Given's Rotation SVD example

Percentage Accurate: 79.5% → 99.7%

Time: 7.1s

Alternatives: 6

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: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

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)
   (/ (- 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 = -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 = -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 = -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(-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 = -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[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 16.6%

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

   2. Taylor expanded in x around -inf 41.4%

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

      1. unpow2 41.4%

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

      2. unpow2 41.4%

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

   4. Simplified 41.4%

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

   5. Taylor expanded in p around -inf 53.2%

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

      1. associate-*r/ 53.2%

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

      2. neg-mul-1 53.2%

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

   7. Simplified 53.2%

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

   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 46.3%

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

      7. add-sqr-sqrt 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 87.9%

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

Alternative 2: 67.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 1.25 \cdot 10^{-136}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.9 \cdot 10^{-94}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= p 1.25e-136)
   (/ (- p) x)
   (if (<= p 3.9e-94)
     1.0
     (if (<= p 9.5e-76)
       (sqrt 0.5)
       (if (<= p 1.9e-5) 1.0 (sqrt (+ 0.5 (* 0.25 (/ x p)))))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 1.25e-136) {
		tmp = -p / x;
	} else if (p <= 3.9e-94) {
		tmp = 1.0;
	} else if (p <= 9.5e-76) {
		tmp = sqrt(0.5);
	} else if (p <= 1.9e-5) {
		tmp = 1.0;
	} 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) :: tmp
    if (p <= 1.25d-136) then
        tmp = -p / x
    else if (p <= 3.9d-94) then
        tmp = 1.0d0
    else if (p <= 9.5d-76) then
        tmp = sqrt(0.5d0)
    else if (p <= 1.9d-5) then
        tmp = 1.0d0
    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 tmp;
	if (p <= 1.25e-136) {
		tmp = -p / x;
	} else if (p <= 3.9e-94) {
		tmp = 1.0;
	} else if (p <= 9.5e-76) {
		tmp = Math.sqrt(0.5);
	} else if (p <= 1.9e-5) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.25 * (x / p))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 1.25e-136:
		tmp = -p / x
	elif p <= 3.9e-94:
		tmp = 1.0
	elif p <= 9.5e-76:
		tmp = math.sqrt(0.5)
	elif p <= 1.9e-5:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.25 * (x / p))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 1.25e-136)
		tmp = Float64(Float64(-p) / x);
	elseif (p <= 3.9e-94)
		tmp = 1.0;
	elseif (p <= 9.5e-76)
		tmp = sqrt(0.5);
	elseif (p <= 1.9e-5)
		tmp = 1.0;
	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)
	tmp = 0.0;
	if (p <= 1.25e-136)
		tmp = -p / x;
	elseif (p <= 3.9e-94)
		tmp = 1.0;
	elseif (p <= 9.5e-76)
		tmp = sqrt(0.5);
	elseif (p <= 1.9e-5)
		tmp = 1.0;
	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_] := If[LessEqual[p, 1.25e-136], N[((-p) / x), $MachinePrecision], If[LessEqual[p, 3.9e-94], 1.0, If[LessEqual[p, 9.5e-76], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, 1.9e-5], 1.0, N[Sqrt[N[(0.5 + N[(0.25 * N[(x / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 1.25e-136

   1. Initial program 74.3%

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

   2. Taylor expanded in x around -inf 14.9%

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

      1. unpow2 14.9%

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

      2. unpow2 14.9%

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

   4. Simplified 14.9%

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

   5. Taylor expanded in p around -inf 16.9%

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

      1. associate-*r/ 16.9%

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

      2. neg-mul-1 16.9%

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

   7. Simplified 16.9%

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

   if 1.25e-136 < p < 3.9000000000000002e-94 or 9.49999999999999984e-76 < p < 1.9000000000000001e-5

   1. Initial program 75.0%

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

      1. add-sqr-sqrt 75.0%

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

      2. hypot-def 75.0%

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

      3. associate-*l* 75.0%

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

      4. sqrt-prod 75.0%

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

      5. metadata-eval 75.0%

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

      6. sqrt-unprod 75.0%

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

      7. add-sqr-sqrt 75.0%

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

   3. Applied egg-rr 75.0%

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

      1. add-sqr-sqrt 74.7%

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

      2. pow2 74.7%

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

      3. pow1/2 74.7%

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

      4. sqrt-pow1 74.7%

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

      5. distribute-lft-in 74.7%

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

      6. metadata-eval 74.7%

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

      7. associate-*r/ 74.7%

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

      8. metadata-eval 74.7%

         \[\leadsto {\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   5. Applied egg-rr 74.7%

      \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{0.25}\right)}^{2}} \]

   6. Taylor expanded in x around inf 54.5%

      \[\leadsto {\color{blue}{1}}^{2} \]

   if 3.9000000000000002e-94 < p < 9.49999999999999984e-76

   1. Initial program 52.3%

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

   2. Taylor expanded in x around 0 52.6%

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

   if 1.9000000000000001e-5 < p

   1. Initial program 98.1%

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

      1. add-sqr-sqrt 98.1%

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

      2. hypot-def 98.1%

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

      3. associate-*l* 98.1%

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

      4. sqrt-prod 98.1%

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

      5. metadata-eval 98.1%

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

      6. sqrt-unprod 98.1%

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

      7. add-sqr-sqrt 98.1%

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

   3. Applied egg-rr 98.1%

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

   4. Taylor expanded in x around 0 88.4%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.25 \cdot 10^{-136}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.9 \cdot 10^{-94}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \end{array} \]

Alternative 3: 68.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 1.35 \cdot 10^{-136}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.9 \cdot 10^{-94}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.2 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.8 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= p 1.35e-136)
   (/ (- p) x)
   (if (<= p 3.9e-94)
     1.0
     (if (<= p 1.2e-75) (sqrt 0.5) (if (<= p 1.8e-24) 1.0 (sqrt 0.5))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 1.35e-136) {
		tmp = -p / x;
	} else if (p <= 3.9e-94) {
		tmp = 1.0;
	} else if (p <= 1.2e-75) {
		tmp = sqrt(0.5);
	} else if (p <= 1.8e-24) {
		tmp = 1.0;
	} 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) :: tmp
    if (p <= 1.35d-136) then
        tmp = -p / x
    else if (p <= 3.9d-94) then
        tmp = 1.0d0
    else if (p <= 1.2d-75) then
        tmp = sqrt(0.5d0)
    else if (p <= 1.8d-24) then
        tmp = 1.0d0
    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 <= 1.35e-136) {
		tmp = -p / x;
	} else if (p <= 3.9e-94) {
		tmp = 1.0;
	} else if (p <= 1.2e-75) {
		tmp = Math.sqrt(0.5);
	} else if (p <= 1.8e-24) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 1.35e-136:
		tmp = -p / x
	elif p <= 3.9e-94:
		tmp = 1.0
	elif p <= 1.2e-75:
		tmp = math.sqrt(0.5)
	elif p <= 1.8e-24:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 1.35e-136)
		tmp = Float64(Float64(-p) / x);
	elseif (p <= 3.9e-94)
		tmp = 1.0;
	elseif (p <= 1.2e-75)
		tmp = sqrt(0.5);
	elseif (p <= 1.8e-24)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 1.35e-136)
		tmp = -p / x;
	elseif (p <= 3.9e-94)
		tmp = 1.0;
	elseif (p <= 1.2e-75)
		tmp = sqrt(0.5);
	elseif (p <= 1.8e-24)
		tmp = 1.0;
	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, 1.35e-136], N[((-p) / x), $MachinePrecision], If[LessEqual[p, 3.9e-94], 1.0, If[LessEqual[p, 1.2e-75], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, 1.8e-24], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.3499999999999999e-136

   1. Initial program 74.3%

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

   2. Taylor expanded in x around -inf 14.9%

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

      1. unpow2 14.9%

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

      2. unpow2 14.9%

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

   4. Simplified 14.9%

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

   5. Taylor expanded in p around -inf 16.9%

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

      1. associate-*r/ 16.9%

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

      2. neg-mul-1 16.9%

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

   7. Simplified 16.9%

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

   if 1.3499999999999999e-136 < p < 3.9000000000000002e-94 or 1.2000000000000001e-75 < p < 1.8e-24

   1. Initial program 77.4%

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

      1. add-sqr-sqrt 77.4%

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

      2. hypot-def 77.4%

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

      3. associate-*l* 77.4%

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

      4. sqrt-prod 77.4%

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

      5. metadata-eval 77.4%

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

      6. sqrt-unprod 77.4%

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

      7. add-sqr-sqrt 77.4%

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

   3. Applied egg-rr 77.4%

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

      1. add-sqr-sqrt 77.3%

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

      2. pow2 77.3%

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

      3. pow1/2 77.3%

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

      4. sqrt-pow1 77.3%

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

      5. distribute-lft-in 77.3%

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

      6. metadata-eval 77.3%

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

      7. associate-*r/ 77.3%

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

      8. metadata-eval 77.3%

         \[\leadsto {\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   5. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{0.25}\right)}^{2}} \]

   6. Taylor expanded in x around inf 63.7%

      \[\leadsto {\color{blue}{1}}^{2} \]

   if 3.9000000000000002e-94 < p < 1.2000000000000001e-75 or 1.8e-24 < p

   1. Initial program 90.8%

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

   2. Taylor expanded in x around 0 81.2%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.35 \cdot 10^{-136}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.9 \cdot 10^{-94}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.2 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.8 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4: 67.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 5.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= p 5.5e-109) (/ (- p) x) (sqrt 0.5)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 5.5e-109) {
		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 <= 5.5d-109) 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 <= 5.5e-109) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 5.5e-109:
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 5.5e-109)
		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 <= 5.5e-109)
		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, 5.5e-109], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 5.5000000000000003e-109

   1. Initial program 74.2%

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

   2. Taylor expanded in x around -inf 15.6%

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

      1. unpow2 15.6%

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

      2. unpow2 15.6%

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

   4. Simplified 15.6%

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

   5. Taylor expanded in p around -inf 17.5%

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

      1. associate-*r/ 17.5%

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

      2. neg-mul-1 17.5%

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

   7. Simplified 17.5%

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

   if 5.5000000000000003e-109 < p

   1. Initial program 89.4%

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

   2. Taylor expanded in x around 0 73.4%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 5.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 28.6% 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(Float64(-p) / 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 51.3%

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

   2. Taylor expanded in x around -inf 26.4%

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

      1. unpow2 26.4%

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

      2. unpow2 26.4%

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

   4. Simplified 26.4%

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

   5. Taylor expanded in p around -inf 32.7%

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

      1. associate-*r/ 32.7%

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

      2. neg-mul-1 32.7%

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

   7. Simplified 32.7%

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

   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 4.8%

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

      1. unpow2 4.8%

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

      2. unpow2 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in p around 0 3.4%

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

4. Final simplification 16.3%

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

Alternative 6: 6.6% 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 78.5%

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

2. Taylor expanded in x around -inf 14.4%

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

   1. unpow2 14.4%

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

   2. unpow2 14.4%

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

4. Simplified 14.4%

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

5. Taylor expanded in p around 0 18.2%

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

6. Final simplification 18.2%

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

Developer target: 79.5% 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 2023214 
(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))))))))