Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.8%

Time: 9.8s

Alternatives: 9

Speedup: 0.7×

Specification

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.4% 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, 0.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.98)
   (/ p (- x))
   (sqrt (+ 0.5 (* x (/ 0.5 (hypot x (* p 2.0))))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98) {
		tmp = p / -x;
	} else {
		tmp = sqrt((0.5 + (x * (0.5 / hypot(x, (p * 2.0))))));
	}
	return tmp;
}

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98) {
		tmp = p / -x;
	} else {
		tmp = Math.sqrt((0.5 + (x * (0.5 / Math.hypot(x, (p * 2.0))))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98:
		tmp = p / -x
	else:
		tmp = math.sqrt((0.5 + (x * (0.5 / math.hypot(x, (p * 2.0))))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.98)
		tmp = Float64(p / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 + Float64(x * Float64(0.5 / hypot(x, Float64(p * 2.0))))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98)
		tmp = p / -x;
	else
		tmp = sqrt((0.5 + (x * (0.5 / hypot(x, (p * 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.98], N[(p / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.97999999999999998

   1. Initial program 23.7%

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

   2. Taylor expanded in x around -inf 56.9%

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

      1. unpow2 56.9%

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

      2. unpow2 56.9%

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

      3. times-frac 63.2%

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

   4. Simplified 63.2%

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

   5. Taylor expanded in p around -inf 65.1%

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

      1. mul-1-neg 65.1%

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

      2. *-lft-identity 65.1%

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

      3. distribute-rgt-neg-in 65.1%

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

      4. metadata-eval 65.1%

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

      5. distribute-frac-neg 65.1%

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

      6. times-frac 65.1%

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

      7. neg-mul-1 65.1%

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

      8. remove-double-neg 65.1%

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

      9. neg-mul-1 65.1%

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

   7. Simplified 65.1%

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

   if -0.97999999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. +-commutative 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. hypot-def 100.0%

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

      6. associate-*l* 100.0%

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

      7. sqrt-prod 100.0%

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

      8. metadata-eval 100.0%

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

      9. sqrt-unprod 51.3%

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

      10. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. *-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. div-inv 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 98.9%

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

      2. expm1-udef 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. expm1-def 98.9%

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

      2. expm1-log1p 100.0%

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

      3. associate-/l* 100.0%

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

   9. Simplified 100.0%

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

       1. associate-/r/ 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]

Alternative 2: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+50} \lor \neg \left(x \leq -3.4 \cdot 10^{-7}\right) \land x \leq -1.35 \cdot 10^{-95}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (or (<= x -3.1e+50) (and (not (<= x -3.4e-7)) (<= x -1.35e-95)))
   (/ p (- x))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x <= -3.1e+50) || (!(x <= -3.4e-7) && (x <= -1.35e-95))) {
		tmp = p / -x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	return tmp;
}

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if ((x <= -3.1e+50) || (!(x <= -3.4e-7) && (x <= -1.35e-95))) {
		tmp = p / -x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x <= -3.1e+50) or (not (x <= -3.4e-7) and (x <= -1.35e-95)):
		tmp = p / -x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if ((x <= -3.1e+50) || (!(x <= -3.4e-7) && (x <= -1.35e-95)))
		tmp = Float64(p / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x <= -3.1e+50) || (~((x <= -3.4e-7)) && (x <= -1.35e-95)))
		tmp = p / -x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[Or[LessEqual[x, -3.1e+50], And[N[Not[LessEqual[x, -3.4e-7]], $MachinePrecision], LessEqual[x, -1.35e-95]]], N[(p / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000003e50 or -3.39999999999999974e-7 < x < -1.35e-95

   1. Initial program 49.1%

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

   2. Taylor expanded in x around -inf 39.0%

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

      1. unpow2 39.0%

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

      2. unpow2 39.0%

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

      3. times-frac 44.5%

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

   4. Simplified 44.5%

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

   5. Taylor expanded in p around -inf 46.7%

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

      1. mul-1-neg 46.7%

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

      2. *-lft-identity 46.7%

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

      3. distribute-rgt-neg-in 46.7%

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

      4. metadata-eval 46.7%

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

      5. distribute-frac-neg 46.7%

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

      6. times-frac 46.7%

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

      7. neg-mul-1 46.7%

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

      8. remove-double-neg 46.7%

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

      9. neg-mul-1 46.7%

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

   7. Simplified 46.7%

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

   if -3.10000000000000003e50 < x < -3.39999999999999974e-7 or -1.35e-95 < x

   1. Initial program 93.5%

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

      1. add-sqr-sqrt 93.5%

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

      2. hypot-def 93.5%

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

      3. associate-*l* 93.5%

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

      4. sqrt-prod 93.5%

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

      5. metadata-eval 93.5%

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

      6. sqrt-unprod 48.8%

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

      7. add-sqr-sqrt 93.5%

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

   3. Applied egg-rr 93.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+50} \lor \neg \left(x \leq -3.4 \cdot 10^{-7}\right) \land x \leq -1.35 \cdot 10^{-95}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 3: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{p}{-x}\\ \mathbf{if}\;x \leq -6 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + 2 \cdot \frac{p \cdot p}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ p (- x))))
   (if (<= x -6e+50)
     t_0
     (if (<= x -5.2e-7)
       (sqrt 0.5)
       (if (<= x -1.35e-95)
         t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (+ x (* 2.0 (/ (* p p) x))))))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = p / -x;
	double tmp;
	if (x <= -6e+50) {
		tmp = t_0;
	} else if (x <= -5.2e-7) {
		tmp = sqrt(0.5);
	} else if (x <= -1.35e-95) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = p / -x
    if (x <= (-6d+50)) then
        tmp = t_0
    else if (x <= (-5.2d-7)) then
        tmp = sqrt(0.5d0)
    else if (x <= (-1.35d-95)) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (x + (2.0d0 * ((p * p) / x)))))))
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = p / -x;
	double tmp;
	if (x <= -6e+50) {
		tmp = t_0;
	} else if (x <= -5.2e-7) {
		tmp = Math.sqrt(0.5);
	} else if (x <= -1.35e-95) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = p / -x
	tmp = 0
	if x <= -6e+50:
		tmp = t_0
	elif x <= -5.2e-7:
		tmp = math.sqrt(0.5)
	elif x <= -1.35e-95:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(p / Float64(-x))
	tmp = 0.0
	if (x <= -6e+50)
		tmp = t_0;
	elseif (x <= -5.2e-7)
		tmp = sqrt(0.5);
	elseif (x <= -1.35e-95)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(x + Float64(2.0 * Float64(Float64(p * p) / x)))))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = p / -x;
	tmp = 0.0;
	if (x <= -6e+50)
		tmp = t_0;
	elseif (x <= -5.2e-7)
		tmp = sqrt(0.5);
	elseif (x <= -1.35e-95)
		tmp = t_0;
	else
		tmp = sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(p / (-x)), $MachinePrecision]}, If[LessEqual[x, -6e+50], t$95$0, If[LessEqual[x, -5.2e-7], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[x, -1.35e-95], t$95$0, N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(x + N[(2.0 * N[(N[(p * p), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.9999999999999996e50 or -5.19999999999999998e-7 < x < -1.35e-95

   1. Initial program 49.1%

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

   2. Taylor expanded in x around -inf 39.0%

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

      1. unpow2 39.0%

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

      2. unpow2 39.0%

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

      3. times-frac 44.5%

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

   4. Simplified 44.5%

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

   5. Taylor expanded in p around -inf 46.7%

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

      1. mul-1-neg 46.7%

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

      2. *-lft-identity 46.7%

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

      3. distribute-rgt-neg-in 46.7%

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

      4. metadata-eval 46.7%

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

      5. distribute-frac-neg 46.7%

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

      6. times-frac 46.7%

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

      7. neg-mul-1 46.7%

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

      8. remove-double-neg 46.7%

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

      9. neg-mul-1 46.7%

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

   7. Simplified 46.7%

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

   if -5.9999999999999996e50 < x < -5.19999999999999998e-7

   1. Initial program 68.2%

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

   2. Taylor expanded in x around 0 65.2%

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

   if -1.35e-95 < x

   1. Initial program 98.7%

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

   2. Taylor expanded in p around 0 96.4%

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

      1. unpow2 96.4%

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

   4. Simplified 96.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+50}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-95}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + 2 \cdot \frac{p \cdot p}{x}}\right)}\\ \end{array} \]

Alternative 4: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{p}{-x}\\ \mathbf{if}\;x \leq -1.72 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{2 \cdot \frac{p}{x} + 0.25 \cdot \frac{x}{p}}\right)}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + 2 \cdot \frac{p \cdot p}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ p (- x))))
   (if (<= x -1.72e+50)
     t_0
     (if (<= x -4.2e-7)
       (sqrt (* 0.5 (+ 1.0 (/ 1.0 (+ (* 2.0 (/ p x)) (* 0.25 (/ x p)))))))
       (if (<= x -1.3e-95)
         t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (+ x (* 2.0 (/ (* p p) x))))))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = p / -x;
	double tmp;
	if (x <= -1.72e+50) {
		tmp = t_0;
	} else if (x <= -4.2e-7) {
		tmp = sqrt((0.5 * (1.0 + (1.0 / ((2.0 * (p / x)) + (0.25 * (x / p)))))));
	} else if (x <= -1.3e-95) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = p / -x
    if (x <= (-1.72d+50)) then
        tmp = t_0
    else if (x <= (-4.2d-7)) then
        tmp = sqrt((0.5d0 * (1.0d0 + (1.0d0 / ((2.0d0 * (p / x)) + (0.25d0 * (x / p)))))))
    else if (x <= (-1.3d-95)) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (x + (2.0d0 * ((p * p) / x)))))))
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = p / -x;
	double tmp;
	if (x <= -1.72e+50) {
		tmp = t_0;
	} else if (x <= -4.2e-7) {
		tmp = Math.sqrt((0.5 * (1.0 + (1.0 / ((2.0 * (p / x)) + (0.25 * (x / p)))))));
	} else if (x <= -1.3e-95) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = p / -x
	tmp = 0
	if x <= -1.72e+50:
		tmp = t_0
	elif x <= -4.2e-7:
		tmp = math.sqrt((0.5 * (1.0 + (1.0 / ((2.0 * (p / x)) + (0.25 * (x / p)))))))
	elif x <= -1.3e-95:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(p / Float64(-x))
	tmp = 0.0
	if (x <= -1.72e+50)
		tmp = t_0;
	elseif (x <= -4.2e-7)
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / Float64(Float64(2.0 * Float64(p / x)) + Float64(0.25 * Float64(x / p)))))));
	elseif (x <= -1.3e-95)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(x + Float64(2.0 * Float64(Float64(p * p) / x)))))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = p / -x;
	tmp = 0.0;
	if (x <= -1.72e+50)
		tmp = t_0;
	elseif (x <= -4.2e-7)
		tmp = sqrt((0.5 * (1.0 + (1.0 / ((2.0 * (p / x)) + (0.25 * (x / p)))))));
	elseif (x <= -1.3e-95)
		tmp = t_0;
	else
		tmp = sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(p / (-x)), $MachinePrecision]}, If[LessEqual[x, -1.72e+50], t$95$0, If[LessEqual[x, -4.2e-7], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[(N[(2.0 * N[(p / x), $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[(x / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -1.3e-95], t$95$0, N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(x + N[(2.0 * N[(N[(p * p), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.72e50 or -4.2e-7 < x < -1.3e-95

   1. Initial program 49.1%

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

   2. Taylor expanded in x around -inf 39.0%

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

      1. unpow2 39.0%

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

      2. unpow2 39.0%

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

      3. times-frac 44.5%

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

   4. Simplified 44.5%

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

   5. Taylor expanded in p around -inf 46.7%

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

      1. mul-1-neg 46.7%

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

      2. *-lft-identity 46.7%

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

      3. distribute-rgt-neg-in 46.7%

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

      4. metadata-eval 46.7%

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

      5. distribute-frac-neg 46.7%

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

      6. times-frac 46.7%

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

      7. neg-mul-1 46.7%

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

      8. remove-double-neg 46.7%

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

      9. neg-mul-1 46.7%

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

   7. Simplified 46.7%

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

   if -1.72e50 < x < -4.2e-7

   1. Initial program 68.2%

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

      1. clear-num 68.2%

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

      2. inv-pow 68.2%

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

      3. +-commutative 68.2%

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

      4. add-sqr-sqrt 68.2%

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

      5. hypot-def 68.2%

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

      6. associate-*l* 68.2%

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

      7. sqrt-prod 68.2%

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

      8. metadata-eval 68.2%

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

      9. sqrt-unprod 27.4%

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

      10. add-sqr-sqrt 68.2%

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

   3. Applied egg-rr 68.2%

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

      1. unpow-1 68.2%

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

      2. *-commutative 68.2%

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

      3. metadata-eval 68.2%

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

      4. div-inv 68.2%

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

   5. Applied egg-rr 68.2%

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

   6. Taylor expanded in x around 0 65.3%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\color{blue}{0.25 \cdot \frac{x}{p} + 2 \cdot \frac{p}{x}}}\right)} \]

   if -1.3e-95 < x

   1. Initial program 98.7%

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

   2. Taylor expanded in p around 0 96.4%

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

      1. unpow2 96.4%

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

   4. Simplified 96.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+50}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{2 \cdot \frac{p}{x} + 0.25 \cdot \frac{x}{p}}\right)}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + 2 \cdot \frac{p \cdot p}{x}}\right)}\\ \end{array} \]

Alternative 5: 68.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{p}{-x}\\ \mathbf{if}\;p \leq 3.9 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-133}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.9 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7.5 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ p (- x))))
   (if (<= p 3.9e-192)
     t_0
     (if (<= p 1.55e-133)
       1.0
       (if (<= p 2.9e-69)
         t_0
         (if (<= p 3.8e-53)
           1.0
           (if (<= p 7.5e-35) t_0 (sqrt (+ 0.5 (* 0.25 (/ x p)))))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = p / -x;
	double tmp;
	if (p <= 3.9e-192) {
		tmp = t_0;
	} else if (p <= 1.55e-133) {
		tmp = 1.0;
	} else if (p <= 2.9e-69) {
		tmp = t_0;
	} else if (p <= 3.8e-53) {
		tmp = 1.0;
	} else if (p <= 7.5e-35) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 + (0.25 * (x / p))));
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = p / -x
    if (p <= 3.9d-192) then
        tmp = t_0
    else if (p <= 1.55d-133) then
        tmp = 1.0d0
    else if (p <= 2.9d-69) then
        tmp = t_0
    else if (p <= 3.8d-53) then
        tmp = 1.0d0
    else if (p <= 7.5d-35) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 + (0.25d0 * (x / p))))
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = p / -x;
	double tmp;
	if (p <= 3.9e-192) {
		tmp = t_0;
	} else if (p <= 1.55e-133) {
		tmp = 1.0;
	} else if (p <= 2.9e-69) {
		tmp = t_0;
	} else if (p <= 3.8e-53) {
		tmp = 1.0;
	} else if (p <= 7.5e-35) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 + (0.25 * (x / p))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = p / -x
	tmp = 0
	if p <= 3.9e-192:
		tmp = t_0
	elif p <= 1.55e-133:
		tmp = 1.0
	elif p <= 2.9e-69:
		tmp = t_0
	elif p <= 3.8e-53:
		tmp = 1.0
	elif p <= 7.5e-35:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 + (0.25 * (x / p))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(p / Float64(-x))
	tmp = 0.0
	if (p <= 3.9e-192)
		tmp = t_0;
	elseif (p <= 1.55e-133)
		tmp = 1.0;
	elseif (p <= 2.9e-69)
		tmp = t_0;
	elseif (p <= 3.8e-53)
		tmp = 1.0;
	elseif (p <= 7.5e-35)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(x / p))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = p / -x;
	tmp = 0.0;
	if (p <= 3.9e-192)
		tmp = t_0;
	elseif (p <= 1.55e-133)
		tmp = 1.0;
	elseif (p <= 2.9e-69)
		tmp = t_0;
	elseif (p <= 3.8e-53)
		tmp = 1.0;
	elseif (p <= 7.5e-35)
		tmp = t_0;
	else
		tmp = sqrt((0.5 + (0.25 * (x / p))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(p / (-x)), $MachinePrecision]}, If[LessEqual[p, 3.9e-192], t$95$0, If[LessEqual[p, 1.55e-133], 1.0, If[LessEqual[p, 2.9e-69], t$95$0, If[LessEqual[p, 3.8e-53], 1.0, If[LessEqual[p, 7.5e-35], t$95$0, N[Sqrt[N[(0.5 + N[(0.25 * N[(x / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 3.9000000000000003e-192 or 1.55000000000000008e-133 < p < 2.8999999999999998e-69 or 3.7999999999999998e-53 < p < 7.5e-35

   1. Initial program 71.9%

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

   2. Taylor expanded in x around -inf 23.6%

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

      1. unpow2 23.6%

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

      2. unpow2 23.6%

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

      3. times-frac 26.3%

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

   4. Simplified 26.3%

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

   5. Taylor expanded in p around -inf 25.5%

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

      1. mul-1-neg 25.5%

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

      2. *-lft-identity 25.5%

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

      3. distribute-rgt-neg-in 25.5%

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

      4. metadata-eval 25.5%

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

      5. distribute-frac-neg 25.5%

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

      6. times-frac 25.5%

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

      7. neg-mul-1 25.5%

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

      8. remove-double-neg 25.5%

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

      9. neg-mul-1 25.5%

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

   7. Simplified 25.5%

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

   if 3.9000000000000003e-192 < p < 1.55000000000000008e-133 or 2.8999999999999998e-69 < p < 3.7999999999999998e-53

   1. Initial program 83.2%

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

      1. clear-num 83.2%

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

      2. inv-pow 83.2%

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

      3. +-commutative 83.2%

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

      4. add-sqr-sqrt 83.2%

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

      5. hypot-def 83.2%

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

      6. associate-*l* 83.2%

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

      7. sqrt-prod 83.2%

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

      8. metadata-eval 83.2%

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

      9. sqrt-unprod 83.2%

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

      10. add-sqr-sqrt 83.2%

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

   3. Applied egg-rr 83.2%

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

      1. unpow-1 83.2%

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

      2. *-commutative 83.2%

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

      3. metadata-eval 83.2%

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

      4. div-inv 83.2%

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

   5. Applied egg-rr 83.2%

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

      1. expm1-log1p-u 83.2%

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

      2. expm1-udef 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. expm1-def 83.2%

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

      2. expm1-log1p 83.2%

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

      3. associate-/l* 83.2%

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

   9. Simplified 83.2%

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

   10. Taylor expanded in x around inf 78.5%

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

   if 7.5e-35 < p

   1. Initial program 94.3%

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

   2. Taylor expanded in x around 0 88.1%

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

      1. *-commutative 88.1%

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

      2. associate-*l/ 88.1%

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

   4. Simplified 88.1%

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

      1. expm1-log1p-u 86.8%

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

      2. expm1-udef 86.8%

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

      3. distribute-rgt-in 86.8%

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

      4. metadata-eval 86.8%

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

      5. associate-/l* 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. expm1-def 86.8%

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

      2. expm1-log1p 88.1%

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

      3. associate-/r/ 88.1%

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

      4. associate-*l* 88.1%

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

      5. metadata-eval 88.1%

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

   8. Simplified 88.1%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.9 \cdot 10^{-192}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-133}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \end{array} \]

Alternative 6: 69.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{p}{-x}\\ \mathbf{if}\;p \leq 7.6 \cdot 10^{-193}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.15 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.6 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 4.3 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ p (- x))))
   (if (<= p 7.6e-193)
     t_0
     (if (<= p 2.15e-134)
       1.0
       (if (<= p 2.6e-71)
         t_0
         (if (<= p 4.3e-53) 1.0 (if (<= p 9.5e-35) t_0 (sqrt 0.5))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = p / -x;
	double tmp;
	if (p <= 7.6e-193) {
		tmp = t_0;
	} else if (p <= 2.15e-134) {
		tmp = 1.0;
	} else if (p <= 2.6e-71) {
		tmp = t_0;
	} else if (p <= 4.3e-53) {
		tmp = 1.0;
	} else if (p <= 9.5e-35) {
		tmp = t_0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = p / -x
    if (p <= 7.6d-193) then
        tmp = t_0
    else if (p <= 2.15d-134) then
        tmp = 1.0d0
    else if (p <= 2.6d-71) then
        tmp = t_0
    else if (p <= 4.3d-53) then
        tmp = 1.0d0
    else if (p <= 9.5d-35) then
        tmp = t_0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = p / -x;
	double tmp;
	if (p <= 7.6e-193) {
		tmp = t_0;
	} else if (p <= 2.15e-134) {
		tmp = 1.0;
	} else if (p <= 2.6e-71) {
		tmp = t_0;
	} else if (p <= 4.3e-53) {
		tmp = 1.0;
	} else if (p <= 9.5e-35) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = p / -x
	tmp = 0
	if p <= 7.6e-193:
		tmp = t_0
	elif p <= 2.15e-134:
		tmp = 1.0
	elif p <= 2.6e-71:
		tmp = t_0
	elif p <= 4.3e-53:
		tmp = 1.0
	elif p <= 9.5e-35:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(p / Float64(-x))
	tmp = 0.0
	if (p <= 7.6e-193)
		tmp = t_0;
	elseif (p <= 2.15e-134)
		tmp = 1.0;
	elseif (p <= 2.6e-71)
		tmp = t_0;
	elseif (p <= 4.3e-53)
		tmp = 1.0;
	elseif (p <= 9.5e-35)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = p / -x;
	tmp = 0.0;
	if (p <= 7.6e-193)
		tmp = t_0;
	elseif (p <= 2.15e-134)
		tmp = 1.0;
	elseif (p <= 2.6e-71)
		tmp = t_0;
	elseif (p <= 4.3e-53)
		tmp = 1.0;
	elseif (p <= 9.5e-35)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(p / (-x)), $MachinePrecision]}, If[LessEqual[p, 7.6e-193], t$95$0, If[LessEqual[p, 2.15e-134], 1.0, If[LessEqual[p, 2.6e-71], t$95$0, If[LessEqual[p, 4.3e-53], 1.0, If[LessEqual[p, 9.5e-35], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 7.60000000000000007e-193 or 2.14999999999999993e-134 < p < 2.5999999999999999e-71 or 4.3e-53 < p < 9.5000000000000003e-35

   1. Initial program 71.9%

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

   2. Taylor expanded in x around -inf 23.6%

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

      1. unpow2 23.6%

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

      2. unpow2 23.6%

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

      3. times-frac 26.3%

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

   4. Simplified 26.3%

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

   5. Taylor expanded in p around -inf 25.5%

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

      1. mul-1-neg 25.5%

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

      2. *-lft-identity 25.5%

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

      3. distribute-rgt-neg-in 25.5%

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

      4. metadata-eval 25.5%

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

      5. distribute-frac-neg 25.5%

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

      6. times-frac 25.5%

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

      7. neg-mul-1 25.5%

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

      8. remove-double-neg 25.5%

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

      9. neg-mul-1 25.5%

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

   7. Simplified 25.5%

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

   if 7.60000000000000007e-193 < p < 2.14999999999999993e-134 or 2.5999999999999999e-71 < p < 4.3e-53

   1. Initial program 83.2%

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

      1. clear-num 83.2%

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

      2. inv-pow 83.2%

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

      3. +-commutative 83.2%

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

      4. add-sqr-sqrt 83.2%

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

      5. hypot-def 83.2%

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

      6. associate-*l* 83.2%

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

      7. sqrt-prod 83.2%

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

      8. metadata-eval 83.2%

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

      9. sqrt-unprod 83.2%

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

      10. add-sqr-sqrt 83.2%

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

   3. Applied egg-rr 83.2%

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

      1. unpow-1 83.2%

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

      2. *-commutative 83.2%

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

      3. metadata-eval 83.2%

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

      4. div-inv 83.2%

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

   5. Applied egg-rr 83.2%

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

      1. expm1-log1p-u 83.2%

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

      2. expm1-udef 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. expm1-def 83.2%

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

      2. expm1-log1p 83.2%

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

      3. associate-/l* 83.2%

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

   9. Simplified 83.2%

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

   10. Taylor expanded in x around inf 78.5%

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

   if 9.5000000000000003e-35 < p

   1. Initial program 94.3%

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

   2. Taylor expanded in x around 0 86.4%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 7.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 2.15 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 4.3 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 68.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 1.55 \cdot 10^{-34}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= p 1.55e-34) (/ p (- x)) (sqrt 0.5)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 1.55e-34) {
		tmp = p / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 1.55d-34) then
        tmp = p / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (p <= 1.55e-34) {
		tmp = p / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 1.55e-34:
		tmp = p / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 1.55e-34)
		tmp = Float64(p / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 1.55e-34)
		tmp = p / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[p, 1.55e-34], N[(p / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.5499999999999999e-34

   1. Initial program 73.0%

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

   2. Taylor expanded in x around -inf 22.6%

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

      1. unpow2 22.6%

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

      2. unpow2 22.6%

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

      3. times-frac 25.3%

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

   4. Simplified 25.3%

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

   5. Taylor expanded in p around -inf 24.9%

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

      1. mul-1-neg 24.9%

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

      2. *-lft-identity 24.9%

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

      3. distribute-rgt-neg-in 24.9%

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

      4. metadata-eval 24.9%

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

      5. distribute-frac-neg 24.9%

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

      6. times-frac 24.9%

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

      7. neg-mul-1 24.9%

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

      8. remove-double-neg 24.9%

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

      9. neg-mul-1 24.9%

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

   7. Simplified 24.9%

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

   if 1.5499999999999999e-34 < p

   1. Initial program 94.3%

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

   2. Taylor expanded in x around 0 86.4%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.55 \cdot 10^{-34}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8: 28.3% accurate, 35.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -2e-310) (/ p (- x)) (/ p x)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = p / -x;
	} else {
		tmp = p / x;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = p / -x
    else
        tmp = p / x
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = p / -x;
	} else {
		tmp = p / x;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -2e-310:
		tmp = p / -x
	else:
		tmp = p / x
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(p / Float64(-x));
	else
		tmp = Float64(p / x);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = p / -x;
	else
		tmp = p / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -2e-310], N[(p / (-x)), $MachinePrecision], N[(p / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 61.5%

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

   2. Taylor expanded in x around -inf 31.3%

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

      1. unpow2 31.3%

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

      2. unpow2 31.3%

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

      3. times-frac 34.6%

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

   4. Simplified 34.6%

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

   5. Taylor expanded in p around -inf 34.5%

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

      1. mul-1-neg 34.5%

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

      2. *-lft-identity 34.5%

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

      3. distribute-rgt-neg-in 34.5%

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

      4. metadata-eval 34.5%

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

      5. distribute-frac-neg 34.5%

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

      6. times-frac 34.5%

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

      7. neg-mul-1 34.5%

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

      8. remove-double-neg 34.5%

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

      9. neg-mul-1 34.5%

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

   7. Simplified 34.5%

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

   if -1.999999999999994e-310 < 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. Taylor expanded in x around -inf 5.1%

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

      1. unpow2 5.1%

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

      2. unpow2 5.1%

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

      3. times-frac 5.5%

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

   4. Simplified 5.5%

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

   5. Taylor expanded in p around 0 3.7%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]

Alternative 9: 6.1% accurate, 71.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \frac{p}{x} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (/ p x))

C

p = abs(p);
double code(double p, double x) {
	return p / x;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = p / x
end function

Java

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

Python

p = abs(p)
def code(p, x):
	return p / x

Julia

p = abs(p)
function code(p, x)
	return Float64(p / x)
end

MATLAB

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

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := N[(p / x), $MachinePrecision]

Derivation

1. Initial program 79.4%

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

2. Taylor expanded in x around -inf 19.1%

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

   1. unpow2 19.1%

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

   2. unpow2 19.1%

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

   3. times-frac 21.1%

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

4. Simplified 21.1%

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

5. Taylor expanded in p around 0 17.9%

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

6. Final simplification 17.9%

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

Developer target: 79.4% 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 2023283 
(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))))))))