Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.9%

Time: 8.7s

Alternatives: 5

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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;-0.25 \cdot \frac{-6}{{\left(\frac{x}{p}\right)}^{3}} - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.5)
   (- (* -0.25 (/ -6.0 (pow (/ x p) 3.0))) (/ p x))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.5) {
		tmp = (-0.25 * (-6.0 / pow((x / p), 3.0))) - (p / x);
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	return tmp;
}

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.5) {
		tmp = (-0.25 * (-6.0 / Math.pow((x / p), 3.0))) - (p / x);
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.5:
		tmp = (-0.25 * (-6.0 / math.pow((x / p), 3.0))) - (p / x)
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.5)
		tmp = Float64(Float64(-0.25 * Float64(-6.0 / (Float64(x / p) ^ 3.0))) - Float64(p / x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.5)
		tmp = (-0.25 * (-6.0 / ((x / p) ^ 3.0))) - (p / x);
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(-0.25 * N[(-6.0 / N[Power[N[(x / p), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(p / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.5

   1. Initial program 28.6%

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

   3. Simplified 28.2%

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

      1. clear-num 28.2%

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

      2. fma-udef 28.6%

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

      3. *-commutative 28.6%

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

      4. +-commutative 28.6%

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

      5. inv-pow 28.6%

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

   5. Applied egg-rr 28.3%

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

      1. pow1/2 28.3%

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

      2. distribute-lft-in 28.3%

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

      3. metadata-eval 28.3%

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

      4. unpow-1 28.3%

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

      5. un-div-inv 28.3%

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

   7. Applied egg-rr 28.3%

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

   8. Taylor expanded in x around -inf 43.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x} + -0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}}} \]
   9. Step-by-step derivation

      1. +-commutative 43.5%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}} + -1 \cdot \frac{p}{x}} \]

      2. mul-1-neg 43.5%

         \[\leadsto -0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}} + \color{blue}{\left(-\frac{p}{x}\right)} \]

      3. unsub-neg 43.5%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}} - \frac{p}{x}} \]

      4. distribute-rgt-out 43.5%

         \[\leadsto -0.25 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-4 + -2\right)}}{p \cdot {x}^{3}} - \frac{p}{x} \]

      5. metadata-eval 43.5%

         \[\leadsto -0.25 \cdot \frac{{p}^{4} \cdot \color{blue}{-6}}{p \cdot {x}^{3}} - \frac{p}{x} \]

      6. *-commutative 43.5%

         \[\leadsto -0.25 \cdot \frac{{p}^{4} \cdot -6}{\color{blue}{{x}^{3} \cdot p}} - \frac{p}{x} \]

      7. times-frac 53.9%

         \[\leadsto -0.25 \cdot \color{blue}{\left(\frac{{p}^{4}}{{x}^{3}} \cdot \frac{-6}{p}\right)} - \frac{p}{x} \]

   10. Simplified 53.9%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{{p}^{4}}{{x}^{3}} \cdot \frac{-6}{p}\right) - \frac{p}{x}} \]

   11. Taylor expanded in p around 0 57.9%

       \[\leadsto -0.25 \cdot \color{blue}{\left(-6 \cdot \frac{{p}^{3}}{{x}^{3}}\right)} - \frac{p}{x} \]
   12. Step-by-step derivation

       1. associate-*r/ 57.9%

          \[\leadsto -0.25 \cdot \color{blue}{\frac{-6 \cdot {p}^{3}}{{x}^{3}}} - \frac{p}{x} \]

       2. associate-/l* 57.9%

          \[\leadsto -0.25 \cdot \color{blue}{\frac{-6}{\frac{{x}^{3}}{{p}^{3}}}} - \frac{p}{x} \]

       3. cube-div 60.0%

          \[\leadsto -0.25 \cdot \frac{-6}{\color{blue}{{\left(\frac{x}{p}\right)}^{3}}} - \frac{p}{x} \]

   13. Simplified 60.0%

       \[\leadsto -0.25 \cdot \color{blue}{\frac{-6}{{\left(\frac{x}{p}\right)}^{3}}} - \frac{p}{x} \]

   if -0.5 < (/.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. 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-def 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 52.7%

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

      7. add-sqr-sqrt 100.0%

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

   3. 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 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;-0.25 \cdot \frac{-6}{{\left(\frac{x}{p}\right)}^{3}} - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 2: 68.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 7.6 \cdot 10^{-179}:\\ \;\;\;\;{1}^{0.5}\\ \mathbf{elif}\;p \leq 5.4 \cdot 10^{-87}:\\ \;\;\;\;-0.25 \cdot \frac{-6}{{\left(\frac{x}{p}\right)}^{3}} - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= p 7.6e-179)
   (pow 1.0 0.5)
   (if (<= p 5.4e-87)
     (- (* -0.25 (/ -6.0 (pow (/ x p) 3.0))) (/ p x))
     (sqrt 0.5))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 7.6e-179) {
		tmp = pow(1.0, 0.5);
	} else if (p <= 5.4e-87) {
		tmp = (-0.25 * (-6.0 / pow((x / p), 3.0))) - (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 <= 7.6d-179) then
        tmp = 1.0d0 ** 0.5d0
    else if (p <= 5.4d-87) then
        tmp = ((-0.25d0) * ((-6.0d0) / ((x / p) ** 3.0d0))) - (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 <= 7.6e-179) {
		tmp = Math.pow(1.0, 0.5);
	} else if (p <= 5.4e-87) {
		tmp = (-0.25 * (-6.0 / Math.pow((x / p), 3.0))) - (p / x);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 7.6e-179:
		tmp = math.pow(1.0, 0.5)
	elif p <= 5.4e-87:
		tmp = (-0.25 * (-6.0 / math.pow((x / p), 3.0))) - (p / x)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 7.6e-179)
		tmp = 1.0 ^ 0.5;
	elseif (p <= 5.4e-87)
		tmp = Float64(Float64(-0.25 * Float64(-6.0 / (Float64(x / p) ^ 3.0))) - Float64(p / x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 7.6e-179)
		tmp = 1.0 ^ 0.5;
	elseif (p <= 5.4e-87)
		tmp = (-0.25 * (-6.0 / ((x / p) ^ 3.0))) - (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, 7.6e-179], N[Power[1.0, 0.5], $MachinePrecision], If[LessEqual[p, 5.4e-87], N[(N[(-0.25 * N[(-6.0 / N[Power[N[(x / p), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(p / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 7.59999999999999947e-179

   1. Initial program 85.3%

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

   3. Simplified 85.3%

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

      1. clear-num 85.3%

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

      2. fma-udef 85.3%

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

      3. *-commutative 85.3%

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

      4. +-commutative 85.3%

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

      5. inv-pow 85.3%

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

   5. Applied egg-rr 85.3%

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

      1. pow1/2 85.3%

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

      2. distribute-lft-in 85.3%

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

      3. metadata-eval 85.3%

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

      4. unpow-1 85.3%

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

      5. un-div-inv 85.3%

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

   7. Applied egg-rr 85.3%

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

   8. Taylor expanded in x around inf 38.5%

      \[\leadsto {\color{blue}{1}}^{0.5} \]

   if 7.59999999999999947e-179 < p < 5.39999999999999967e-87

   1. Initial program 52.4%

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

   3. Simplified 52.4%

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

      1. clear-num 52.4%

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

      2. fma-udef 52.4%

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

      3. *-commutative 52.4%

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

      4. +-commutative 52.4%

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

      5. inv-pow 52.4%

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

   5. Applied egg-rr 52.4%

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

      1. pow1/2 52.4%

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

      2. distribute-lft-in 52.4%

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

      3. metadata-eval 52.4%

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

      4. unpow-1 52.4%

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

      5. un-div-inv 52.4%

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

   7. Applied egg-rr 52.4%

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

   8. Taylor expanded in x around -inf 32.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x} + -0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}}} \]
   9. Step-by-step derivation

      1. +-commutative 32.6%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}} + -1 \cdot \frac{p}{x}} \]

      2. mul-1-neg 32.6%

         \[\leadsto -0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}} + \color{blue}{\left(-\frac{p}{x}\right)} \]

      3. unsub-neg 32.6%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}} - \frac{p}{x}} \]

      4. distribute-rgt-out 32.6%

         \[\leadsto -0.25 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-4 + -2\right)}}{p \cdot {x}^{3}} - \frac{p}{x} \]

      5. metadata-eval 32.6%

         \[\leadsto -0.25 \cdot \frac{{p}^{4} \cdot \color{blue}{-6}}{p \cdot {x}^{3}} - \frac{p}{x} \]

      6. *-commutative 32.6%

         \[\leadsto -0.25 \cdot \frac{{p}^{4} \cdot -6}{\color{blue}{{x}^{3} \cdot p}} - \frac{p}{x} \]

      7. times-frac 46.5%

         \[\leadsto -0.25 \cdot \color{blue}{\left(\frac{{p}^{4}}{{x}^{3}} \cdot \frac{-6}{p}\right)} - \frac{p}{x} \]

   10. Simplified 46.5%

       \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{{p}^{4}}{{x}^{3}} \cdot \frac{-6}{p}\right) - \frac{p}{x}} \]

   11. Taylor expanded in p around 0 46.5%

       \[\leadsto -0.25 \cdot \color{blue}{\left(-6 \cdot \frac{{p}^{3}}{{x}^{3}}\right)} - \frac{p}{x} \]
   12. Step-by-step derivation

       1. associate-*r/ 46.5%

          \[\leadsto -0.25 \cdot \color{blue}{\frac{-6 \cdot {p}^{3}}{{x}^{3}}} - \frac{p}{x} \]

       2. associate-/l* 46.5%

          \[\leadsto -0.25 \cdot \color{blue}{\frac{-6}{\frac{{x}^{3}}{{p}^{3}}}} - \frac{p}{x} \]

       3. cube-div 51.4%

          \[\leadsto -0.25 \cdot \frac{-6}{\color{blue}{{\left(\frac{x}{p}\right)}^{3}}} - \frac{p}{x} \]

   13. Simplified 51.4%

       \[\leadsto -0.25 \cdot \color{blue}{\frac{-6}{{\left(\frac{x}{p}\right)}^{3}}} - \frac{p}{x} \]

   if 5.39999999999999967e-87 < p

   1. Initial program 96.0%

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

   2. Taylor expanded in x around 0 86.9%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 7.6 \cdot 10^{-179}:\\ \;\;\;\;{1}^{0.5}\\ \mathbf{elif}\;p \leq 5.4 \cdot 10^{-87}:\\ \;\;\;\;-0.25 \cdot \frac{-6}{{\left(\frac{x}{p}\right)}^{3}} - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3: 68.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 2.15 \cdot 10^{-179}:\\ \;\;\;\;{1}^{0.5}\\ \mathbf{elif}\;p \leq 5 \cdot 10^{-87}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= p 2.15e-179) (pow 1.0 0.5) (if (<= p 5e-87) (/ (- p) x) (sqrt 0.5))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 2.15e-179) {
		tmp = pow(1.0, 0.5);
	} else if (p <= 5e-87) {
		tmp = -p / x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 2.15d-179) then
        tmp = 1.0d0 ** 0.5d0
    else if (p <= 5d-87) then
        tmp = -p / x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (p <= 2.15e-179) {
		tmp = Math.pow(1.0, 0.5);
	} else if (p <= 5e-87) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 2.15e-179:
		tmp = math.pow(1.0, 0.5)
	elif p <= 5e-87:
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 2.15e-179)
		tmp = 1.0 ^ 0.5;
	elseif (p <= 5e-87)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 2.15e-179)
		tmp = 1.0 ^ 0.5;
	elseif (p <= 5e-87)
		tmp = -p / x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[p, 2.15e-179], N[Power[1.0, 0.5], $MachinePrecision], If[LessEqual[p, 5e-87], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.15000000000000013e-179

   1. Initial program 85.3%

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

   3. Simplified 85.3%

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

      1. clear-num 85.3%

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

      2. fma-udef 85.3%

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

      3. *-commutative 85.3%

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

      4. +-commutative 85.3%

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

      5. inv-pow 85.3%

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

   5. Applied egg-rr 85.3%

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

      1. pow1/2 85.3%

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

      2. distribute-lft-in 85.3%

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

      3. metadata-eval 85.3%

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

      4. unpow-1 85.3%

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

      5. un-div-inv 85.3%

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

   7. Applied egg-rr 85.3%

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

   8. Taylor expanded in x around inf 38.5%

      \[\leadsto {\color{blue}{1}}^{0.5} \]

   if 2.15000000000000013e-179 < p < 5.00000000000000042e-87

   1. Initial program 52.4%

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

   3. Simplified 52.4%

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

      1. clear-num 52.4%

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

      2. fma-udef 52.4%

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

      3. *-commutative 52.4%

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

      4. +-commutative 52.4%

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

      5. inv-pow 52.4%

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

   5. Applied egg-rr 52.4%

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

      1. pow1/2 52.4%

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

      2. distribute-lft-in 52.4%

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

      3. metadata-eval 52.4%

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

      4. unpow-1 52.4%

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

      5. un-div-inv 52.4%

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

   7. Applied egg-rr 52.4%

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

   8. Taylor expanded in x around -inf 51.1%

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

      1. associate-*r/ 51.1%

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

      2. mul-1-neg 51.1%

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

   10. Simplified 51.1%

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

   if 5.00000000000000042e-87 < p

   1. Initial program 96.0%

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

   2. Taylor expanded in x around 0 86.9%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.15 \cdot 10^{-179}:\\ \;\;\;\;{1}^{0.5}\\ \mathbf{elif}\;p \leq 5 \cdot 10^{-87}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4: 68.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 4.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= p 4.2e-87) (/ (- p) x) (sqrt 0.5)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 4.2e-87) {
		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 <= 4.2d-87) 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 <= 4.2e-87) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 4.2e-87:
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 4.2e-87)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 4.2e-87)
		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, 4.2e-87], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 4.20000000000000014e-87

   1. Initial program 80.9%

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

   3. Simplified 80.9%

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

      1. clear-num 80.9%

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

      2. fma-udef 80.9%

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

      3. *-commutative 80.9%

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

      4. +-commutative 80.9%

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

      5. inv-pow 80.9%

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

   5. Applied egg-rr 80.9%

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

      1. pow1/2 80.9%

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

      2. distribute-lft-in 80.9%

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

      3. metadata-eval 80.9%

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

      4. unpow-1 80.9%

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

      5. un-div-inv 80.9%

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

   7. Applied egg-rr 80.9%

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

   8. Taylor expanded in x around -inf 17.5%

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

      1. associate-*r/ 17.5%

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

      2. mul-1-neg 17.5%

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

   10. Simplified 17.5%

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

   if 4.20000000000000014e-87 < p

   1. Initial program 96.0%

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

   2. Taylor expanded in x around 0 86.9%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 26.5% accurate, 53.8× 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(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 86.3%

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

3. Simplified 86.3%

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

   1. clear-num 86.3%

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

   2. fma-udef 86.3%

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

   3. *-commutative 86.3%

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

   4. +-commutative 86.3%

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

   5. inv-pow 86.3%

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

5. Applied egg-rr 86.3%

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

   1. pow1/2 86.3%

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

   2. distribute-lft-in 86.3%

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

   3. metadata-eval 86.3%

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

   4. unpow-1 86.3%

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

   5. un-div-inv 86.3%

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

7. Applied egg-rr 86.3%

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

8. Taylor expanded in x around -inf 13.9%

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

   1. associate-*r/ 13.9%

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

   2. mul-1-neg 13.9%

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

10. Simplified 13.9%

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

11. Final simplification 13.9%

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

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 2023320 
(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))))))))