math.sqrt on complex, real part

Percentage Accurate: 41.4% → 86.7%

Time: 9.7s

Alternatives: 8

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 41.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -3e-17)
   (* 0.5 (pow (pow (exp 0.25) (fma 2.0 (log im) (log (/ -1.0 re)))) 2.0))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -3e-17) {
		tmp = 0.5 * pow(pow(exp(0.25), fma(2.0, log(im), log((-1.0 / re)))), 2.0);
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -3e-17)
		tmp = Float64(0.5 * ((exp(0.25) ^ fma(2.0, log(im), log(Float64(-1.0 / re)))) ^ 2.0));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -3e-17], N[(0.5 * N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(2.0 * N[Log[im], $MachinePrecision] + N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.00000000000000006e-17

   1. Initial program 15.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 36.4%

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

      1. add-sqr-sqrt 36.3%

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

      2. pow2 36.3%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}}\right)}^{2}} \]

      3. pow1/2 36.3%

         \[\leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 36.3%

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

      5. metadata-eval 36.3%

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

   5. Applied egg-rr 36.3%

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

   6. Taylor expanded in re around -inf 59.5%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}}^{2} \]
   7. Step-by-step derivation

      1. exp-prod 57.5%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(e^{0.25}\right)}^{\left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}}^{2} \]

      2. +-commutative 57.5%

         \[\leadsto 0.5 \cdot {\left({\left(e^{0.25}\right)}^{\color{blue}{\left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}}\right)}^{2} \]

      3. log-pow 36.4%

         \[\leadsto 0.5 \cdot {\left({\left(e^{0.25}\right)}^{\left(\color{blue}{2 \cdot \log im} + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2} \]

      4. fma-def 36.4%

         \[\leadsto 0.5 \cdot {\left({\left(e^{0.25}\right)}^{\color{blue}{\left(\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)\right)}}\right)}^{2} \]

   8. Simplified 36.4%

      \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)\right)}\right)}}^{2} \]

   if -3.00000000000000006e-17 < re

   1. Initial program 57.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 96.1%

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

      1. add-sqr-sqrt 95.2%

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

      2. sqrt-unprod 96.1%

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

      3. *-commutative 96.1%

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

      4. *-commutative 96.1%

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

      5. swap-sqr 96.1%

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

      6. add-sqr-sqrt 96.1%

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

      7. metadata-eval 96.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot \color{blue}{0.25}} \]

   5. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot 0.25}} \]
   6. Step-by-step derivation

      1. *-commutative 96.1%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}} \]

      2. associate-*r* 96.1%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

      3. metadata-eval 96.1%

         \[\leadsto \sqrt{\color{blue}{0.5} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]

   7. Simplified 96.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+121}:\\ \;\;\;\;0.5 \cdot {\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -1.5e+121)
   (* 0.5 (pow (exp (* 0.25 (+ (log (/ -1.0 re)) (log (pow im 2.0))))) 2.0))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -1.5e+121) {
		tmp = 0.5 * pow(exp((0.25 * (log((-1.0 / re)) + log(pow(im, 2.0))))), 2.0);
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -1.5e+121) {
		tmp = 0.5 * Math.pow(Math.exp((0.25 * (Math.log((-1.0 / re)) + Math.log(Math.pow(im, 2.0))))), 2.0);
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -1.5e+121:
		tmp = 0.5 * math.pow(math.exp((0.25 * (math.log((-1.0 / re)) + math.log(math.pow(im, 2.0))))), 2.0)
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -1.5e+121)
		tmp = Float64(0.5 * (exp(Float64(0.25 * Float64(log(Float64(-1.0 / re)) + log((im ^ 2.0))))) ^ 2.0));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.5e+121)
		tmp = 0.5 * (exp((0.25 * (log((-1.0 / re)) + log((im ^ 2.0))))) ^ 2.0);
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -1.5e+121], N[(0.5 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] + N[Log[N[Power[im, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.5000000000000001e121

   1. Initial program 9.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 31.8%

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

      1. add-sqr-sqrt 31.8%

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

      2. pow2 31.8%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}}\right)}^{2}} \]

      3. pow1/2 31.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 31.8%

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

      5. metadata-eval 31.8%

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

   5. Applied egg-rr 31.8%

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

   6. Taylor expanded in re around -inf 74.1%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}}^{2} \]

   if -1.5000000000000001e121 < re

   1. Initial program 53.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 89.7%

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

      1. add-sqr-sqrt 88.9%

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

      2. sqrt-unprod 89.7%

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

      3. *-commutative 89.7%

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

      4. *-commutative 89.7%

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

      5. swap-sqr 89.7%

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

      6. add-sqr-sqrt 89.7%

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

      7. metadata-eval 89.7%

         \[\leadsto \sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot \color{blue}{0.25}} \]

   5. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot 0.25}} \]
   6. Step-by-step derivation

      1. *-commutative 89.7%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}} \]

      2. associate-*r* 89.7%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

      3. metadata-eval 89.7%

         \[\leadsto \sqrt{\color{blue}{0.5} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]

   7. Simplified 89.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+121}:\\ \;\;\;\;0.5 \cdot {\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+162}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-0.5 \cdot \left(im \cdot \left(im \cdot \frac{1}{re}\right)\right)} \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -2.2e+162)
   (* 0.5 (* (sqrt (* -0.5 (* im (* im (/ 1.0 re))))) (sqrt 2.0)))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+162) {
		tmp = 0.5 * (sqrt((-0.5 * (im * (im * (1.0 / re))))) * sqrt(2.0));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -2.2e+162) {
		tmp = 0.5 * (Math.sqrt((-0.5 * (im * (im * (1.0 / re))))) * Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -2.2e+162:
		tmp = 0.5 * (math.sqrt((-0.5 * (im * (im * (1.0 / re))))) * math.sqrt(2.0))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -2.2e+162)
		tmp = Float64(0.5 * Float64(sqrt(Float64(-0.5 * Float64(im * Float64(im * Float64(1.0 / re))))) * sqrt(2.0)));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.2e+162)
		tmp = 0.5 * (sqrt((-0.5 * (im * (im * (1.0 / re))))) * sqrt(2.0));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -2.2e+162], N[(0.5 * N[(N[Sqrt[N[(-0.5 * N[(im * N[(im * N[(1.0 / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.2000000000000002e162

   1. Initial program 2.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 28.4%

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

      1. sqrt-prod 28.3%

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

      2. *-commutative 28.3%

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

   5. Applied egg-rr 28.3%

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

   6. Taylor expanded in re around -inf 48.1%

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

      1. div-inv 48.1%

         \[\leadsto 0.5 \cdot \left(\sqrt{-0.5 \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{re}\right)}} \cdot \sqrt{2}\right) \]

      2. unpow2 48.1%

         \[\leadsto 0.5 \cdot \left(\sqrt{-0.5 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{re}\right)} \cdot \sqrt{2}\right) \]

      3. associate-*l* 53.9%

         \[\leadsto 0.5 \cdot \left(\sqrt{-0.5 \cdot \color{blue}{\left(im \cdot \left(im \cdot \frac{1}{re}\right)\right)}} \cdot \sqrt{2}\right) \]

   8. Applied egg-rr 53.9%

      \[\leadsto 0.5 \cdot \left(\sqrt{-0.5 \cdot \color{blue}{\left(im \cdot \left(im \cdot \frac{1}{re}\right)\right)}} \cdot \sqrt{2}\right) \]

   if -2.2000000000000002e162 < re

   1. Initial program 53.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 88.7%

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

      1. add-sqr-sqrt 87.9%

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

      2. sqrt-unprod 88.7%

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

      3. *-commutative 88.7%

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

      4. *-commutative 88.7%

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

      5. swap-sqr 88.7%

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

      6. add-sqr-sqrt 88.7%

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

      7. metadata-eval 88.7%

         \[\leadsto \sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot \color{blue}{0.25}} \]

   5. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot 0.25}} \]
   6. Step-by-step derivation

      1. *-commutative 88.7%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}} \]

      2. associate-*r* 88.7%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

      3. metadata-eval 88.7%

         \[\leadsto \sqrt{\color{blue}{0.5} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]

   7. Simplified 88.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+162}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-0.5 \cdot \left(im \cdot \left(im \cdot \frac{1}{re}\right)\right)} \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 4: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+157}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{-0.5 \cdot \left(im \cdot \frac{im}{re}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e+157)
   (* 0.5 (* (sqrt 2.0) (sqrt (* -0.5 (* im (/ im re))))))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -4.5e+157) {
		tmp = 0.5 * (sqrt(2.0) * sqrt((-0.5 * (im * (im / re)))));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5e+157) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt((-0.5 * (im * (im / re)))));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -4.5e+157:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt((-0.5 * (im * (im / re)))))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -4.5e+157)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(-0.5 * Float64(im * Float64(im / re))))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5e+157)
		tmp = 0.5 * (sqrt(2.0) * sqrt((-0.5 * (im * (im / re)))));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -4.5e+157], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.49999999999999985e157

   1. Initial program 2.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 28.4%

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

      1. sqrt-prod 28.3%

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

      2. *-commutative 28.3%

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

   5. Applied egg-rr 28.3%

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

   6. Taylor expanded in re around -inf 48.1%

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

      1. unpow2 48.1%

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

      2. *-un-lft-identity 48.1%

         \[\leadsto 0.5 \cdot \left(\sqrt{-0.5 \cdot \frac{im \cdot im}{\color{blue}{1 \cdot re}}} \cdot \sqrt{2}\right) \]

      3. times-frac 53.9%

         \[\leadsto 0.5 \cdot \left(\sqrt{-0.5 \cdot \color{blue}{\left(\frac{im}{1} \cdot \frac{im}{re}\right)}} \cdot \sqrt{2}\right) \]

      4. /-rgt-identity 53.9%

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

   8. Applied egg-rr 53.9%

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

   if -4.49999999999999985e157 < re

   1. Initial program 53.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 88.7%

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

      1. add-sqr-sqrt 87.9%

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

      2. sqrt-unprod 88.7%

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

      3. *-commutative 88.7%

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

      4. *-commutative 88.7%

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

      5. swap-sqr 88.7%

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

      6. add-sqr-sqrt 88.7%

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

      7. metadata-eval 88.7%

         \[\leadsto \sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot \color{blue}{0.25}} \]

   5. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot 0.25}} \]
   6. Step-by-step derivation

      1. *-commutative 88.7%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}} \]

      2. associate-*r* 88.7%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

      3. metadata-eval 88.7%

         \[\leadsto \sqrt{\color{blue}{0.5} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]

   7. Simplified 88.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+157}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{-0.5 \cdot \left(im \cdot \frac{im}{re}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 5: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{+181}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -2.1e+181)
   (* 0.5 (sqrt (- (/ (pow im 2.0) re))))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -2.1e+181) {
		tmp = 0.5 * sqrt(-(pow(im, 2.0) / re));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -2.1e+181) {
		tmp = 0.5 * Math.sqrt(-(Math.pow(im, 2.0) / re));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -2.1e+181:
		tmp = 0.5 * math.sqrt(-(math.pow(im, 2.0) / re))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -2.1e+181)
		tmp = Float64(0.5 * sqrt(Float64(-Float64((im ^ 2.0) / re))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.1e+181)
		tmp = 0.5 * sqrt(-((im ^ 2.0) / re));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -2.1e+181], N[(0.5 * N[Sqrt[(-N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.09999999999999997e181

   1. Initial program 2.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 24.7%

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

   4. Taylor expanded in re around -inf 48.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 48.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{{im}^{2}}{re}}} \]

   6. Simplified 48.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{{im}^{2}}{re}}} \]

   if -2.09999999999999997e181 < re

   1. Initial program 52.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 87.6%

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

      1. add-sqr-sqrt 86.9%

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

      2. sqrt-unprod 87.6%

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

      3. *-commutative 87.6%

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

      4. *-commutative 87.6%

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

      5. swap-sqr 87.6%

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

      6. add-sqr-sqrt 87.6%

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

      7. metadata-eval 87.6%

         \[\leadsto \sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot \color{blue}{0.25}} \]

   5. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot 0.25}} \]
   6. Step-by-step derivation

      1. *-commutative 87.6%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}} \]

      2. associate-*r* 87.6%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

      3. metadata-eval 87.6%

         \[\leadsto \sqrt{\color{blue}{0.5} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]

   7. Simplified 87.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{+181}:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 6: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (sqrt (* 0.5 (+ re (hypot re im)))))

C

im = abs(im);
double code(double re, double im) {
	return sqrt((0.5 * (re + hypot(re, im))));
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return Math.sqrt((0.5 * (re + Math.hypot(re, im))));
}

Python

im = abs(im)
def code(re, im):
	return math.sqrt((0.5 * (re + math.hypot(re, im))))

Julia

im = abs(im)
function code(re, im)
	return sqrt(Float64(0.5 * Float64(re + hypot(re, im))))
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = sqrt((0.5 * (re + hypot(re, im))));
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 46.4%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

3. Simplified 80.0%

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

   1. add-sqr-sqrt 79.3%

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

   2. sqrt-unprod 80.0%

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

   3. *-commutative 80.0%

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

   4. *-commutative 80.0%

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

   5. swap-sqr 80.0%

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

   6. add-sqr-sqrt 80.0%

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

   7. metadata-eval 80.0%

      \[\leadsto \sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot \color{blue}{0.25}} \]

5. Applied egg-rr 80.0%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot 0.25}} \]
6. Step-by-step derivation

   1. *-commutative 80.0%

      \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}} \]

   2. associate-*r* 80.0%

      \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   3. metadata-eval 80.0%

      \[\leadsto \sqrt{\color{blue}{0.5} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]

7. Simplified 80.0%

   \[\leadsto \color{blue}{\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

8. Final simplification 80.0%

   \[\leadsto \sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]

Alternative 7: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re 4.4e-42) (* 0.5 (sqrt (* 2.0 im))) (sqrt re)))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= 4.4e-42) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 4.4d-42) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= 4.4e-42) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= 4.4e-42:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= 4.4e-42)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.4e-42)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, 4.4e-42], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.4000000000000001e-42

   1. Initial program 45.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 70.7%

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

   4. Taylor expanded in re around 0 29.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   5. Step-by-step derivation

      1. *-commutative 29.6%

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

   6. Simplified 29.6%

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

   if 4.4000000000000001e-42 < re

   1. Initial program 48.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 83.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 83.6%

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

      2. unpow2 83.6%

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

      3. rem-square-sqrt 85.1%

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

      4. associate-*r* 85.1%

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

      5. metadata-eval 85.1%

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

      6. *-lft-identity 85.1%

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

   6. Simplified 85.1%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 8: 26.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \sqrt{re} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (sqrt re))

C

im = abs(im);
double code(double re, double im) {
	return sqrt(re);
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(re)
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return Math.sqrt(re);
}

Python

im = abs(im)
def code(re, im):
	return math.sqrt(re)

Julia

im = abs(im)
function code(re, im)
	return sqrt(re)
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = sqrt(re);
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 46.4%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]

3. Simplified 80.0%

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

4. Taylor expanded in im around 0 30.9%

   \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. *-commutative 30.9%

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

   2. unpow2 30.9%

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

   3. rem-square-sqrt 31.5%

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

   4. associate-*r* 31.5%

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

   5. metadata-eval 31.5%

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

   6. *-lft-identity 31.5%

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

6. Simplified 31.5%

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

7. Final simplification 31.5%

   \[\leadsto \sqrt{re} \]

Developer target: 48.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023314 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))