Given's Rotation SVD example

Percentage Accurate: 79.1% → 99.7%

Time: 10.9s

Alternatives: 5

Speedup: 1.0×

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: 79.1% 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

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.9:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}\right)\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.9)
   (/ p_m (- x))
   (sqrt (* 0.5 (+ -1.0 (+ 2.0 (/ 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)))) <= -0.9) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 * (-1.0 + (2.0 + (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)))) <= -0.9) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 * (-1.0 + (2.0 + (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)))) <= -0.9:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 * (-1.0 + (2.0 + (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)))) <= -0.9)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(-1.0 + Float64(2.0 + 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)))) <= -0.9)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 * (-1.0 + (2.0 + (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], -0.9], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(-1.0 + N[(2.0 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 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 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.900000000000000022

   1. Initial program 18.8%

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

   3. Taylor expanded in x around -inf 55.6%

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

      1. mul-1-neg 55.6%

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

   5. Simplified 55.6%

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

      1. distribute-neg-frac 55.6%

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

      2. sqrt-unprod 56.4%

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

      3. metadata-eval 56.4%

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

      4. metadata-eval 56.4%

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

      5. *-rgt-identity 56.4%

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

   7. Applied egg-rr 56.4%

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

   if -0.900000000000000022 < (/.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. expm1-log1p-u 99.4%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}} \]

      2. expm1-undefine 99.4%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}} \]

      3. +-commutative 99.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)} - 1\right)} \]

      4. add-sqr-sqrt 99.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)} - 1\right)} \]

      5. hypot-define 99.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)} - 1\right)} \]

      6. associate-*l* 99.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)} - 1\right)} \]

      7. sqrt-prod 99.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)} - 1\right)} \]

      8. metadata-eval 99.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)} - 1\right)} \]

      9. sqrt-unprod 49.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)} - 1\right)} \]

      10. add-sqr-sqrt 99.4%

          \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)} - 1\right)} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} - 1\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 99.4%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \left(-1\right)\right)}} \]

      2. metadata-eval 99.4%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \color{blue}{-1}\right)} \]

      3. +-commutative 99.4%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}} \]

      4. log1p-undefine 100.0%

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

      5. rem-exp-log 100.0%

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

      6. associate-+r+ 100.0%

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

      7. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+47} \lor \neg \left(x \leq -1.45 \cdot 10^{-23}\right) \land x \leq -9 \cdot 10^{-55}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (or (<= x -4.8e+47) (and (not (<= x -1.45e-23)) (<= x -9e-55)))
   (/ p_m (- x))
   (sqrt (+ 0.5 (/ (* x 0.5) (hypot x (* p_m 2.0)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x <= -4.8e+47) || (!(x <= -1.45e-23) && (x <= -9e-55))) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 + ((x * 0.5) / 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 <= -4.8e+47) || (!(x <= -1.45e-23) && (x <= -9e-55))) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.5) / Math.hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x <= -4.8e+47) or (not (x <= -1.45e-23) and (x <= -9e-55)):
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 + ((x * 0.5) / math.hypot(x, (p_m * 2.0)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if ((x <= -4.8e+47) || (!(x <= -1.45e-23) && (x <= -9e-55)))
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.5) / 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 <= -4.8e+47) || (~((x <= -1.45e-23)) && (x <= -9e-55)))
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[Or[LessEqual[x, -4.8e+47], And[N[Not[LessEqual[x, -1.45e-23]], $MachinePrecision], LessEqual[x, -9e-55]]], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(x * 0.5), $MachinePrecision] / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000037e47 or -1.4500000000000001e-23 < x < -8.99999999999999941e-55

   1. Initial program 48.9%

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

   3. Taylor expanded in x around -inf 40.3%

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

      1. mul-1-neg 40.3%

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

   5. Simplified 40.3%

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

      1. distribute-neg-frac 40.3%

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

      2. sqrt-unprod 40.9%

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

      3. metadata-eval 40.9%

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

      4. metadata-eval 40.9%

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

      5. *-rgt-identity 40.9%

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

   7. Applied egg-rr 40.9%

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

   if -4.80000000000000037e47 < x < -1.4500000000000001e-23 or -8.99999999999999941e-55 < x

   1. Initial program 94.4%

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

      1. +-commutative 94.4%

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

      2. distribute-lft-in 94.4%

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

      3. associate-*r/ 94.4%

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

      4. +-commutative 94.4%

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

      5. add-sqr-sqrt 94.4%

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

      6. hypot-define 94.4%

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

      7. associate-*l* 94.4%

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

      8. sqrt-prod 94.4%

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

      9. metadata-eval 94.4%

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

      10. sqrt-unprod 47.6%

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

      11. add-sqr-sqrt 94.4%

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

      12. metadata-eval 94.4%

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

   4. Applied egg-rr 94.4%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+47} \lor \neg \left(x \leq -1.45 \cdot 10^{-23}\right) \land x \leq -9 \cdot 10^{-55}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 7 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 5.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 1.28 \cdot 10^{-48}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 7e-115)
   1.0
   (if (<= p_m 5.4e-109) (/ p_m (- x)) (if (<= p_m 1.28e-48) 1.0 (sqrt 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 7e-115) {
		tmp = 1.0;
	} else if (p_m <= 5.4e-109) {
		tmp = p_m / -x;
	} else if (p_m <= 1.28e-48) {
		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) :: tmp
    if (p_m <= 7d-115) then
        tmp = 1.0d0
    else if (p_m <= 5.4d-109) then
        tmp = p_m / -x
    else if (p_m <= 1.28d-48) 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 tmp;
	if (p_m <= 7e-115) {
		tmp = 1.0;
	} else if (p_m <= 5.4e-109) {
		tmp = p_m / -x;
	} else if (p_m <= 1.28e-48) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 7e-115:
		tmp = 1.0
	elif p_m <= 5.4e-109:
		tmp = p_m / -x
	elif p_m <= 1.28e-48:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 7e-115)
		tmp = 1.0;
	elseif (p_m <= 5.4e-109)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 1.28e-48)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 7e-115)
		tmp = 1.0;
	elseif (p_m <= 5.4e-109)
		tmp = p_m / -x;
	elseif (p_m <= 1.28e-48)
		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_] := If[LessEqual[p$95$m, 7e-115], 1.0, If[LessEqual[p$95$m, 5.4e-109], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 1.28e-48], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 7.0000000000000004e-115 or 5.4000000000000001e-109 < p < 1.28000000000000001e-48

   1. Initial program 83.6%

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

      1. +-commutative 83.6%

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

      2. distribute-lft-in 83.6%

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

      3. associate-*r/ 83.6%

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

      4. +-commutative 83.6%

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

      5. add-sqr-sqrt 83.6%

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

      6. hypot-define 83.6%

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

      7. associate-*l* 83.6%

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

      8. sqrt-prod 83.6%

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

      9. metadata-eval 83.6%

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

      10. sqrt-unprod 21.6%

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

      11. add-sqr-sqrt 83.6%

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

      12. metadata-eval 83.6%

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

   4. Applied egg-rr 83.6%

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

   5. Taylor expanded in x around inf 48.6%

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

   if 7.0000000000000004e-115 < p < 5.4000000000000001e-109

   1. Initial program 4.3%

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

   3. Taylor expanded in x around -inf 98.0%

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

      1. mul-1-neg 98.0%

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

   5. Simplified 98.0%

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

      1. distribute-neg-frac 98.0%

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

      2. sqrt-unprod 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 1.28000000000000001e-48 < p

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 0 82.3%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 7 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.28 \cdot 10^{-48}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.8% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -3.2e-103) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.2e-103) {
		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 <= (-3.2d-103)) 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 <= -3.2e-103) {
		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 <= -3.2e-103:
		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 <= -3.2e-103)
		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 <= -3.2e-103)
		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, -3.2e-103], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999976e-103

   1. Initial program 58.2%

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

   3. Taylor expanded in x around -inf 29.7%

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

      1. mul-1-neg 29.7%

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

   5. Simplified 29.7%

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

      1. distribute-neg-frac 29.7%

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

      2. sqrt-unprod 30.2%

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

      3. metadata-eval 30.2%

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

      4. metadata-eval 30.2%

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

      5. *-rgt-identity 30.2%

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

   7. Applied egg-rr 30.2%

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

   if -3.19999999999999976e-103 < x

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. associate-*r/ 99.4%

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

      4. +-commutative 99.4%

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

      5. add-sqr-sqrt 99.4%

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

      6. hypot-define 99.4%

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

      7. associate-*l* 99.4%

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

      8. sqrt-prod 99.4%

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

      9. metadata-eval 99.4%

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

      10. sqrt-unprod 49.1%

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

      11. add-sqr-sqrt 99.4%

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

      12. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around inf 56.7%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.8% accurate, 215.0× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 1.0)

C

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

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

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

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 1.0

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 1.0

Derivation

1. Initial program 84.4%

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

   1. +-commutative 84.4%

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

   2. distribute-lft-in 84.4%

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

   3. associate-*r/ 84.4%

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

   4. +-commutative 84.4%

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

   5. add-sqr-sqrt 84.4%

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

   6. hypot-define 84.4%

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

   7. associate-*l* 84.4%

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

   8. sqrt-prod 84.4%

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

   9. metadata-eval 84.4%

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

   10. sqrt-unprod 41.4%

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

   11. add-sqr-sqrt 84.4%

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

   12. metadata-eval 84.4%

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

4. Applied egg-rr 84.4%

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

5. Taylor expanded in x around inf 40.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer target: 79.1% 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 2024110 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (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))))))))