Given's Rotation SVD example

Percentage Accurate: 79.3% → 99.8%

Time: 10.4s

Alternatives: 7

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: 79.3% 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

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

FPCore

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

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)))) <= -0.5) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p_m * 2.0), x)))));
	}
	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)))) <= -0.5) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p_m * 2.0), x)))));
	}
	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)))) <= -0.5:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p_m * 2.0), x)))))
	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)))) <= -0.5)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p_m * 2.0), x)))));
	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)))) <= -0.5)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p_m * 2.0), x)))));
	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], -0.5], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p$95$m * 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 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.5

   1. Initial program 19.5%

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

   3. Taylor expanded in x around -inf 58.2%

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

      1. mul-1-neg 58.2%

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

      2. distribute-neg-frac2 58.2%

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

      3. associate-*r* 58.4%

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

      4. *-commutative 58.4%

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

   5. Simplified 58.4%

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

   6. Taylor expanded in p around 0 58.2%

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

      1. mul-1-neg 58.2%

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

      2. associate-/l* 58.2%

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

      3. distribute-rgt-neg-in 58.2%

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

      4. associate-/l* 58.5%

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

   8. Simplified 58.5%

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

      1. distribute-rgt-neg-out 58.5%

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

      2. neg-sub0 58.5%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 58.2%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 58.8%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 58.8%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 58.8%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 59.0%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 59.0%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 59.0%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   10. Applied egg-rr 59.0%

       \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   11. Step-by-step derivation

       1. neg-sub0 59.0%

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

       2. distribute-neg-frac 59.0%

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

   12. Simplified 59.0%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 50.8%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 90.2%

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

Alternative 2: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 4.1 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 3.4 \cdot 10^{-257}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 8.2 \cdot 10^{-124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 9.2 \cdot 10^{-17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 4.1e-286)
     t_0
     (if (<= p_m 3.4e-257)
       1.0
       (if (<= p_m 8.2e-124) t_0 (if (<= p_m 9.2e-17) 1.0 (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 4.1e-286) {
		tmp = t_0;
	} else if (p_m <= 3.4e-257) {
		tmp = 1.0;
	} else if (p_m <= 8.2e-124) {
		tmp = t_0;
	} else if (p_m <= 9.2e-17) {
		tmp = 1.0;
	} 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) :: tmp
    t_0 = p_m / -x
    if (p_m <= 4.1d-286) then
        tmp = t_0
    else if (p_m <= 3.4d-257) then
        tmp = 1.0d0
    else if (p_m <= 8.2d-124) then
        tmp = t_0
    else if (p_m <= 9.2d-17) then
        tmp = 1.0d0
    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 tmp;
	if (p_m <= 4.1e-286) {
		tmp = t_0;
	} else if (p_m <= 3.4e-257) {
		tmp = 1.0;
	} else if (p_m <= 8.2e-124) {
		tmp = t_0;
	} else if (p_m <= 9.2e-17) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if p_m <= 4.1e-286:
		tmp = t_0
	elif p_m <= 3.4e-257:
		tmp = 1.0
	elif p_m <= 8.2e-124:
		tmp = t_0
	elif p_m <= 9.2e-17:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 4.1e-286)
		tmp = t_0;
	elseif (p_m <= 3.4e-257)
		tmp = 1.0;
	elseif (p_m <= 8.2e-124)
		tmp = t_0;
	elseif (p_m <= 9.2e-17)
		tmp = 1.0;
	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;
	tmp = 0.0;
	if (p_m <= 4.1e-286)
		tmp = t_0;
	elseif (p_m <= 3.4e-257)
		tmp = 1.0;
	elseif (p_m <= 8.2e-124)
		tmp = t_0;
	elseif (p_m <= 9.2e-17)
		tmp = 1.0;
	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]}, If[LessEqual[p$95$m, 4.1e-286], t$95$0, If[LessEqual[p$95$m, 3.4e-257], 1.0, If[LessEqual[p$95$m, 8.2e-124], t$95$0, If[LessEqual[p$95$m, 9.2e-17], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 4.1e-286 or 3.3999999999999998e-257 < p < 8.2000000000000008e-124

   1. Initial program 74.4%

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

   3. Taylor expanded in x around -inf 19.3%

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

      1. mul-1-neg 19.3%

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

      2. distribute-neg-frac2 19.3%

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

      3. associate-*r* 19.4%

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

      4. *-commutative 19.4%

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

   5. Simplified 19.4%

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

   6. Taylor expanded in p around 0 19.3%

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

      1. mul-1-neg 19.3%

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

      2. associate-/l* 19.3%

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

      3. distribute-rgt-neg-in 19.3%

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

      4. associate-/l* 19.4%

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

   8. Simplified 19.4%

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

      1. distribute-rgt-neg-out 19.4%

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

      2. neg-sub0 19.4%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 19.3%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 19.4%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 19.4%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 19.4%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 19.5%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 19.5%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 19.5%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   10. Applied egg-rr 19.5%

       \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   11. Step-by-step derivation

       1. neg-sub0 19.5%

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

       2. distribute-neg-frac 19.5%

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

   12. Simplified 19.5%

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

   if 4.1e-286 < p < 3.3999999999999998e-257 or 8.2000000000000008e-124 < p < 9.20000000000000035e-17

   1. Initial program 84.1%

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

   3. Taylor expanded in x around inf 71.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

      1. metadata-eval 71.8%

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

      2. metadata-eval 71.8%

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

   5. Applied egg-rr 71.8%

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

   if 9.20000000000000035e-17 < p

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0 85.9%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.1 \cdot 10^{-286}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 3.4 \cdot 10^{-257}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8.2 \cdot 10^{-124}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 9.2 \cdot 10^{-17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.2% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -4.8e-113) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -4.8e-113) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	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 <= (-4.8d-113)) then
        tmp = p_m / -x
    else
        tmp = 1.0d0
    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 <= -4.8e-113) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -4.8e-113:
		tmp = p_m / -x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -4.8e-113)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -4.8e-113)
		tmp = p_m / -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -4.8e-113], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000024e-113

   1. Initial program 53.8%

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

   3. Taylor expanded in x around -inf 35.8%

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

      1. mul-1-neg 35.8%

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

      2. distribute-neg-frac2 35.8%

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

      3. associate-*r* 35.9%

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

      4. *-commutative 35.9%

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

   5. Simplified 35.9%

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

   6. Taylor expanded in p around 0 35.8%

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

      1. mul-1-neg 35.8%

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

      2. associate-/l* 35.8%

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

      3. distribute-rgt-neg-in 35.8%

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

      4. associate-/l* 36.0%

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

   8. Simplified 36.0%

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

      1. distribute-rgt-neg-out 36.0%

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

      2. neg-sub0 36.0%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 35.8%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 36.1%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 36.1%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 36.1%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 36.3%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 36.3%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 36.3%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   10. Applied egg-rr 36.3%

       \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   11. Step-by-step derivation

       1. neg-sub0 36.3%

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

       2. distribute-neg-frac 36.3%

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

   12. Simplified 36.3%

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

   if -4.80000000000000024e-113 < x

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf 47.4%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

      1. metadata-eval 47.4%

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

      2. metadata-eval 47.4%

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

   5. Applied egg-rr 47.4%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.4% accurate, 26.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+34}:\\ \;\;\;\;p\_m \cdot p\_m\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -5.8e+34) (* p_m p_m) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5.8e+34) {
		tmp = p_m * p_m;
	} else {
		tmp = 1.0;
	}
	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 <= (-5.8d+34)) then
        tmp = p_m * p_m
    else
        tmp = 1.0d0
    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 <= -5.8e+34) {
		tmp = p_m * p_m;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5.8e+34:
		tmp = p_m * p_m
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -5.8e+34)
		tmp = Float64(p_m * p_m);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -5.8e+34)
		tmp = p_m * p_m;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -5.8e+34], N[(p$95$m * p$95$m), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -5.8000000000000003e34

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 43.5%

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

      1. mul-1-neg 43.5%

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

      2. distribute-neg-frac2 43.5%

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

      3. associate-*r* 43.6%

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

      4. *-commutative 43.6%

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

   5. Simplified 43.6%

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

   7. Applied egg-rr 24.8%

      \[\leadsto \color{blue}{p \cdot p} \]

   if -5.8000000000000003e34 < x

   1. Initial program 85.5%

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

   3. Taylor expanded in x around inf 38.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

      1. metadata-eval 38.0%

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

      2. metadata-eval 38.0%

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

   5. Applied egg-rr 38.0%

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

Alternative 5: 37.2% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.66 \cdot 10^{+34}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -1.66e+34) 0.0 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.66e+34) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	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 <= (-1.66d+34)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0
    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 <= -1.66e+34) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1.66e+34:
		tmp = 0.0
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1.66e+34)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1.66e+34)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1.66e+34], 0.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.6599999999999999e34

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 24.0%

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

      1. neg-mul-1 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in x around 0 24.0%

      \[\leadsto \color{blue}{0} \]

   if -1.6599999999999999e34 < x

   1. Initial program 85.5%

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

   3. Taylor expanded in x around inf 38.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

      1. metadata-eval 38.0%

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

      2. metadata-eval 38.0%

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

   5. Applied egg-rr 38.0%

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

Alternative 6: 15.5% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 7.6 \cdot 10^{-220}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.125\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= p_m 7.6e-220) 0.0 0.125))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 7.6e-220) {
		tmp = 0.0;
	} else {
		tmp = 0.125;
	}
	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 (p_m <= 7.6d-220) then
        tmp = 0.0d0
    else
        tmp = 0.125d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 7.6e-220) {
		tmp = 0.0;
	} else {
		tmp = 0.125;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 7.6e-220:
		tmp = 0.0
	else:
		tmp = 0.125
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 7.6e-220)
		tmp = 0.0;
	else
		tmp = 0.125;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 7.6e-220)
		tmp = 0.0;
	else
		tmp = 0.125;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 7.6e-220], 0.0, 0.125]

Derivation

1. Split input into 2 regimes

2. if p < 7.60000000000000018e-220

   1. Initial program 79.8%

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

   3. Taylor expanded in x around -inf 9.7%

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

      1. neg-mul-1 9.7%

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

   5. Simplified 9.7%

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

   6. Taylor expanded in x around 0 9.7%

      \[\leadsto \color{blue}{0} \]

   if 7.60000000000000018e-220 < p

   1. Initial program 82.1%

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

   3. Taylor expanded in x around 0 65.4%

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

   5. Applied egg-rr 14.6%

      \[\leadsto \color{blue}{0.125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 6.6% accurate, 215.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ 0 \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 0.0)

C

p_m = fabs(p);
double code(double p_m, double x) {
	return 0.0;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return 0.0;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 0.0

Julia

p_m = abs(p)
function code(p_m, x)
	return 0.0
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = 0.0;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 0.0

Derivation

1. Initial program 80.8%

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

3. Taylor expanded in x around -inf 6.9%

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

   1. neg-mul-1 6.9%

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

5. Simplified 6.9%

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

6. Taylor expanded in x around 0 6.9%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Developer Target 1: 79.3% 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 2024136 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (! :herbie-platform default (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x))))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))