LessEqual[p$95$m, 2.5e-38], N[(1.0 + N[(-0.5 * N[(N[(x * p$95$m), $MachinePrecision] / N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 46.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-149}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(p_m \cdot \left(x \cdot \left(x \cdot p_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -5e-149) (/ (- p_m) x) (+ 1.0 (* -0.5 (* p_m (* x (* x p_m)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-149) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0 + (-0.5 * (p_m * (x * (x * p_m))));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-149)) then
        tmp = -p_m / x
    else
        tmp = 1.0d0 + ((-0.5d0) * (p_m * (x * (x * p_m))))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-149) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0 + (-0.5 * (p_m * (x * (x * p_m))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5e-149:
		tmp = -p_m / x
	else:
		tmp = 1.0 + (-0.5 * (p_m * (x * (x * p_m))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -5e-149)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(p_m * Float64(x * Float64(x * p_m)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -5e-149)
		tmp = -p_m / x;
	else
		tmp = 1.0 + (-0.5 * (p_m * (x * (x * p_m))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -5e-149], N[((-p$95$m) / x), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(p$95$m * N[(x * N[(x * p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 48.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-266}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{x \cdot p_m}{\frac{x}{p_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -7.8e-266) (/ (- p_m) x) (+ 1.0 (* -0.5 (/ (* x p_m) (/ x p_m))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -7.8e-266) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0 + (-0.5 * ((x * p_m) / (x / p_m)));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-7.8d-266)) then
        tmp = -p_m / x
    else
        tmp = 1.0d0 + ((-0.5d0) * ((x * p_m) / (x / p_m)))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -7.8e-266) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0 + (-0.5 * ((x * p_m) / (x / p_m)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -7.8e-266:
		tmp = -p_m / x
	else:
		tmp = 1.0 + (-0.5 * ((x * p_m) / (x / p_m)))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -7.8e-266)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(x * p_m) / Float64(x / p_m))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -7.8e-266)
		tmp = -p_m / x;
	else
		tmp = 1.0 + (-0.5 * ((x * p_m) / (x / p_m)));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -7.8e-266], N[((-p$95$m) / x), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(x * p$95$m), $MachinePrecision] / N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 28.5% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-149}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;p_m\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -5e-149) (/ (- p_m) x) p_m))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-149) {
		tmp = -p_m / x;
	} else {
		tmp = p_m;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-149)) then
        tmp = -p_m / x
    else
        tmp = p_m
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-149) {
		tmp = -p_m / x;
	} else {
		tmp = p_m;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5e-149:
		tmp = -p_m / x
	else:
		tmp = p_m
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -5e-149)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = p_m;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -5e-149)
		tmp = -p_m / x;
	else
		tmp = p_m;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -5e-149], N[((-p$95$m) / x), $MachinePrecision], p$95$m]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 5.8% accurate, 215.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ p_m \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 p_m)

C

p_m = fabs(p);
double code(double p_m, double x) {
	return p_m;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = p_m
end function

Java

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

Python

p_m = math.fabs(p)
def code(p_m, x):
	return p_m

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := p$95$m

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Developer target: 79.6% 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 2024006 
(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))))))))