Given's Rotation SVD example

Percentage Accurate: 79.3% → 99.7%

Time: 10.6s

Alternatives: 9

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: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x / t$95$0), $MachinePrecision], 1/3], $MachinePrecision] + 1.0), $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 14.1%

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

   2. Taylor expanded in x around -inf 52.3%

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

      1. unpow2 52.3%

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

      2. associate-*r/ 52.3%

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

      3. unpow2 52.3%

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

   4. Simplified 52.3%

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

   5. Taylor expanded in p around -inf 57.7%

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

      1. associate-*r/ 57.7%

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

      2. mul-1-neg 57.7%

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

   7. Simplified 57.7%

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

   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. add-cbrt-cube 80.9%

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

      3. cbrt-prod 99.6%

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

      4. add-cbrt-cube 88.6%

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

      5. add-sqr-sqrt 88.6%

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

      6. cbrt-prod 99.7%

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

      7. times-frac 99.7%

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

      8. cbrt-div 99.7%

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

      9. fma-def 99.7%

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

   3. Applied egg-rr 99.8%

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

4. Final simplification 89.2%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
   (/ (- p) x)
   (exp (* 0.5 (log (fma (/ x (hypot x (* p 2.0))) 0.5 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 = exp((0.5 * log(fma((x / hypot(x, (p * 2.0))), 0.5, 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 = exp(Float64(0.5 * log(fma(Float64(x / hypot(x, Float64(p * 2.0))), 0.5, 0.5))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 14.1%

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

   2. Taylor expanded in x around -inf 52.3%

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

      1. unpow2 52.3%

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

      2. associate-*r/ 52.3%

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

      3. unpow2 52.3%

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

   4. Simplified 52.3%

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

   5. Taylor expanded in p around -inf 57.7%

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

      1. associate-*r/ 57.7%

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

      2. mul-1-neg 57.7%

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

   7. Simplified 57.7%

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

   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. pow1/2 99.7%

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

      2. pow-to-exp 99.7%

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

   3. Applied egg-rr 99.8%

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

4. Final simplification 89.2%

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

Alternative 3: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 14.1%

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

   2. Taylor expanded in x around -inf 52.3%

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

      1. unpow2 52.3%

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

      2. associate-*r/ 52.3%

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

      3. unpow2 52.3%

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

   4. Simplified 52.3%

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

   5. Taylor expanded in p around -inf 57.7%

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

      1. associate-*r/ 57.7%

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

      2. mul-1-neg 57.7%

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

   7. Simplified 57.7%

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

   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. add-sqr-sqrt 99.7%

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

      2. hypot-def 99.7%

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

      3. associate-*l* 99.7%

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

      4. sqrt-prod 99.7%

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

      5. metadata-eval 99.7%

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

      6. sqrt-unprod 52.5%

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

      7. add-sqr-sqrt 99.7%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 89.2%

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

Alternative 4: 67.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 3.2 \cdot 10^{-267}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 0.00146:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 29000000000000:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= p 3.2e-267)
   1.0
   (if (<= p 1.8e-81)
     (/ (- p) x)
     (if (<= p 0.00146)
       (sqrt 0.5)
       (if (<= p 29000000000000.0)
         (sqrt (* 0.5 (* 2.0 (* (/ p x) (/ p x)))))
         (sqrt (* 0.5 (+ 1.0 (/ x (* p 2.0))))))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 3.2e-267) {
		tmp = 1.0;
	} else if (p <= 1.8e-81) {
		tmp = -p / x;
	} else if (p <= 0.00146) {
		tmp = sqrt(0.5);
	} else if (p <= 29000000000000.0) {
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	}
	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 (p <= 3.2d-267) then
        tmp = 1.0d0
    else if (p <= 1.8d-81) then
        tmp = -p / x
    else if (p <= 0.00146d0) then
        tmp = sqrt(0.5d0)
    else if (p <= 29000000000000.0d0) then
        tmp = sqrt((0.5d0 * (2.0d0 * ((p / x) * (p / x)))))
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (p * 2.0d0)))))
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (p <= 3.2e-267) {
		tmp = 1.0;
	} else if (p <= 1.8e-81) {
		tmp = -p / x;
	} else if (p <= 0.00146) {
		tmp = Math.sqrt(0.5);
	} else if (p <= 29000000000000.0) {
		tmp = Math.sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 3.2e-267:
		tmp = 1.0
	elif p <= 1.8e-81:
		tmp = -p / x
	elif p <= 0.00146:
		tmp = math.sqrt(0.5)
	elif p <= 29000000000000.0:
		tmp = math.sqrt((0.5 * (2.0 * ((p / x) * (p / x)))))
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (p * 2.0)))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 3.2e-267)
		tmp = 1.0;
	elseif (p <= 1.8e-81)
		tmp = Float64(Float64(-p) / x);
	elseif (p <= 0.00146)
		tmp = sqrt(0.5);
	elseif (p <= 29000000000000.0)
		tmp = sqrt(Float64(0.5 * Float64(2.0 * Float64(Float64(p / x) * Float64(p / x)))));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(p * 2.0)))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 3.2e-267)
		tmp = 1.0;
	elseif (p <= 1.8e-81)
		tmp = -p / x;
	elseif (p <= 0.00146)
		tmp = sqrt(0.5);
	elseif (p <= 29000000000000.0)
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	else
		tmp = sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[p, 3.2e-267], 1.0, If[LessEqual[p, 1.8e-81], N[((-p) / x), $MachinePrecision], If[LessEqual[p, 0.00146], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, 29000000000000.0], N[Sqrt[N[(0.5 * N[(2.0 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(p * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if p < 3.19999999999999986e-267

   1. Initial program 77.8%

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

      1. add-sqr-sqrt 77.8%

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

      2. hypot-def 77.9%

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

      3. associate-*l* 77.9%

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

      4. sqrt-prod 77.9%

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

      5. metadata-eval 77.9%

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

      6. sqrt-unprod 6.2%

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

      7. add-sqr-sqrt 77.9%

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

   3. Applied egg-rr 77.9%

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

      1. add-cbrt-cube 77.8%

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

      2. pow1/3 77.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 77.9%

         \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      4. pow1 77.9%

         \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 77.9%

         \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 77.8%

         \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. distribute-lft-in 77.8%

         \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 77.8%

         \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 77.8%

         \[\leadsto {\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{{\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around inf 35.1%

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

   if 3.19999999999999986e-267 < p < 1.7999999999999999e-81

   1. Initial program 48.8%

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

   2. Taylor expanded in x around -inf 22.7%

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

      1. unpow2 22.7%

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

      2. associate-*r/ 22.7%

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

      3. unpow2 22.7%

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

   4. Simplified 22.7%

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

   5. Taylor expanded in p around -inf 57.8%

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

      1. associate-*r/ 57.8%

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

      2. mul-1-neg 57.8%

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

   7. Simplified 57.8%

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

   if 1.7999999999999999e-81 < p < 0.0014599999999999999

   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. Taylor expanded in x around 0 50.8%

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

   if 0.0014599999999999999 < p < 2.9e13

   1. Initial program 52.0%

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

   2. Taylor expanded in x around -inf 53.3%

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

      1. unpow2 53.3%

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

      2. unpow2 53.3%

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

      3. times-frac 53.3%

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

   4. Simplified 53.3%

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

   if 2.9e13 < p

   1. Initial program 95.6%

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

   2. Taylor expanded in p around inf 89.3%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{2 \cdot p}}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.2 \cdot 10^{-267}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 0.00146:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 29000000000000:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2}\right)}\\ \end{array} \]

Alternative 5: 67.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq 7 \cdot 10^{-268}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.05 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 0.000205:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= p 7e-268)
     1.0
     (if (<= p 1.05e-80)
       t_0
       (if (<= p 0.000205)
         (sqrt 0.5)
         (if (<= p 1.3e+14) t_0 (sqrt (* 0.5 (+ 1.0 (/ x (* p 2.0)))))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= 7e-268) {
		tmp = 1.0;
	} else if (p <= 1.05e-80) {
		tmp = t_0;
	} else if (p <= 0.000205) {
		tmp = sqrt(0.5);
	} else if (p <= 1.3e+14) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -p / x
    if (p <= 7d-268) then
        tmp = 1.0d0
    else if (p <= 1.05d-80) then
        tmp = t_0
    else if (p <= 0.000205d0) then
        tmp = sqrt(0.5d0)
    else if (p <= 1.3d+14) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (p * 2.0d0)))))
    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 <= 7e-268) {
		tmp = 1.0;
	} else if (p <= 1.05e-80) {
		tmp = t_0;
	} else if (p <= 0.000205) {
		tmp = Math.sqrt(0.5);
	} else if (p <= 1.3e+14) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -p / x
	tmp = 0
	if p <= 7e-268:
		tmp = 1.0
	elif p <= 1.05e-80:
		tmp = t_0
	elif p <= 0.000205:
		tmp = math.sqrt(0.5)
	elif p <= 1.3e+14:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (p * 2.0)))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (p <= 7e-268)
		tmp = 1.0;
	elseif (p <= 1.05e-80)
		tmp = t_0;
	elseif (p <= 0.000205)
		tmp = sqrt(0.5);
	elseif (p <= 1.3e+14)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(p * 2.0)))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -p / x;
	tmp = 0.0;
	if (p <= 7e-268)
		tmp = 1.0;
	elseif (p <= 1.05e-80)
		tmp = t_0;
	elseif (p <= 0.000205)
		tmp = sqrt(0.5);
	elseif (p <= 1.3e+14)
		tmp = t_0;
	else
		tmp = sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	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, 7e-268], 1.0, If[LessEqual[p, 1.05e-80], t$95$0, If[LessEqual[p, 0.000205], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, 1.3e+14], t$95$0, N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(p * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 7.00000000000000011e-268

   1. Initial program 77.8%

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

      1. add-sqr-sqrt 77.8%

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

      2. hypot-def 77.9%

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

      3. associate-*l* 77.9%

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

      4. sqrt-prod 77.9%

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

      5. metadata-eval 77.9%

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

      6. sqrt-unprod 6.2%

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

      7. add-sqr-sqrt 77.9%

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

   3. Applied egg-rr 77.9%

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

      1. add-cbrt-cube 77.8%

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

      2. pow1/3 77.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 77.9%

         \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      4. pow1 77.9%

         \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 77.9%

         \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 77.8%

         \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. distribute-lft-in 77.8%

         \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 77.8%

         \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 77.8%

         \[\leadsto {\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{{\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around inf 35.1%

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

   if 7.00000000000000011e-268 < p < 1.05000000000000001e-80 or 2.05e-4 < p < 1.3e14

   1. Initial program 49.1%

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

   2. Taylor expanded in x around -inf 25.7%

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

      1. unpow2 25.7%

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

      2. associate-*r/ 25.7%

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

      3. unpow2 25.7%

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

   4. Simplified 25.7%

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

   5. Taylor expanded in p around -inf 57.1%

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

      1. associate-*r/ 57.1%

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

      2. mul-1-neg 57.1%

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

   7. Simplified 57.1%

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

   if 1.05000000000000001e-80 < p < 2.05e-4

   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. Taylor expanded in x around 0 50.8%

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

   if 1.3e14 < p

   1. Initial program 95.6%

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

   2. Taylor expanded in p around inf 89.3%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{2 \cdot p}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 7 \cdot 10^{-268}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.05 \cdot 10^{-80}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 0.000205:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2}\right)}\\ \end{array} \]

Alternative 6: 67.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 2.6 \cdot 10^{-269}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 6.5 \cdot 10^{-82} \lor \neg \left(p \leq 0.00083\right) \land p \leq 29000000000000:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= p 2.6e-269)
   1.0
   (if (or (<= p 6.5e-82) (and (not (<= p 0.00083)) (<= p 29000000000000.0)))
     (/ (- p) x)
     (sqrt 0.5))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 2.6e-269) {
		tmp = 1.0;
	} else if ((p <= 6.5e-82) || (!(p <= 0.00083) && (p <= 29000000000000.0))) {
		tmp = -p / x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 2.6d-269) then
        tmp = 1.0d0
    else if ((p <= 6.5d-82) .or. (.not. (p <= 0.00083d0)) .and. (p <= 29000000000000.0d0)) then
        tmp = -p / x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (p <= 2.6e-269) {
		tmp = 1.0;
	} else if ((p <= 6.5e-82) || (!(p <= 0.00083) && (p <= 29000000000000.0))) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 2.6e-269:
		tmp = 1.0
	elif (p <= 6.5e-82) or (not (p <= 0.00083) and (p <= 29000000000000.0)):
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 2.6e-269)
		tmp = 1.0;
	elseif ((p <= 6.5e-82) || (!(p <= 0.00083) && (p <= 29000000000000.0)))
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 2.6e-269)
		tmp = 1.0;
	elseif ((p <= 6.5e-82) || (~((p <= 0.00083)) && (p <= 29000000000000.0)))
		tmp = -p / x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[p, 2.6e-269], 1.0, If[Or[LessEqual[p, 6.5e-82], And[N[Not[LessEqual[p, 0.00083]], $MachinePrecision], LessEqual[p, 29000000000000.0]]], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.6e-269

   1. Initial program 77.8%

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

      1. add-sqr-sqrt 77.8%

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

      2. hypot-def 77.9%

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

      3. associate-*l* 77.9%

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

      4. sqrt-prod 77.9%

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

      5. metadata-eval 77.9%

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

      6. sqrt-unprod 6.2%

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

      7. add-sqr-sqrt 77.9%

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

   3. Applied egg-rr 77.9%

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

      1. add-cbrt-cube 77.8%

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

      2. pow1/3 77.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 77.9%

         \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      4. pow1 77.9%

         \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 77.9%

         \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 77.8%

         \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. distribute-lft-in 77.8%

         \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 77.8%

         \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 77.8%

         \[\leadsto {\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{{\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around inf 35.1%

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

   if 2.6e-269 < p < 6.4999999999999997e-82 or 8.3000000000000001e-4 < p < 2.9e13

   1. Initial program 49.1%

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

   2. Taylor expanded in x around -inf 25.7%

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

      1. unpow2 25.7%

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

      2. associate-*r/ 25.7%

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

      3. unpow2 25.7%

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

   4. Simplified 25.7%

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

   5. Taylor expanded in p around -inf 57.1%

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

      1. associate-*r/ 57.1%

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

      2. mul-1-neg 57.1%

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

   7. Simplified 57.1%

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

   if 6.4999999999999997e-82 < p < 8.3000000000000001e-4 or 2.9e13 < p

   1. Initial program 93.9%

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

   2. Taylor expanded in x around 0 83.5%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.6 \cdot 10^{-269}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 6.5 \cdot 10^{-82} \lor \neg \left(p \leq 0.00083\right) \land p \leq 29000000000000:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 55.8% accurate, 35.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-127}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -1.65e-127) (/ (- p) x) 1.0))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -1.65e-127) {
		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 <= (-1.65d-127)) 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 <= -1.65e-127) {
		tmp = -p / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -1.65e-127:
		tmp = -p / x
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -1.65e-127)
		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 <= -1.65e-127)
		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, -1.65e-127], N[((-p) / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.6499999999999999e-127

   1. Initial program 58.7%

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

   2. Taylor expanded in x around -inf 28.7%

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

      1. unpow2 28.7%

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

      2. associate-*r/ 28.7%

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

      3. unpow2 28.7%

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

   4. Simplified 28.7%

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

   5. Taylor expanded in p around -inf 30.0%

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

      1. associate-*r/ 30.0%

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

      2. mul-1-neg 30.0%

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

   7. Simplified 30.0%

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

   if -1.6499999999999999e-127 < x

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.2%

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

      2. hypot-def 99.2%

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

      3. associate-*l* 99.2%

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

      4. sqrt-prod 99.2%

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

      5. metadata-eval 99.2%

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

      6. sqrt-unprod 49.2%

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

      7. add-sqr-sqrt 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. add-cbrt-cube 99.2%

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

      2. pow1/3 99.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 99.2%

         \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      4. pow1 99.2%

         \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 99.2%

         \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 99.2%

         \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. distribute-lft-in 99.2%

         \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 99.2%

         \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 99.2%

         \[\leadsto {\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{{\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around inf 56.5%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-127}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 37.6% accurate, 42.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -3.6e+85) (/ p x) 1.0))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -3.6e+85) {
		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 <= (-3.6d+85)) 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 <= -3.6e+85) {
		tmp = p / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -3.6e+85:
		tmp = p / x
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -3.6e+85)
		tmp = 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 <= -3.6e+85)
		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, -3.6e+85], N[(p / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.5999999999999998e85

   1. Initial program 49.5%

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

   2. Taylor expanded in x around -inf 55.9%

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

      1. unpow2 55.9%

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

      2. associate-*r/ 55.9%

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

      3. unpow2 55.9%

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

   4. Simplified 55.9%

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

   5. Taylor expanded in p around 0 44.8%

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

   if -3.5999999999999998e85 < x

   1. Initial program 81.9%

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

      1. add-sqr-sqrt 81.9%

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

      2. hypot-def 81.9%

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

      3. associate-*l* 81.9%

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

      4. sqrt-prod 81.9%

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

      5. metadata-eval 81.9%

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

      6. sqrt-unprod 42.6%

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

      7. add-sqr-sqrt 81.9%

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

   3. Applied egg-rr 81.9%

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

      1. add-cbrt-cube 81.8%

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

      2. pow1/3 81.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 81.9%

         \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      4. pow1 81.9%

         \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 81.9%

         \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 81.9%

         \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. distribute-lft-in 81.9%

         \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 81.9%

         \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 81.9%

         \[\leadsto {\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{{\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in x around inf 37.1%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 36.2% 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 78.3%

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

   1. add-sqr-sqrt 78.3%

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

   2. hypot-def 78.3%

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

   3. associate-*l* 78.3%

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

   4. sqrt-prod 78.3%

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

   5. metadata-eval 78.3%

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

   6. sqrt-unprod 40.9%

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

   7. add-sqr-sqrt 78.3%

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

3. Applied egg-rr 78.3%

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

   1. add-cbrt-cube 78.3%

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

   2. pow1/3 78.3%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

   3. add-sqr-sqrt 78.3%

      \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

   4. pow1 78.3%

      \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

   5. pow1/2 78.3%

      \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

   6. pow-prod-up 78.3%

      \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

   7. distribute-lft-in 78.3%

      \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

   8. metadata-eval 78.3%

      \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

   9. metadata-eval 78.3%

      \[\leadsto {\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

5. Applied egg-rr 78.3%

   \[\leadsto \color{blue}{{\left({\left(0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in x around inf 33.9%

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

7. Final simplification 33.9%

   \[\leadsto 1 \]

Developer target: 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 2023229 
(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))))))))