Given's Rotation SVD example

Percentage Accurate: 78.9% → 99.8%

Time: 10.0s

Alternatives: 7

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: 78.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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[Abs[N[(p / x), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[Log[N[Exp[N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $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 17.4%

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

      1. *-commutative 17.4%

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

      2. sqrt-prod 17.4%

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

      3. +-commutative 17.4%

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

      4. add-sqr-sqrt 17.4%

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

      5. hypot-def 17.4%

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

      6. associate-*l* 17.4%

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

      7. sqrt-prod 17.4%

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

      8. metadata-eval 17.4%

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

      9. sqrt-unprod 8.2%

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

      10. add-sqr-sqrt 17.4%

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

   3. Applied egg-rr 17.4%

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

   4. Taylor expanded in x around -inf 61.6%

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

      1. mul-1-neg 61.6%

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

      2. associate-/l* 61.5%

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

      3. distribute-neg-frac 61.5%

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

   6. Simplified 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-/r/ 61.4%

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

      3. add-sqr-sqrt 8.6%

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

      4. sqrt-unprod 16.5%

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

      5. sqr-neg 16.5%

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

      6. unpow2 16.5%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-prod 70.6%

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

      9. unpow2 70.6%

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

      10. sqrt-div 58.2%

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

      11. sqrt-prod 58.5%

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

      12. *-commutative 58.5%

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

      13. pow1/2 58.5%

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

      14. pow1/2 58.5%

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

      15. pow-prod-down 58.7%

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

   8. Applied egg-rr 66.7%

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

      1. unpow1/2 66.7%

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

      2. associate-*r* 66.7%

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

      3. metadata-eval 66.7%

         \[\leadsto \sqrt{\color{blue}{1} \cdot {\left(\frac{p}{x}\right)}^{2}} \]

      4. *-lft-identity 66.7%

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

      5. unpow2 66.7%

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

      6. rem-sqrt-square 100.0%

         \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

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

   1. Initial program 99.9%

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

      1. add-log-exp 99.9%

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

      2. +-commutative 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. hypot-def 99.9%

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

      5. associate-*l* 99.9%

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

      6. sqrt-prod 99.9%

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

      7. metadata-eval 99.9%

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

      8. sqrt-unprod 45.9%

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

      9. add-sqr-sqrt 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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[Abs[N[(p / x), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(0.5 * N[(2.0 + N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $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 17.4%

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

      1. *-commutative 17.4%

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

      2. sqrt-prod 17.4%

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

      3. +-commutative 17.4%

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

      4. add-sqr-sqrt 17.4%

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

      5. hypot-def 17.4%

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

      6. associate-*l* 17.4%

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

      7. sqrt-prod 17.4%

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

      8. metadata-eval 17.4%

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

      9. sqrt-unprod 8.2%

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

      10. add-sqr-sqrt 17.4%

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

   3. Applied egg-rr 17.4%

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

   4. Taylor expanded in x around -inf 61.6%

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

      1. mul-1-neg 61.6%

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

      2. associate-/l* 61.5%

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

      3. distribute-neg-frac 61.5%

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

   6. Simplified 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-/r/ 61.4%

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

      3. add-sqr-sqrt 8.6%

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

      4. sqrt-unprod 16.5%

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

      5. sqr-neg 16.5%

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

      6. unpow2 16.5%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-prod 70.6%

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

      9. unpow2 70.6%

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

      10. sqrt-div 58.2%

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

      11. sqrt-prod 58.5%

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

      12. *-commutative 58.5%

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

      13. pow1/2 58.5%

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

      14. pow1/2 58.5%

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

      15. pow-prod-down 58.7%

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

   8. Applied egg-rr 66.7%

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

      1. unpow1/2 66.7%

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

      2. associate-*r* 66.7%

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

      3. metadata-eval 66.7%

         \[\leadsto \sqrt{\color{blue}{1} \cdot {\left(\frac{p}{x}\right)}^{2}} \]

      4. *-lft-identity 66.7%

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

      5. unpow2 66.7%

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

      6. rem-sqrt-square 100.0%

         \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

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

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sqrt-prod 99.4%

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

      3. +-commutative 99.4%

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

      4. add-sqr-sqrt 99.4%

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

      5. hypot-def 99.4%

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

      6. associate-*l* 99.4%

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

      7. sqrt-prod 99.4%

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

      8. metadata-eval 99.4%

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

      9. sqrt-unprod 45.7%

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

      10. add-sqr-sqrt 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. expm1-log1p-u 99.4%

         \[\leadsto \sqrt{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}} \cdot \sqrt{0.5} \]

      2. expm1-udef 99.4%

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

      3. log1p-udef 99.4%

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

      4. add-exp-log 99.4%

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

      5. rem-log-exp 99.4%

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

      6. rem-log-exp 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

      1. expm1-log1p-u 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + \left(1 + \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + -1\right)\right)} \cdot \sqrt{0.5}\right)\right)} \]

      2. expm1-udef 98.8%

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

      3. sqrt-unprod 98.8%

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

      4. associate-+r+ 98.8%

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

      5. metadata-eval 98.8%

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

   9. Applied egg-rr 98.8%

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

       1. expm1-def 98.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(2 + \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + -1\right)\right) \cdot 0.5}\right)\right)} \]

       2. expm1-log1p 99.9%

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

       3. *-commutative 99.9%

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

       4. associate-+r+ 99.9%

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

       5. distribute-lft-in 99.9%

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

       6. metadata-eval 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+104} \lor \neg \left(x \leq -7.2 \cdot 10^{+47}\right) \land x \leq -9.2 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (or (<= x -2.4e+104) (and (not (<= x -7.2e+47)) (<= x -9.2e-29)))
   (fabs (/ p x))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

C

double code(double p, double x) {
	double tmp;
	if ((x <= -2.4e+104) || (!(x <= -7.2e+47) && (x <= -9.2e-29))) {
		tmp = fabs((p / x));
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	return tmp;
}

Java

public static double code(double p, double x) {
	double tmp;
	if ((x <= -2.4e+104) || (!(x <= -7.2e+47) && (x <= -9.2e-29))) {
		tmp = Math.abs((p / x));
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if (x <= -2.4e+104) or (not (x <= -7.2e+47) and (x <= -9.2e-29)):
		tmp = math.fabs((p / x))
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if ((x <= -2.4e+104) || (!(x <= -7.2e+47) && (x <= -9.2e-29)))
		tmp = abs(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

function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x <= -2.4e+104) || (~((x <= -7.2e+47)) && (x <= -9.2e-29)))
		tmp = abs((p / x));
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[Or[LessEqual[x, -2.4e+104], And[N[Not[LessEqual[x, -7.2e+47]], $MachinePrecision], LessEqual[x, -9.2e-29]]], N[Abs[N[(p / x), $MachinePrecision]], $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 x < -2.4e104 or -7.20000000000000015e47 < x < -9.19999999999999965e-29

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

      2. sqrt-prod 33.6%

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

      3. +-commutative 33.6%

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

      4. add-sqr-sqrt 33.6%

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

      5. hypot-def 33.6%

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

      6. associate-*l* 33.6%

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

      7. sqrt-prod 33.6%

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

      8. metadata-eval 33.6%

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

      9. sqrt-unprod 13.6%

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

      10. add-sqr-sqrt 33.6%

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

   3. Applied egg-rr 33.6%

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

   4. Taylor expanded in x around -inf 47.8%

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

      1. mul-1-neg 47.8%

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

      2. associate-/l* 47.7%

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

      3. distribute-neg-frac 47.7%

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

   6. Simplified 47.7%

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

      1. *-commutative 47.7%

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

      2. associate-/r/ 47.7%

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

      3. add-sqr-sqrt 4.5%

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

      4. sqrt-unprod 6.4%

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

      5. sqr-neg 6.4%

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

      6. unpow2 6.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-prod 58.4%

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

      9. unpow2 58.4%

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

      10. sqrt-div 47.0%

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

      11. sqrt-prod 47.4%

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

      12. *-commutative 47.4%

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

      13. pow1/2 47.4%

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

      14. pow1/2 47.4%

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

      15. pow-prod-down 47.5%

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

   8. Applied egg-rr 47.8%

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

      1. unpow1/2 47.8%

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

      2. associate-*r* 47.8%

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

      3. metadata-eval 47.8%

         \[\leadsto \sqrt{\color{blue}{1} \cdot {\left(\frac{p}{x}\right)}^{2}} \]

      4. *-lft-identity 47.8%

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

      5. unpow2 47.8%

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

      6. rem-sqrt-square 75.2%

         \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

   10. Simplified 75.2%

       \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

   if -2.4e104 < x < -7.20000000000000015e47 or -9.19999999999999965e-29 < x

   1. Initial program 90.8%

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

      1. add-sqr-sqrt 90.8%

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

      2. hypot-def 90.8%

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

      3. associate-*l* 90.8%

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

      4. sqrt-prod 90.8%

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

      5. metadata-eval 90.8%

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

      6. sqrt-unprod 42.3%

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

      7. add-sqr-sqrt 90.8%

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

   3. Applied egg-rr 90.8%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+104} \lor \neg \left(x \leq -7.2 \cdot 10^{+47}\right) \land x \leq -9.2 \cdot 10^{-29}:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 4: 48.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{p}{x}\right|\\ \mathbf{if}\;p \leq 5 \cdot 10^{-254}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 4.9 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.46 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 5 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 10^{-37}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x}{p} \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (fabs (/ p x))))
   (if (<= p 5e-254)
     t_0
     (if (<= p 4.9e-132)
       1.0
       (if (<= p 1.46e-120)
         t_0
         (if (<= p 5e-89)
           1.0
           (if (<= p 1e-37)
             (sqrt 0.5)
             (if (<= p 3.3e+28) t_0 (sqrt (+ 0.5 (* (/ x p) 0.25)))))))))))

C

double code(double p, double x) {
	double t_0 = fabs((p / x));
	double tmp;
	if (p <= 5e-254) {
		tmp = t_0;
	} else if (p <= 4.9e-132) {
		tmp = 1.0;
	} else if (p <= 1.46e-120) {
		tmp = t_0;
	} else if (p <= 5e-89) {
		tmp = 1.0;
	} else if (p <= 1e-37) {
		tmp = sqrt(0.5);
	} else if (p <= 3.3e+28) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 + ((x / p) * 0.25)));
	}
	return tmp;
}

Fortran

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 = abs((p / x))
    if (p <= 5d-254) then
        tmp = t_0
    else if (p <= 4.9d-132) then
        tmp = 1.0d0
    else if (p <= 1.46d-120) then
        tmp = t_0
    else if (p <= 5d-89) then
        tmp = 1.0d0
    else if (p <= 1d-37) then
        tmp = sqrt(0.5d0)
    else if (p <= 3.3d+28) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 + ((x / p) * 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double t_0 = Math.abs((p / x));
	double tmp;
	if (p <= 5e-254) {
		tmp = t_0;
	} else if (p <= 4.9e-132) {
		tmp = 1.0;
	} else if (p <= 1.46e-120) {
		tmp = t_0;
	} else if (p <= 5e-89) {
		tmp = 1.0;
	} else if (p <= 1e-37) {
		tmp = Math.sqrt(0.5);
	} else if (p <= 3.3e+28) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 + ((x / p) * 0.25)));
	}
	return tmp;
}

Python

def code(p, x):
	t_0 = math.fabs((p / x))
	tmp = 0
	if p <= 5e-254:
		tmp = t_0
	elif p <= 4.9e-132:
		tmp = 1.0
	elif p <= 1.46e-120:
		tmp = t_0
	elif p <= 5e-89:
		tmp = 1.0
	elif p <= 1e-37:
		tmp = math.sqrt(0.5)
	elif p <= 3.3e+28:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 + ((x / p) * 0.25)))
	return tmp

Julia

function code(p, x)
	t_0 = abs(Float64(p / x))
	tmp = 0.0
	if (p <= 5e-254)
		tmp = t_0;
	elseif (p <= 4.9e-132)
		tmp = 1.0;
	elseif (p <= 1.46e-120)
		tmp = t_0;
	elseif (p <= 5e-89)
		tmp = 1.0;
	elseif (p <= 1e-37)
		tmp = sqrt(0.5);
	elseif (p <= 3.3e+28)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x / p) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	t_0 = abs((p / x));
	tmp = 0.0;
	if (p <= 5e-254)
		tmp = t_0;
	elseif (p <= 4.9e-132)
		tmp = 1.0;
	elseif (p <= 1.46e-120)
		tmp = t_0;
	elseif (p <= 5e-89)
		tmp = 1.0;
	elseif (p <= 1e-37)
		tmp = sqrt(0.5);
	elseif (p <= 3.3e+28)
		tmp = t_0;
	else
		tmp = sqrt((0.5 + ((x / p) * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := Block[{t$95$0 = N[Abs[N[(p / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[p, 5e-254], t$95$0, If[LessEqual[p, 4.9e-132], 1.0, If[LessEqual[p, 1.46e-120], t$95$0, If[LessEqual[p, 5e-89], 1.0, If[LessEqual[p, 1e-37], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, 3.3e+28], t$95$0, N[Sqrt[N[(0.5 + N[(N[(x / p), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 5.0000000000000003e-254 or 4.89999999999999981e-132 < p < 1.4599999999999999e-120 or 1.00000000000000007e-37 < p < 3.3e28

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

      2. sqrt-prod 73.5%

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

      3. +-commutative 73.5%

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

      4. add-sqr-sqrt 73.5%

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

      5. hypot-def 73.5%

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

      6. associate-*l* 73.5%

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

      7. sqrt-prod 73.5%

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

      8. metadata-eval 73.5%

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

      9. sqrt-unprod 9.5%

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

      10. add-sqr-sqrt 73.5%

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

   3. Applied egg-rr 73.5%

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

   4. Taylor expanded in x around -inf 19.1%

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

      1. mul-1-neg 19.1%

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

      2. associate-/l* 19.1%

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

      3. distribute-neg-frac 19.1%

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

   6. Simplified 19.1%

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

      1. *-commutative 19.1%

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

      2. associate-/r/ 19.1%

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

      3. add-sqr-sqrt 5.8%

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

      4. sqrt-unprod 8.3%

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

      5. sqr-neg 8.3%

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

      6. unpow2 8.3%

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

      7. add-sqr-sqrt 2.2%

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

      8. sqrt-prod 25.3%

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

      9. unpow2 25.3%

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

      10. sqrt-div 21.8%

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

      11. sqrt-prod 21.9%

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

      12. *-commutative 21.9%

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

      13. pow1/2 21.9%

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

      14. pow1/2 21.9%

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

      15. pow-prod-down 21.9%

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

   8. Applied egg-rr 25.4%

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

      1. unpow1/2 25.4%

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

      2. associate-*r* 25.4%

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

      3. metadata-eval 25.4%

         \[\leadsto \sqrt{\color{blue}{1} \cdot {\left(\frac{p}{x}\right)}^{2}} \]

      4. *-lft-identity 25.4%

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

      5. unpow2 25.4%

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

      6. rem-sqrt-square 36.0%

         \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

   10. Simplified 36.0%

       \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

   if 5.0000000000000003e-254 < p < 4.89999999999999981e-132 or 1.4599999999999999e-120 < p < 4.99999999999999967e-89

   1. Initial program 65.1%

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

      1. add-log-exp 65.1%

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

      2. +-commutative 65.1%

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

      3. add-sqr-sqrt 65.1%

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

      4. hypot-def 65.1%

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

      5. associate-*l* 65.1%

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

      6. sqrt-prod 65.1%

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

      7. metadata-eval 65.1%

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

      8. sqrt-unprod 65.1%

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

      9. add-sqr-sqrt 65.1%

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

   3. Applied egg-rr 65.1%

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

      1. sqrt-prod 64.2%

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

      2. rem-log-exp 64.2%

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

      3. hypot-udef 64.2%

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

      4. +-commutative 64.2%

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

      5. hypot-udef 64.2%

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

      6. add-sqr-sqrt 57.0%

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

      7. add-sqr-sqrt 64.2%

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

      8. sqrt-prod 65.1%

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

      9. add-cbrt-cube 65.1%

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

      10. pow1/3 65.1%

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

   5. Applied egg-rr 65.1%

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

   6. Taylor expanded in x around inf 59.6%

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

   if 4.99999999999999967e-89 < p < 1.00000000000000007e-37

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0 50.4%

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

   if 3.3e28 < p

   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. Step-by-step derivation

      1. add-sqr-sqrt 94.4%

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

      2. hypot-def 94.4%

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

      3. associate-*l* 94.4%

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

      4. sqrt-prod 94.4%

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

      5. metadata-eval 94.4%

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

      6. sqrt-unprod 94.4%

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

      7. add-sqr-sqrt 94.4%

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

   3. Applied egg-rr 94.4%

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

   4. Taylor expanded in x around 0 92.7%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{x}{p}}} \]
   5. Step-by-step derivation

      1. *-commutative 92.7%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{x}{p} \cdot 0.25}} \]

   6. Simplified 92.7%

      \[\leadsto \sqrt{\color{blue}{0.5 + \frac{x}{p} \cdot 0.25}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 5 \cdot 10^{-254}:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{elif}\;p \leq 4.9 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.46 \cdot 10^{-120}:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{elif}\;p \leq 5 \cdot 10^{-89}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 10^{-37}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x}{p} \cdot 0.25}\\ \end{array} \]

Alternative 5: 49.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{p}{x}\right|\\ \mathbf{if}\;p \leq 4.5 \cdot 10^{-254}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.6 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-87}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 3.4 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (fabs (/ p x))))
   (if (<= p 4.5e-254)
     t_0
     (if (<= p 1.3e-131)
       1.0
       (if (<= p 2.6e-120)
         t_0
         (if (<= p 7e-87)
           1.0
           (if (<= p 1.2e-38)
             (sqrt 0.5)
             (if (<= p 3.4e-10) t_0 (sqrt 0.5)))))))))

C

double code(double p, double x) {
	double t_0 = fabs((p / x));
	double tmp;
	if (p <= 4.5e-254) {
		tmp = t_0;
	} else if (p <= 1.3e-131) {
		tmp = 1.0;
	} else if (p <= 2.6e-120) {
		tmp = t_0;
	} else if (p <= 7e-87) {
		tmp = 1.0;
	} else if (p <= 1.2e-38) {
		tmp = sqrt(0.5);
	} else if (p <= 3.4e-10) {
		tmp = t_0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

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 = abs((p / x))
    if (p <= 4.5d-254) then
        tmp = t_0
    else if (p <= 1.3d-131) then
        tmp = 1.0d0
    else if (p <= 2.6d-120) then
        tmp = t_0
    else if (p <= 7d-87) then
        tmp = 1.0d0
    else if (p <= 1.2d-38) then
        tmp = sqrt(0.5d0)
    else if (p <= 3.4d-10) then
        tmp = t_0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double t_0 = Math.abs((p / x));
	double tmp;
	if (p <= 4.5e-254) {
		tmp = t_0;
	} else if (p <= 1.3e-131) {
		tmp = 1.0;
	} else if (p <= 2.6e-120) {
		tmp = t_0;
	} else if (p <= 7e-87) {
		tmp = 1.0;
	} else if (p <= 1.2e-38) {
		tmp = Math.sqrt(0.5);
	} else if (p <= 3.4e-10) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	t_0 = math.fabs((p / x))
	tmp = 0
	if p <= 4.5e-254:
		tmp = t_0
	elif p <= 1.3e-131:
		tmp = 1.0
	elif p <= 2.6e-120:
		tmp = t_0
	elif p <= 7e-87:
		tmp = 1.0
	elif p <= 1.2e-38:
		tmp = math.sqrt(0.5)
	elif p <= 3.4e-10:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	t_0 = abs(Float64(p / x))
	tmp = 0.0
	if (p <= 4.5e-254)
		tmp = t_0;
	elseif (p <= 1.3e-131)
		tmp = 1.0;
	elseif (p <= 2.6e-120)
		tmp = t_0;
	elseif (p <= 7e-87)
		tmp = 1.0;
	elseif (p <= 1.2e-38)
		tmp = sqrt(0.5);
	elseif (p <= 3.4e-10)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	t_0 = abs((p / x));
	tmp = 0.0;
	if (p <= 4.5e-254)
		tmp = t_0;
	elseif (p <= 1.3e-131)
		tmp = 1.0;
	elseif (p <= 2.6e-120)
		tmp = t_0;
	elseif (p <= 7e-87)
		tmp = 1.0;
	elseif (p <= 1.2e-38)
		tmp = sqrt(0.5);
	elseif (p <= 3.4e-10)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := Block[{t$95$0 = N[Abs[N[(p / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[p, 4.5e-254], t$95$0, If[LessEqual[p, 1.3e-131], 1.0, If[LessEqual[p, 2.6e-120], t$95$0, If[LessEqual[p, 7e-87], 1.0, If[LessEqual[p, 1.2e-38], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, 3.4e-10], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 4.5e-254 or 1.29999999999999998e-131 < p < 2.6000000000000001e-120 or 1.20000000000000011e-38 < p < 3.40000000000000015e-10

   1. Initial program 75.9%

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

      1. *-commutative 75.9%

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

      2. sqrt-prod 75.5%

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

      3. +-commutative 75.5%

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

      4. add-sqr-sqrt 75.5%

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

      5. hypot-def 75.5%

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

      6. associate-*l* 75.5%

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

      7. sqrt-prod 75.5%

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

      8. metadata-eval 75.5%

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

      9. sqrt-unprod 7.4%

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

      10. add-sqr-sqrt 75.5%

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

   3. Applied egg-rr 75.5%

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

   4. Taylor expanded in x around -inf 16.4%

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

      1. mul-1-neg 16.4%

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

      2. associate-/l* 16.4%

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

      3. distribute-neg-frac 16.4%

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

   6. Simplified 16.4%

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

      1. *-commutative 16.4%

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

      2. associate-/r/ 16.4%

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

      3. add-sqr-sqrt 6.2%

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

      4. sqrt-unprod 8.7%

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

      5. sqr-neg 8.7%

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

      6. unpow2 8.7%

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

      7. add-sqr-sqrt 2.3%

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

      8. sqrt-prod 23.0%

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

      9. unpow2 23.0%

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

      10. sqrt-div 19.2%

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

      11. sqrt-prod 19.3%

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

      12. *-commutative 19.3%

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

      13. pow1/2 19.3%

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

      14. pow1/2 19.3%

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

      15. pow-prod-down 19.3%

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

   8. Applied egg-rr 23.0%

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

      1. unpow1/2 23.0%

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

      2. associate-*r* 23.0%

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

      3. metadata-eval 23.0%

         \[\leadsto \sqrt{\color{blue}{1} \cdot {\left(\frac{p}{x}\right)}^{2}} \]

      4. *-lft-identity 23.0%

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

      5. unpow2 23.0%

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

      6. rem-sqrt-square 34.3%

         \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

   10. Simplified 34.3%

       \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

   if 4.5e-254 < p < 1.29999999999999998e-131 or 2.6000000000000001e-120 < p < 7.00000000000000023e-87

   1. Initial program 65.1%

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

      1. add-log-exp 65.1%

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

      2. +-commutative 65.1%

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

      3. add-sqr-sqrt 65.1%

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

      4. hypot-def 65.1%

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

      5. associate-*l* 65.1%

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

      6. sqrt-prod 65.1%

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

      7. metadata-eval 65.1%

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

      8. sqrt-unprod 65.1%

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

      9. add-sqr-sqrt 65.1%

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

   3. Applied egg-rr 65.1%

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

      1. sqrt-prod 64.2%

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

      2. rem-log-exp 64.2%

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

      3. hypot-udef 64.2%

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

      4. +-commutative 64.2%

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

      5. hypot-udef 64.2%

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

      6. add-sqr-sqrt 57.0%

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

      7. add-sqr-sqrt 64.2%

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

      8. sqrt-prod 65.1%

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

      9. add-cbrt-cube 65.1%

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

      10. pow1/3 65.1%

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

   5. Applied egg-rr 65.1%

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

   6. Taylor expanded in x around inf 59.6%

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

   if 7.00000000000000023e-87 < p < 1.20000000000000011e-38 or 3.40000000000000015e-10 < p

   1. Initial program 84.0%

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

   2. Taylor expanded in x around 0 77.4%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.5 \cdot 10^{-254}:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.6 \cdot 10^{-120}:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-87}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 3.4 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6: 53.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq 1.25 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (if (<= p 1.25e-85) 1.0 (sqrt 0.5)))

C

double code(double p, double x) {
	double tmp;
	if (p <= 1.25e-85) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 1.25d-85) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= 1.25e-85) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= 1.25e-85:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= 1.25e-85)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 1.25e-85)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, 1.25e-85], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.25e-85

   1. Initial program 75.9%

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

      1. add-log-exp 75.9%

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

      2. +-commutative 75.9%

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

      3. add-sqr-sqrt 75.9%

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

      4. hypot-def 75.9%

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

      5. associate-*l* 75.9%

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

      6. sqrt-prod 75.9%

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

      7. metadata-eval 75.9%

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

      8. sqrt-unprod 12.4%

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

      9. add-sqr-sqrt 75.9%

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

   3. Applied egg-rr 75.9%

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

      1. sqrt-prod 75.5%

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

      2. rem-log-exp 75.5%

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

      3. hypot-udef 75.5%

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

      4. +-commutative 75.5%

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

      5. hypot-udef 75.5%

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

      6. add-sqr-sqrt 51.6%

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

      7. add-sqr-sqrt 75.5%

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

      8. sqrt-prod 75.9%

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

      9. add-cbrt-cube 75.9%

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

      10. pow1/3 75.9%

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

   5. Applied egg-rr 75.9%

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

   6. Taylor expanded in x around inf 37.9%

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

   if 1.25e-85 < p

   1. Initial program 80.9%

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

   2. Taylor expanded in x around 0 72.9%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.25 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 35.7% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (p x) :precision binary64 1.0)

C

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

Fortran

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

Java

public static double code(double p, double x) {
	return 1.0;
}

Python

def code(p, x):
	return 1.0

Julia

function code(p, x)
	return 1.0
end

MATLAB

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

Wolfram

code[p_, x_] := 1.0

Derivation

1. Initial program 77.6%

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

   1. add-log-exp 77.6%

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

   2. +-commutative 77.6%

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

   3. add-sqr-sqrt 77.6%

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

   4. hypot-def 77.6%

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

   5. associate-*l* 77.6%

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

   6. sqrt-prod 77.6%

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

   7. metadata-eval 77.6%

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

   8. sqrt-unprod 35.7%

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

   9. add-sqr-sqrt 77.6%

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

3. Applied egg-rr 77.6%

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

   1. sqrt-prod 77.3%

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

   2. rem-log-exp 77.3%

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

   3. hypot-udef 77.3%

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

   4. +-commutative 77.3%

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

   5. hypot-udef 77.3%

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

   6. add-sqr-sqrt 46.6%

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

   7. add-sqr-sqrt 77.3%

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

   8. sqrt-prod 77.6%

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

   9. add-cbrt-cube 77.6%

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

   10. pow1/3 77.6%

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

5. Applied egg-rr 77.6%

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

6. Taylor expanded in x around inf 32.5%

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

7. Final simplification 32.5%

   \[\leadsto 1 \]

Developer target: 78.9% 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 2023310 
(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))))))))