Given's Rotation SVD example

Percentage Accurate: 79.6% → 99.7%

Time: 14.1s

Alternatives: 4

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: 99.7% accurate, 0.7× speedup

Math

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

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 (* (/ x (hypot x (* p 2.0))) 0.5)))))

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 + ((x / hypot(x, (p * 2.0))) * 0.5)));
	}
	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 + ((x / Math.hypot(x, (p * 2.0))) * 0.5)));
	}
	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 + ((x / math.hypot(x, (p * 2.0))) * 0.5)))
	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(Float64(x / hypot(x, Float64(p * 2.0))) * 0.5)));
	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 + ((x / hypot(x, (p * 2.0))) * 0.5)));
	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[(N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * 0.5), $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 19.6%

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

      1. +-commutative 19.6%

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

      2. distribute-rgt-in 19.6%

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

      3. +-commutative 19.6%

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

      4. add-sqr-sqrt 19.6%

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

      5. hypot-def 19.6%

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

      6. associate-*l* 19.6%

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

      7. sqrt-prod 19.6%

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

      8. metadata-eval 19.6%

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

      9. sqrt-unprod 9.0%

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

      10. add-sqr-sqrt 19.6%

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

      11. metadata-eval 19.6%

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

   3. Applied egg-rr 19.6%

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

   4. Taylor expanded in x around -inf 64.0%

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

      1. associate-*r/ 64.0%

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

      2. neg-mul-1 64.0%

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

   6. Simplified 64.0%

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

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

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. +-commutative 99.7%

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

      4. add-sqr-sqrt 99.7%

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

      5. hypot-def 99.7%

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

      6. associate-*l* 99.7%

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

      7. sqrt-prod 99.7%

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

      8. metadata-eval 99.7%

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

      9. sqrt-unprod 50.0%

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

      10. add-sqr-sqrt 99.7%

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

      11. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 89.1%

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

Alternative 2: 68.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq 6.5 \cdot 10^{-189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.3 \cdot 10^{-126}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.9 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 5.3 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= p 6.5e-189)
     t_0
     (if (<= p 2.3e-126)
       1.0
       (if (<= p 3.3e-93)
         t_0
         (if (<= p 3.8e-48)
           (sqrt 0.5)
           (if (<= p 1.9e-29) t_0 (if (<= p 5.3e-14) 1.0 (sqrt 0.5)))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= 6.5e-189) {
		tmp = t_0;
	} else if (p <= 2.3e-126) {
		tmp = 1.0;
	} else if (p <= 3.3e-93) {
		tmp = t_0;
	} else if (p <= 3.8e-48) {
		tmp = sqrt(0.5);
	} else if (p <= 1.9e-29) {
		tmp = t_0;
	} else if (p <= 5.3e-14) {
		tmp = 1.0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -p / x
    if (p <= 6.5d-189) then
        tmp = t_0
    else if (p <= 2.3d-126) then
        tmp = 1.0d0
    else if (p <= 3.3d-93) then
        tmp = t_0
    else if (p <= 3.8d-48) then
        tmp = sqrt(0.5d0)
    else if (p <= 1.9d-29) then
        tmp = t_0
    else if (p <= 5.3d-14) then
        tmp = 1.0d0
    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 t_0 = -p / x;
	double tmp;
	if (p <= 6.5e-189) {
		tmp = t_0;
	} else if (p <= 2.3e-126) {
		tmp = 1.0;
	} else if (p <= 3.3e-93) {
		tmp = t_0;
	} else if (p <= 3.8e-48) {
		tmp = Math.sqrt(0.5);
	} else if (p <= 1.9e-29) {
		tmp = t_0;
	} else if (p <= 5.3e-14) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -p / x
	tmp = 0
	if p <= 6.5e-189:
		tmp = t_0
	elif p <= 2.3e-126:
		tmp = 1.0
	elif p <= 3.3e-93:
		tmp = t_0
	elif p <= 3.8e-48:
		tmp = math.sqrt(0.5)
	elif p <= 1.9e-29:
		tmp = t_0
	elif p <= 5.3e-14:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (p <= 6.5e-189)
		tmp = t_0;
	elseif (p <= 2.3e-126)
		tmp = 1.0;
	elseif (p <= 3.3e-93)
		tmp = t_0;
	elseif (p <= 3.8e-48)
		tmp = sqrt(0.5);
	elseif (p <= 1.9e-29)
		tmp = t_0;
	elseif (p <= 5.3e-14)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -p / x;
	tmp = 0.0;
	if (p <= 6.5e-189)
		tmp = t_0;
	elseif (p <= 2.3e-126)
		tmp = 1.0;
	elseif (p <= 3.3e-93)
		tmp = t_0;
	elseif (p <= 3.8e-48)
		tmp = sqrt(0.5);
	elseif (p <= 1.9e-29)
		tmp = t_0;
	elseif (p <= 5.3e-14)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	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[p, 6.5e-189], t$95$0, If[LessEqual[p, 2.3e-126], 1.0, If[LessEqual[p, 3.3e-93], t$95$0, If[LessEqual[p, 3.8e-48], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, 1.9e-29], t$95$0, If[LessEqual[p, 5.3e-14], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 6.5000000000000001e-189 or 2.30000000000000011e-126 < p < 3.3000000000000001e-93 or 3.80000000000000002e-48 < p < 1.89999999999999988e-29

   1. Initial program 71.1%

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

      1. +-commutative 71.1%

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

      2. distribute-rgt-in 71.1%

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

      3. +-commutative 71.1%

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

      4. add-sqr-sqrt 71.1%

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

      5. hypot-def 71.1%

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

      6. associate-*l* 71.1%

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

      7. sqrt-prod 71.1%

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

      8. metadata-eval 71.1%

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

      9. sqrt-unprod 11.2%

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

      10. add-sqr-sqrt 71.1%

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

      11. metadata-eval 71.1%

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

   3. Applied egg-rr 71.1%

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

   4. Taylor expanded in x around -inf 23.0%

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

      1. associate-*r/ 23.0%

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

      2. neg-mul-1 23.0%

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

   6. Simplified 23.0%

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

   if 6.5000000000000001e-189 < p < 2.30000000000000011e-126 or 1.89999999999999988e-29 < p < 5.3000000000000001e-14

   1. Initial program 70.4%

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

      1. distribute-lft-in 70.4%

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

      2. metadata-eval 70.4%

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

      3. flip3-+ 70.4%

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

   3. Applied egg-rr 70.4%

      \[\leadsto \sqrt{\color{blue}{\frac{0.125 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)}^{3}}{0.25 + \left(\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) - 0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)\right)}}} \]
   4. Step-by-step derivation

      1. associate-*l/ 70.4%

         \[\leadsto \sqrt{\frac{0.125 + {\color{blue}{\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}^{3}}{0.25 + \left(\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) - 0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)\right)}} \]

      2. distribute-rgt-out-- 70.4%

         \[\leadsto \sqrt{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \color{blue}{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 - 0.5\right)}}} \]

      3. associate-*l/ 70.4%

         \[\leadsto \sqrt{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 - 0.5\right)}} \]

      4. associate-*l/ 70.4%

         \[\leadsto \sqrt{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \left(\color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} - 0.5\right)}} \]

   5. Simplified 70.4%

      \[\leadsto \sqrt{\color{blue}{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 0.5\right)}}} \]

   6. Taylor expanded in x around inf 70.1%

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

   if 3.3000000000000001e-93 < p < 3.80000000000000002e-48 or 5.3000000000000001e-14 < p

   1. Initial program 87.5%

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

   2. Taylor expanded in x around 0 80.1%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 6.5 \cdot 10^{-189}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 2.3 \cdot 10^{-126}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.9 \cdot 10^{-29}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 5.3 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3: 55.1% accurate, 35.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -3.4e-149) (/ (- p) x) 1.0))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -3.4e-149) {
		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 <= (-3.4d-149)) 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 <= -3.4e-149) {
		tmp = -p / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -3.4e-149:
		tmp = -p / x
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -3.4e-149)
		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 <= -3.4e-149)
		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, -3.4e-149], N[((-p) / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.3999999999999999e-149

   1. Initial program 57.6%

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

      1. +-commutative 57.6%

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

      2. distribute-rgt-in 57.6%

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

      3. +-commutative 57.6%

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

      4. add-sqr-sqrt 57.6%

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

      5. hypot-def 57.6%

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

      6. associate-*l* 57.6%

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

      7. sqrt-prod 57.6%

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

      8. metadata-eval 57.6%

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

      9. sqrt-unprod 26.8%

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

      10. add-sqr-sqrt 57.6%

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

      11. metadata-eval 57.6%

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

   3. Applied egg-rr 57.6%

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

   4. Taylor expanded in x around -inf 35.1%

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

      1. associate-*r/ 35.1%

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

      2. neg-mul-1 35.1%

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

   6. Simplified 35.1%

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

   if -3.3999999999999999e-149 < 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. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. flip3-+ 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \sqrt{\color{blue}{\frac{0.125 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)}^{3}}{0.25 + \left(\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) - 0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)\right)}}} \]
   4. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto \sqrt{\frac{0.125 + {\color{blue}{\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}^{3}}{0.25 + \left(\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) - 0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)\right)}} \]

      2. distribute-rgt-out-- 100.0%

         \[\leadsto \sqrt{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \color{blue}{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 - 0.5\right)}}} \]

      3. associate-*l/ 100.0%

         \[\leadsto \sqrt{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 - 0.5\right)}} \]

      4. associate-*l/ 100.0%

         \[\leadsto \sqrt{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \left(\color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} - 0.5\right)}} \]

   5. Simplified 100.0%

      \[\leadsto \sqrt{\color{blue}{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 0.5\right)}}} \]

   6. Taylor expanded in x around inf 55.3%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 35.6% 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 76.0%

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

   1. distribute-lft-in 76.0%

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

   2. metadata-eval 76.0%

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

   3. flip3-+ 76.0%

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

3. Applied egg-rr 76.0%

   \[\leadsto \sqrt{\color{blue}{\frac{0.125 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)}^{3}}{0.25 + \left(\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) - 0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)\right)}}} \]
4. Step-by-step derivation

   1. associate-*l/ 76.0%

      \[\leadsto \sqrt{\frac{0.125 + {\color{blue}{\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}^{3}}{0.25 + \left(\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) - 0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)\right)}} \]

   2. distribute-rgt-out-- 76.0%

      \[\leadsto \sqrt{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \color{blue}{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right) \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 - 0.5\right)}}} \]

   3. associate-*l/ 76.0%

      \[\leadsto \sqrt{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} \cdot \left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 - 0.5\right)}} \]

   4. associate-*l/ 76.0%

      \[\leadsto \sqrt{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \left(\color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} - 0.5\right)}} \]

5. Simplified 76.0%

   \[\leadsto \sqrt{\color{blue}{\frac{0.125 + {\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{0.25 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 0.5\right)}}} \]

6. Taylor expanded in x around inf 30.9%

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

7. Final simplification 30.9%

   \[\leadsto 1 \]

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 2023274 
(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))))))))