Given's Rotation SVD example

Percentage Accurate: 79.7% → 99.6%

Time: 11.5s

Alternatives: 12

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: 79.7% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\left(p\_m \cdot \left({2}^{0.25} \cdot \frac{{2}^{0.25}}{x}\right)\right) \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p\_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (* (* p_m (* (pow 2.0 0.25) (/ (pow 2.0 0.25) x))) (- (sqrt 0.5)))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p_m 2.0) x)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = (p_m * (pow(2.0, 0.25) * (pow(2.0, 0.25) / x))) * -sqrt(0.5);
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p_m * 2.0), x)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = (p_m * (Math.pow(2.0, 0.25) * (Math.pow(2.0, 0.25) / x))) * -Math.sqrt(0.5);
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p_m * 2.0), x)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0:
		tmp = (p_m * (math.pow(2.0, 0.25) * (math.pow(2.0, 0.25) / x))) * -math.sqrt(0.5)
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p_m * 2.0), x)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(p_m * Float64((2.0 ^ 0.25) * Float64((2.0 ^ 0.25) / x))) * Float64(-sqrt(0.5)));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p_m * 2.0), x)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0)
		tmp = (p_m * ((2.0 ^ 0.25) * ((2.0 ^ 0.25) / x))) * -sqrt(0.5);
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p_m * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(p$95$m * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p$95$m * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -1

   1. Initial program 16.0%

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

      1. sqrt-prod 16.0%

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

      2. *-commutative 16.0%

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

      3. +-commutative 16.0%

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

      4. add-sqr-sqrt 16.0%

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

      5. hypot-define 16.0%

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

      6. associate-*l* 16.0%

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

      7. sqrt-prod 16.0%

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

      8. metadata-eval 16.0%

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

      9. sqrt-unprod 9.1%

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

      10. add-sqr-sqrt 16.0%

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

   4. Applied egg-rr 16.0%

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

   5. Taylor expanded in x around -inf 56.4%

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

      1. mul-1-neg 56.4%

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

      2. associate-/l* 56.3%

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

      3. distribute-lft-neg-in 56.3%

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

   7. Simplified 56.3%

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

      1. add-sqr-sqrt 56.8%

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

      2. associate-/l* 56.8%

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

      3. pow1/2 56.8%

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

      4. sqrt-pow1 56.8%

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

      5. metadata-eval 56.8%

         \[\leadsto \left(\left(-p\right) \cdot \left({2}^{\color{blue}{0.25}} \cdot \frac{\sqrt{\sqrt{2}}}{x}\right)\right) \cdot \sqrt{0.5} \]

      6. pow1/2 56.8%

         \[\leadsto \left(\left(-p\right) \cdot \left({2}^{0.25} \cdot \frac{\sqrt{\color{blue}{{2}^{0.5}}}}{x}\right)\right) \cdot \sqrt{0.5} \]

      7. sqrt-pow1 56.8%

         \[\leadsto \left(\left(-p\right) \cdot \left({2}^{0.25} \cdot \frac{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}{x}\right)\right) \cdot \sqrt{0.5} \]

      8. metadata-eval 56.8%

         \[\leadsto \left(\left(-p\right) \cdot \left({2}^{0.25} \cdot \frac{{2}^{\color{blue}{0.25}}}{x}\right)\right) \cdot \sqrt{0.5} \]

   9. Applied egg-rr 56.8%

      \[\leadsto \left(\left(-p\right) \cdot \color{blue}{\left({2}^{0.25} \cdot \frac{{2}^{0.25}}{x}\right)}\right) \cdot \sqrt{0.5} \]

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 55.9%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\left(p \cdot \left({2}^{0.25} \cdot \frac{{2}^{0.25}}{x}\right)\right) \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(p$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p$95$m * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -1

   1. Initial program 16.0%

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

      1. sqrt-prod 16.0%

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

      2. *-commutative 16.0%

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

      3. +-commutative 16.0%

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

      4. add-sqr-sqrt 16.0%

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

      5. hypot-define 16.0%

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

      6. associate-*l* 16.0%

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

      7. sqrt-prod 16.0%

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

      8. metadata-eval 16.0%

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

      9. sqrt-unprod 9.1%

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

      10. add-sqr-sqrt 16.0%

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

   4. Applied egg-rr 16.0%

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

   5. Taylor expanded in x around -inf 56.4%

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

      1. mul-1-neg 56.4%

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

      2. associate-/l* 56.3%

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

      3. distribute-lft-neg-in 56.3%

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

   7. Simplified 56.3%

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

      1. associate-*r/ 56.4%

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

      2. frac-2neg 56.4%

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

      3. add-sqr-sqrt 6.2%

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

      4. sqrt-unprod 15.3%

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

      5. sqr-neg 15.3%

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

      6. sqrt-unprod 8.4%

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

      7. add-sqr-sqrt 57.2%

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

      8. distribute-lft-neg-out 57.2%

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

      9. add-sqr-sqrt 48.7%

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

      10. sqrt-unprod 61.2%

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

      11. sqr-neg 61.2%

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

      12. sqrt-unprod 50.2%

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

      13. add-sqr-sqrt 56.4%

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

   9. Applied egg-rr 56.4%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 55.9%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 89.3%

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

Alternative 3: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.7 \cdot 10^{-278}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 3.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \frac{p\_m}{-x}\right)\\ \mathbf{elif}\;p\_m \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \sqrt{1 + \frac{x \cdot 0.5}{p\_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.7e-278)
   1.0
   (if (<= p_m 3.8e-153)
     (* (sqrt 0.5) (* (sqrt 2.0) (/ p_m (- x))))
     (if (<= p_m 2.9e+20)
       1.0
       (* (sqrt 0.5) (sqrt (+ 1.0 (/ (* x 0.5) p_m))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.7e-278) {
		tmp = 1.0;
	} else if (p_m <= 3.8e-153) {
		tmp = sqrt(0.5) * (sqrt(2.0) * (p_m / -x));
	} else if (p_m <= 2.9e+20) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5) * sqrt((1.0 + ((x * 0.5) / p_m)));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 2.7d-278) then
        tmp = 1.0d0
    else if (p_m <= 3.8d-153) then
        tmp = sqrt(0.5d0) * (sqrt(2.0d0) * (p_m / -x))
    else if (p_m <= 2.9d+20) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0) * sqrt((1.0d0 + ((x * 0.5d0) / p_m)))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.7e-278) {
		tmp = 1.0;
	} else if (p_m <= 3.8e-153) {
		tmp = Math.sqrt(0.5) * (Math.sqrt(2.0) * (p_m / -x));
	} else if (p_m <= 2.9e+20) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5) * Math.sqrt((1.0 + ((x * 0.5) / p_m)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 2.7e-278:
		tmp = 1.0
	elif p_m <= 3.8e-153:
		tmp = math.sqrt(0.5) * (math.sqrt(2.0) * (p_m / -x))
	elif p_m <= 2.9e+20:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5) * math.sqrt((1.0 + ((x * 0.5) / p_m)))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 2.7e-278)
		tmp = 1.0;
	elseif (p_m <= 3.8e-153)
		tmp = Float64(sqrt(0.5) * Float64(sqrt(2.0) * Float64(p_m / Float64(-x))));
	elseif (p_m <= 2.9e+20)
		tmp = 1.0;
	else
		tmp = Float64(sqrt(0.5) * sqrt(Float64(1.0 + Float64(Float64(x * 0.5) / p_m))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 2.7e-278)
		tmp = 1.0;
	elseif (p_m <= 3.8e-153)
		tmp = sqrt(0.5) * (sqrt(2.0) * (p_m / -x));
	elseif (p_m <= 2.9e+20)
		tmp = 1.0;
	else
		tmp = sqrt(0.5) * sqrt((1.0 + ((x * 0.5) / p_m)));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 2.7e-278], 1.0, If[LessEqual[p$95$m, 3.8e-153], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(p$95$m / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 2.9e+20], 1.0, N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] / p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.7000000000000001e-278 or 3.80000000000000023e-153 < p < 2.9e20

   1. Initial program 76.3%

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

   3. Taylor expanded in x around inf 43.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 2.7000000000000001e-278 < p < 3.80000000000000023e-153

   1. Initial program 58.1%

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

      1. sqrt-prod 57.4%

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

      2. *-commutative 57.4%

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

      3. +-commutative 57.4%

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

      4. add-sqr-sqrt 57.4%

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

      5. hypot-define 57.4%

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

      6. associate-*l* 57.4%

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

      7. sqrt-prod 57.4%

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

      8. metadata-eval 57.4%

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

      9. sqrt-unprod 57.4%

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

      10. add-sqr-sqrt 57.4%

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

   4. Applied egg-rr 57.4%

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

   5. Taylor expanded in x around -inf 58.1%

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

      1. mul-1-neg 58.1%

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

      2. associate-/l* 58.1%

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

      3. distribute-lft-neg-in 58.1%

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

   7. Simplified 58.1%

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

   8. Taylor expanded in p around 0 58.1%

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

      1. neg-mul-1 58.1%

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

      2. remove-double-neg 58.1%

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

      3. distribute-neg-frac 58.1%

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

      4. distribute-rgt-neg-out 58.1%

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

      5. distribute-frac-neg2 58.1%

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

      6. *-commutative 58.1%

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

      7. associate-/l* 58.1%

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

      8. distribute-lft-neg-in 58.1%

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

      9. remove-double-neg 58.1%

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

   10. Simplified 58.1%

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

   if 2.9e20 < p

   1. Initial program 95.5%

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

      1. sqrt-prod 95.4%

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

      2. *-commutative 95.4%

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

      3. +-commutative 95.4%

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

      4. add-sqr-sqrt 95.4%

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

      5. hypot-define 95.4%

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

      6. associate-*l* 95.4%

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

      7. sqrt-prod 95.4%

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

      8. metadata-eval 95.4%

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

      9. sqrt-unprod 95.4%

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

      10. add-sqr-sqrt 95.4%

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in x around 0 89.8%

      \[\leadsto \sqrt{\color{blue}{1 + 0.5 \cdot \frac{x}{p}}} \cdot \sqrt{0.5} \]
   6. Step-by-step derivation

      1. associate-*r/ 89.8%

         \[\leadsto \sqrt{1 + \color{blue}{\frac{0.5 \cdot x}{p}}} \cdot \sqrt{0.5} \]

   7. Simplified 89.8%

      \[\leadsto \sqrt{\color{blue}{1 + \frac{0.5 \cdot x}{p}}} \cdot \sqrt{0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.7 \cdot 10^{-278}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \frac{p}{-x}\right)\\ \mathbf{elif}\;p \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \sqrt{1 + \frac{x \cdot 0.5}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 8.8 \cdot 10^{-277}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 9 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \frac{p\_m}{-x}\right)\\ \mathbf{elif}\;p\_m \leq 3.6 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p\_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 8.8e-277)
   1.0
   (if (<= p_m 9e-152)
     (* (sqrt 0.5) (* (sqrt 2.0) (/ p_m (- x))))
     (if (<= p_m 3.6e+22) 1.0 (sqrt (+ 0.5 (/ (* x 0.25) p_m)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 8.8e-277) {
		tmp = 1.0;
	} else if (p_m <= 9e-152) {
		tmp = sqrt(0.5) * (sqrt(2.0) * (p_m / -x));
	} else if (p_m <= 3.6e+22) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 8.8d-277) then
        tmp = 1.0d0
    else if (p_m <= 9d-152) then
        tmp = sqrt(0.5d0) * (sqrt(2.0d0) * (p_m / -x))
    else if (p_m <= 3.6d+22) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + ((x * 0.25d0) / p_m)))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 8.8e-277) {
		tmp = 1.0;
	} else if (p_m <= 9e-152) {
		tmp = Math.sqrt(0.5) * (Math.sqrt(2.0) * (p_m / -x));
	} else if (p_m <= 3.6e+22) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 8.8e-277:
		tmp = 1.0
	elif p_m <= 9e-152:
		tmp = math.sqrt(0.5) * (math.sqrt(2.0) * (p_m / -x))
	elif p_m <= 3.6e+22:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + ((x * 0.25) / p_m)))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 8.8e-277)
		tmp = 1.0;
	elseif (p_m <= 9e-152)
		tmp = Float64(sqrt(0.5) * Float64(sqrt(2.0) * Float64(p_m / Float64(-x))));
	elseif (p_m <= 3.6e+22)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.25) / p_m)));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 8.8e-277)
		tmp = 1.0;
	elseif (p_m <= 9e-152)
		tmp = sqrt(0.5) * (sqrt(2.0) * (p_m / -x));
	elseif (p_m <= 3.6e+22)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 8.8e-277], 1.0, If[LessEqual[p$95$m, 9e-152], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(p$95$m / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 3.6e+22], 1.0, N[Sqrt[N[(0.5 + N[(N[(x * 0.25), $MachinePrecision] / p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 8.79999999999999983e-277 or 9.0000000000000008e-152 < p < 3.6e22

   1. Initial program 76.3%

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

   3. Taylor expanded in x around inf 43.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 8.79999999999999983e-277 < p < 9.0000000000000008e-152

   1. Initial program 58.1%

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

      1. sqrt-prod 57.4%

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

      2. *-commutative 57.4%

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

      3. +-commutative 57.4%

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

      4. add-sqr-sqrt 57.4%

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

      5. hypot-define 57.4%

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

      6. associate-*l* 57.4%

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

      7. sqrt-prod 57.4%

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

      8. metadata-eval 57.4%

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

      9. sqrt-unprod 57.4%

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

      10. add-sqr-sqrt 57.4%

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

   4. Applied egg-rr 57.4%

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

   5. Taylor expanded in x around -inf 58.1%

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

      1. mul-1-neg 58.1%

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

      2. associate-/l* 58.1%

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

      3. distribute-lft-neg-in 58.1%

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

   7. Simplified 58.1%

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

   8. Taylor expanded in p around 0 58.1%

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

      1. neg-mul-1 58.1%

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

      2. remove-double-neg 58.1%

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

      3. distribute-neg-frac 58.1%

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

      4. distribute-rgt-neg-out 58.1%

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

      5. distribute-frac-neg2 58.1%

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

      6. *-commutative 58.1%

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

      7. associate-/l* 58.1%

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

      8. distribute-lft-neg-in 58.1%

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

      9. remove-double-neg 58.1%

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

   10. Simplified 58.1%

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

   if 3.6e22 < p

   1. Initial program 95.5%

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

      1. add-sqr-sqrt 95.5%

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

      2. hypot-define 95.5%

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

      3. associate-*l* 95.5%

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

      4. sqrt-prod 95.5%

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

      5. metadata-eval 95.5%

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

      6. sqrt-unprod 95.5%

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

      7. add-sqr-sqrt 95.5%

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

   4. Applied egg-rr 95.5%

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

   5. Taylor expanded in x around 0 89.8%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

   7. Simplified 89.8%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 8.8 \cdot 10^{-277}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \frac{p}{-x}\right)\\ \mathbf{elif}\;p \leq 3.6 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.2 \cdot 10^{-276}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.25 \cdot 10^{-151}:\\ \;\;\;\;\left(-p\_m\right) \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)\\ \mathbf{elif}\;p\_m \leq 2.2 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p\_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.2e-276)
   1.0
   (if (<= p_m 1.25e-151)
     (* (- p_m) (* (sqrt 0.5) (/ (sqrt 2.0) x)))
     (if (<= p_m 2.2e+20) 1.0 (sqrt (+ 0.5 (/ (* x 0.25) p_m)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.2e-276) {
		tmp = 1.0;
	} else if (p_m <= 1.25e-151) {
		tmp = -p_m * (sqrt(0.5) * (sqrt(2.0) / x));
	} else if (p_m <= 2.2e+20) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 2.2d-276) then
        tmp = 1.0d0
    else if (p_m <= 1.25d-151) then
        tmp = -p_m * (sqrt(0.5d0) * (sqrt(2.0d0) / x))
    else if (p_m <= 2.2d+20) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + ((x * 0.25d0) / p_m)))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.2e-276) {
		tmp = 1.0;
	} else if (p_m <= 1.25e-151) {
		tmp = -p_m * (Math.sqrt(0.5) * (Math.sqrt(2.0) / x));
	} else if (p_m <= 2.2e+20) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 2.2e-276:
		tmp = 1.0
	elif p_m <= 1.25e-151:
		tmp = -p_m * (math.sqrt(0.5) * (math.sqrt(2.0) / x))
	elif p_m <= 2.2e+20:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + ((x * 0.25) / p_m)))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 2.2e-276)
		tmp = 1.0;
	elseif (p_m <= 1.25e-151)
		tmp = Float64(Float64(-p_m) * Float64(sqrt(0.5) * Float64(sqrt(2.0) / x)));
	elseif (p_m <= 2.2e+20)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.25) / p_m)));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 2.2e-276)
		tmp = 1.0;
	elseif (p_m <= 1.25e-151)
		tmp = -p_m * (sqrt(0.5) * (sqrt(2.0) / x));
	elseif (p_m <= 2.2e+20)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 2.2e-276], 1.0, If[LessEqual[p$95$m, 1.25e-151], N[((-p$95$m) * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 2.2e+20], 1.0, N[Sqrt[N[(0.5 + N[(N[(x * 0.25), $MachinePrecision] / p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.19999999999999981e-276 or 1.25000000000000001e-151 < p < 2.2e20

   1. Initial program 76.3%

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

   3. Taylor expanded in x around inf 43.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 2.19999999999999981e-276 < p < 1.25000000000000001e-151

   1. Initial program 58.1%

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

   3. Taylor expanded in x around -inf 58.0%

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

      1. mul-1-neg 58.0%

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

   5. Simplified 58.0%

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

   6. Taylor expanded in p around 0 58.0%

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

      1. mul-1-neg 58.0%

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

      2. associate-/l* 57.9%

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

      3. distribute-rgt-neg-in 57.9%

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

      4. associate-/l* 58.2%

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

      5. distribute-rgt-neg-in 58.2%

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

   8. Simplified 58.2%

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

   if 2.2e20 < p

   1. Initial program 95.5%

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

      1. add-sqr-sqrt 95.5%

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

      2. hypot-define 95.5%

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

      3. associate-*l* 95.5%

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

      4. sqrt-prod 95.5%

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

      5. metadata-eval 95.5%

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

      6. sqrt-unprod 95.5%

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

      7. add-sqr-sqrt 95.5%

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

   4. Applied egg-rr 95.5%

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

   5. Taylor expanded in x around 0 89.8%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

   7. Simplified 89.8%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.2 \cdot 10^{-276}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.25 \cdot 10^{-151}:\\ \;\;\;\;\left(-p\right) \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)\\ \mathbf{elif}\;p \leq 2.2 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.4 \cdot 10^{-277}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.65 \cdot 10^{-153}:\\ \;\;\;\;\frac{p\_m \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{-x}\\ \mathbf{elif}\;p\_m \leq 1.02 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p\_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.4e-277)
   1.0
   (if (<= p_m 1.65e-153)
     (/ (* p_m (* (sqrt 0.5) (sqrt 2.0))) (- x))
     (if (<= p_m 1.02e+21) 1.0 (sqrt (+ 0.5 (/ (* x 0.25) p_m)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.4e-277) {
		tmp = 1.0;
	} else if (p_m <= 1.65e-153) {
		tmp = (p_m * (sqrt(0.5) * sqrt(2.0))) / -x;
	} else if (p_m <= 1.02e+21) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 1.4d-277) then
        tmp = 1.0d0
    else if (p_m <= 1.65d-153) then
        tmp = (p_m * (sqrt(0.5d0) * sqrt(2.0d0))) / -x
    else if (p_m <= 1.02d+21) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + ((x * 0.25d0) / p_m)))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.4e-277) {
		tmp = 1.0;
	} else if (p_m <= 1.65e-153) {
		tmp = (p_m * (Math.sqrt(0.5) * Math.sqrt(2.0))) / -x;
	} else if (p_m <= 1.02e+21) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.4e-277:
		tmp = 1.0
	elif p_m <= 1.65e-153:
		tmp = (p_m * (math.sqrt(0.5) * math.sqrt(2.0))) / -x
	elif p_m <= 1.02e+21:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + ((x * 0.25) / p_m)))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.4e-277)
		tmp = 1.0;
	elseif (p_m <= 1.65e-153)
		tmp = Float64(Float64(p_m * Float64(sqrt(0.5) * sqrt(2.0))) / Float64(-x));
	elseif (p_m <= 1.02e+21)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.25) / p_m)));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.4e-277)
		tmp = 1.0;
	elseif (p_m <= 1.65e-153)
		tmp = (p_m * (sqrt(0.5) * sqrt(2.0))) / -x;
	elseif (p_m <= 1.02e+21)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.4e-277], 1.0, If[LessEqual[p$95$m, 1.65e-153], N[(N[(p$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 1.02e+21], 1.0, N[Sqrt[N[(0.5 + N[(N[(x * 0.25), $MachinePrecision] / p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.39999999999999988e-277 or 1.64999999999999994e-153 < p < 1.02e21

   1. Initial program 76.3%

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

   3. Taylor expanded in x around inf 43.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 1.39999999999999988e-277 < p < 1.64999999999999994e-153

   1. Initial program 58.1%

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

   3. Taylor expanded in x around -inf 58.0%

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

      1. mul-1-neg 58.0%

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

   5. Simplified 58.0%

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

   if 1.02e21 < p

   1. Initial program 95.5%

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

      1. add-sqr-sqrt 95.5%

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

      2. hypot-define 95.5%

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

      3. associate-*l* 95.5%

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

      4. sqrt-prod 95.5%

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

      5. metadata-eval 95.5%

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

      6. sqrt-unprod 95.5%

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

      7. add-sqr-sqrt 95.5%

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

   4. Applied egg-rr 95.5%

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

   5. Taylor expanded in x around 0 89.8%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

   7. Simplified 89.8%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.4 \cdot 10^{-277}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.65 \cdot 10^{-153}:\\ \;\;\;\;\frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{-x}\\ \mathbf{elif}\;p \leq 1.02 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 3.2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p\_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 3.2e+21) 1.0 (sqrt (+ 0.5 (/ (* x 0.25) p_m)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 3.2e+21) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 3.2d+21) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + ((x * 0.25d0) / p_m)))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 3.2e+21) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 3.2e+21:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + ((x * 0.25) / p_m)))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 3.2e+21)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.25) / p_m)));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 3.2e+21)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 3.2e+21], 1.0, N[Sqrt[N[(0.5 + N[(N[(x * 0.25), $MachinePrecision] / p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 3.2e21

   1. Initial program 73.8%

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

   3. Taylor expanded in x around inf 43.7%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 3.2e21 < p

   1. Initial program 95.5%

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

      1. add-sqr-sqrt 95.5%

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

      2. hypot-define 95.5%

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

      3. associate-*l* 95.5%

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

      4. sqrt-prod 95.5%

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

      5. metadata-eval 95.5%

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

      6. sqrt-unprod 95.5%

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

      7. add-sqr-sqrt 95.5%

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

   4. Applied egg-rr 95.5%

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

   5. Taylor expanded in x around 0 89.8%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

   7. Simplified 89.8%

      \[\leadsto \sqrt{\color{blue}{0.5 + \frac{x \cdot 0.25}{p}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.08 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= p_m 1.08e+21) 1.0 (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.08e+21) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 1.08d+21) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.08e+21) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.08e+21:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.08e+21)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.08e+21)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.08e+21], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.08e21

   1. Initial program 73.8%

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

   3. Taylor expanded in x around inf 43.7%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 1.08e21 < p

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0 89.7%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.08 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \sqrt{0.5} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (sqrt 0.5))

C

p_m = fabs(p);
double code(double p_m, double x) {
	return sqrt(0.5);
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = sqrt(0.5d0)
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return Math.sqrt(0.5);
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return math.sqrt(0.5)

Julia

p_m = abs(p)
function code(p_m, x)
	return sqrt(0.5)
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = sqrt(0.5);
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in x around 0 53.0%

   \[\leadsto \color{blue}{\sqrt{0.5}} \]
4. Add Preprocessing

Alternative 10: 17.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+43}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -9.5e+43) 0.0 1.5))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -9.5e+43) {
		tmp = 0.0;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-9.5d+43)) then
        tmp = 0.0d0
    else
        tmp = 1.5d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -9.5e+43) {
		tmp = 0.0;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -9.5e+43:
		tmp = 0.0
	else:
		tmp = 1.5
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -9.5e+43)
		tmp = 0.0;
	else
		tmp = 1.5;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -9.5e+43)
		tmp = 0.0;
	else
		tmp = 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -9.5e+43], 0.0, 1.5]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000004e43

   1. Initial program 46.2%

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

   3. Taylor expanded in x around -inf 23.2%

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

      1. neg-mul-1 23.2%

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

   5. Simplified 23.2%

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

   6. Taylor expanded in x around 0 23.2%

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

   if -9.5000000000000004e43 < x

   1. Initial program 85.1%

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

   3. Taylor expanded in x around 0 58.0%

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

   5. Applied egg-rr 17.1%

      \[\leadsto \color{blue}{1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 14.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+44}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.015625\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -2.2e+44) 0.0 0.015625))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2.2e+44) {
		tmp = 0.0;
	} else {
		tmp = 0.015625;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.2d+44)) then
        tmp = 0.0d0
    else
        tmp = 0.015625d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -2.2e+44) {
		tmp = 0.0;
	} else {
		tmp = 0.015625;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -2.2e+44:
		tmp = 0.0
	else:
		tmp = 0.015625
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -2.2e+44)
		tmp = 0.0;
	else
		tmp = 0.015625;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -2.2e+44)
		tmp = 0.0;
	else
		tmp = 0.015625;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -2.2e+44], 0.0, 0.015625]

Derivation

1. Split input into 2 regimes

2. if x < -2.19999999999999996e44

   1. Initial program 46.2%

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

   3. Taylor expanded in x around -inf 23.2%

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

      1. neg-mul-1 23.2%

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

   5. Simplified 23.2%

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

   6. Taylor expanded in x around 0 23.2%

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

   if -2.19999999999999996e44 < x

   1. Initial program 85.1%

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

   3. Taylor expanded in x around 0 58.0%

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

   5. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{0.015625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 6.0% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in x around -inf 6.3%

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

   1. neg-mul-1 6.3%

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

5. Simplified 6.3%

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

6. Taylor expanded in x around 0 6.3%

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

Developer Target 1: 79.7% 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 2024139 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

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

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