math.sqrt on complex, real part

Percentage Accurate: 41.8% → 89.0%

Time: 10.1s

Alternatives: 7

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: 41.8% 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: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (pow (pow (exp 0.25) (fma 2.0 (log im) (log (/ -1.0 re)))) 2.0))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * pow(pow(exp(0.25), fma(2.0, log(im), log((-1.0 / re)))), 2.0);
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], 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[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 4.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 4.7%

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

      2. +-commutative 4.7%

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

      3. sqr-neg 4.7%

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

      4. distribute-rgt-in 4.7%

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

      5. cancel-sign-sub 4.7%

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

      6. distribute-rgt-out-- 4.7%

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

      7. sub-neg 4.7%

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

      8. remove-double-neg 4.7%

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

      9. hypot-def 4.7%

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

   3. Simplified 4.7%

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

      1. add-sqr-sqrt 4.7%

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

      2. pow2 4.7%

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

      3. pow1/2 4.7%

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

      4. sqrt-pow1 4.7%

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

      5. *-commutative 4.7%

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

      6. metadata-eval 4.7%

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

   5. Applied egg-rr 4.7%

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

   6. Taylor expanded in re around -inf 56.7%

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

      1. exp-prod 54.6%

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

      2. +-commutative 54.6%

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

      3. log-pow 47.3%

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

      4. fma-def 47.3%

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

   8. Simplified 47.3%

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

   if 0.0 < (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 46.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 46.4%

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

      2. +-commutative 46.4%

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

      3. sqr-neg 46.4%

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

      4. distribute-rgt-in 46.4%

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

      5. cancel-sign-sub 46.4%

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

      6. distribute-rgt-out-- 46.4%

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

      7. sub-neg 46.4%

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

      8. remove-double-neg 46.4%

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

      9. hypot-def 89.5%

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

   3. Simplified 89.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot {\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -1.36e-24)
   (* 0.5 (pow (exp (* 0.25 (+ (log (/ -1.0 re)) (log (pow im 2.0))))) 2.0))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -1.36e-24) {
		tmp = 0.5 * pow(exp((0.25 * (log((-1.0 / re)) + log(pow(im, 2.0))))), 2.0);
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -1.36e-24) {
		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 = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -1.36e-24:
		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 = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -1.36e-24)
		tmp = Float64(0.5 * (exp(Float64(0.25 * Float64(log(Float64(-1.0 / re)) + log((im ^ 2.0))))) ^ 2.0));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.36e-24)
		tmp = 0.5 * (exp((0.25 * (log((-1.0 / re)) + log((im ^ 2.0))))) ^ 2.0);
	else
		tmp = 0.5 * sqrt((2.0 * (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, -1.36e-24], 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[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.36000000000000001e-24

   1. Initial program 5.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 5.4%

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

      2. +-commutative 5.4%

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

      3. sqr-neg 5.4%

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

      4. distribute-rgt-in 5.4%

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

      5. cancel-sign-sub 5.4%

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

      6. distribute-rgt-out-- 5.4%

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

      7. sub-neg 5.4%

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

      8. remove-double-neg 5.4%

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

      9. hypot-def 25.5%

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

   3. Simplified 25.5%

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

      1. add-sqr-sqrt 25.4%

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

      2. pow2 25.4%

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

      3. pow1/2 25.4%

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

      4. sqrt-pow1 25.4%

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

      5. *-commutative 25.4%

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

      6. metadata-eval 25.4%

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

   5. Applied egg-rr 25.4%

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

   6. Taylor expanded in re around -inf 59.1%

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

   if -1.36000000000000001e-24 < re

   1. Initial program 50.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 50.4%

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

      2. +-commutative 50.4%

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

      3. sqr-neg 50.4%

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

      4. distribute-rgt-in 50.4%

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

      5. cancel-sign-sub 50.4%

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

      6. distribute-rgt-out-- 50.4%

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

      7. sub-neg 50.4%

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

      8. remove-double-neg 50.4%

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

      9. hypot-def 92.2%

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

   3. Simplified 92.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot {\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3: 84.7% accurate, 0.5× speedup

Math

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (sqrt (/ (* im (- im)) re)))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(N[(im * (-im)), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 4.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 4.7%

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

      2. +-commutative 4.7%

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

      3. sqr-neg 4.7%

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

      4. distribute-rgt-in 4.7%

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

      5. cancel-sign-sub 4.7%

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

      6. distribute-rgt-out-- 4.7%

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

      7. sub-neg 4.7%

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

      8. remove-double-neg 4.7%

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

      9. hypot-def 4.7%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in re around -inf 55.8%

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

      1. associate-*r/ 55.8%

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

      2. neg-mul-1 55.8%

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

      3. unpow2 55.8%

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

      4. distribute-rgt-neg-in 55.8%

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

   6. Simplified 55.8%

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

   if 0.0 < (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 46.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 46.4%

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

      2. +-commutative 46.4%

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

      3. sqr-neg 46.4%

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

      4. distribute-rgt-in 46.4%

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

      5. cancel-sign-sub 46.4%

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

      6. distribute-rgt-out-- 46.4%

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

      7. sub-neg 46.4%

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

      8. remove-double-neg 46.4%

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

      9. hypot-def 89.5%

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

   3. Simplified 89.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 4: 68.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{elif}\;re \leq 5.1 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(im + \frac{re \cdot 0.5}{\frac{im}{re}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -1.36e-24)
   (* 0.5 (sqrt (/ (* im (- im)) re)))
   (if (<= re 5.1e-36)
     (* 0.5 (sqrt (* 2.0 (+ re (+ im (/ (* re 0.5) (/ im re)))))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -1.36e-24) {
		tmp = 0.5 * sqrt(((im * -im) / re));
	} else if (re <= 5.1e-36) {
		tmp = 0.5 * sqrt((2.0 * (re + (im + ((re * 0.5) / (im / re))))));
	} else {
		tmp = 0.5 * (2.0 * 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 <= (-1.36d-24)) then
        tmp = 0.5d0 * sqrt(((im * -im) / re))
    else if (re <= 5.1d-36) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + (im + ((re * 0.5d0) / (im / re))))))
    else
        tmp = 0.5d0 * (2.0d0 * 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 <= -1.36e-24) {
		tmp = 0.5 * Math.sqrt(((im * -im) / re));
	} else if (re <= 5.1e-36) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + (im + ((re * 0.5) / (im / re))))));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -1.36e-24:
		tmp = 0.5 * math.sqrt(((im * -im) / re))
	elif re <= 5.1e-36:
		tmp = 0.5 * math.sqrt((2.0 * (re + (im + ((re * 0.5) / (im / re))))))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -1.36e-24)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * Float64(-im)) / re)));
	elseif (re <= 5.1e-36)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + Float64(im + Float64(Float64(re * 0.5) / Float64(im / re)))))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.36e-24)
		tmp = 0.5 * sqrt(((im * -im) / re));
	elseif (re <= 5.1e-36)
		tmp = 0.5 * sqrt((2.0 * (re + (im + ((re * 0.5) / (im / re))))));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -1.36e-24], N[(0.5 * N[Sqrt[N[(N[(im * (-im)), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.1e-36], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[(im + N[(N[(re * 0.5), $MachinePrecision] / N[(im / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.36000000000000001e-24

   1. Initial program 5.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 5.4%

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

      2. +-commutative 5.4%

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

      3. sqr-neg 5.4%

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

      4. distribute-rgt-in 5.4%

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

      5. cancel-sign-sub 5.4%

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

      6. distribute-rgt-out-- 5.4%

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

      7. sub-neg 5.4%

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

      8. remove-double-neg 5.4%

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

      9. hypot-def 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in re around -inf 52.6%

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

      1. associate-*r/ 52.6%

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

      2. neg-mul-1 52.6%

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

      3. unpow2 52.6%

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

      4. distribute-rgt-neg-in 52.6%

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

   6. Simplified 52.6%

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

   if -1.36000000000000001e-24 < re < 5.09999999999999973e-36

   1. Initial program 52.8%

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

   2. Taylor expanded in re around 0 37.6%

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

      1. unpow2 37.6%

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

   4. Simplified 37.6%

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

      1. *-commutative 37.6%

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

      2. associate-/l* 37.6%

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

      3. associate-*l/ 37.6%

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

   6. Applied egg-rr 37.6%

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

   if 5.09999999999999973e-36 < re

   1. Initial program 46.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 46.9%

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

      2. +-commutative 46.9%

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

      3. sqr-neg 46.9%

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

      4. distribute-rgt-in 46.9%

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

      5. cancel-sign-sub 46.9%

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

      6. distribute-rgt-out-- 46.9%

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

      7. sub-neg 46.9%

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

      8. remove-double-neg 46.9%

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

      9. hypot-def 100.0%

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

   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 79.9%

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

      1. *-commutative 79.9%

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

      2. unpow2 79.9%

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

      3. rem-square-sqrt 81.4%

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

   6. Simplified 81.4%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{elif}\;re \leq 5.1 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(im + \frac{re \cdot 0.5}{\frac{im}{re}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -1.36e-24)
   (* 0.5 (sqrt (/ (* im (- im)) re)))
   (if (<= re 2.7e+21)
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -1.36e-24) {
		tmp = 0.5 * sqrt(((im * -im) / re));
	} else if (re <= 2.7e+21) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * 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 <= (-1.36d-24)) then
        tmp = 0.5d0 * sqrt(((im * -im) / re))
    else if (re <= 2.7d+21) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * 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 <= -1.36e-24) {
		tmp = 0.5 * Math.sqrt(((im * -im) / re));
	} else if (re <= 2.7e+21) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -1.36e-24:
		tmp = 0.5 * math.sqrt(((im * -im) / re))
	elif re <= 2.7e+21:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -1.36e-24)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * Float64(-im)) / re)));
	elseif (re <= 2.7e+21)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.36e-24)
		tmp = 0.5 * sqrt(((im * -im) / re));
	elseif (re <= 2.7e+21)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -1.36e-24], N[(0.5 * N[Sqrt[N[(N[(im * (-im)), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.7e+21], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.36000000000000001e-24

   1. Initial program 5.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 5.4%

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

      2. +-commutative 5.4%

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

      3. sqr-neg 5.4%

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

      4. distribute-rgt-in 5.4%

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

      5. cancel-sign-sub 5.4%

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

      6. distribute-rgt-out-- 5.4%

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

      7. sub-neg 5.4%

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

      8. remove-double-neg 5.4%

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

      9. hypot-def 25.5%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in re around -inf 52.6%

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

      1. associate-*r/ 52.6%

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

      2. neg-mul-1 52.6%

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

      3. unpow2 52.6%

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

      4. distribute-rgt-neg-in 52.6%

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

   6. Simplified 52.6%

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

   if -1.36000000000000001e-24 < re < 2.7e21

   1. Initial program 57.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 57.2%

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

      2. +-commutative 57.2%

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

      3. sqr-neg 57.2%

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

      4. distribute-rgt-in 57.2%

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

      5. cancel-sign-sub 57.2%

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

      6. distribute-rgt-out-- 57.2%

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

      7. sub-neg 57.2%

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

      8. remove-double-neg 57.2%

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

      9. hypot-def 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in re around 0 38.5%

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

      1. distribute-lft-out 38.5%

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

      2. *-commutative 38.5%

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

   6. Simplified 38.5%

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

   if 2.7e21 < re

   1. Initial program 35.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 35.7%

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

      2. +-commutative 35.7%

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

      3. sqr-neg 35.7%

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

      4. distribute-rgt-in 35.7%

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

      5. cancel-sign-sub 35.7%

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

      6. distribute-rgt-out-- 35.7%

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

      7. sub-neg 35.7%

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

      8. remove-double-neg 35.7%

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

      9. hypot-def 100.0%

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

   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 87.2%

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

      1. *-commutative 87.2%

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

      2. unpow2 87.2%

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

      3. rem-square-sqrt 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{elif}\;re \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 6: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 5.1 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re 5.1e-36) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* 2.0 (sqrt re)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= 5.1e-36) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * 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 <= 5.1d-36) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (2.0d0 * 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 <= 5.1e-36) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= 5.1e-36:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= 5.1e-36)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.1e-36)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, 5.1e-36], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.09999999999999973e-36

   1. Initial program 37.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 37.7%

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

      2. +-commutative 37.7%

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

      3. sqr-neg 37.7%

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

      4. distribute-rgt-in 37.7%

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

      5. cancel-sign-sub 37.7%

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

      6. distribute-rgt-out-- 37.7%

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

      7. sub-neg 37.7%

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

      8. remove-double-neg 37.7%

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

      9. hypot-def 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in re around 0 29.0%

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

      1. *-commutative 29.0%

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

   6. Simplified 29.0%

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

   if 5.09999999999999973e-36 < re

   1. Initial program 46.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. sqr-neg 46.9%

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

      2. +-commutative 46.9%

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

      3. sqr-neg 46.9%

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

      4. distribute-rgt-in 46.9%

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

      5. cancel-sign-sub 46.9%

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

      6. distribute-rgt-out-- 46.9%

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

      7. sub-neg 46.9%

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

      8. remove-double-neg 46.9%

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

      9. hypot-def 100.0%

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

   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 79.9%

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

      1. *-commutative 79.9%

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

      2. unpow2 79.9%

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

      3. rem-square-sqrt 81.4%

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

   6. Simplified 81.4%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.1 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 7: 52.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ 0.5 \cdot \sqrt{2 \cdot im} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (* 0.5 (sqrt (* 2.0 im))))

C

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

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 = 0.5d0 * sqrt((2.0d0 * im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
2. Step-by-step derivation

   1. sqr-neg 40.7%

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

   2. +-commutative 40.7%

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

   3. sqr-neg 40.7%

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

   4. distribute-rgt-in 40.7%

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

   5. cancel-sign-sub 40.7%

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

   6. distribute-rgt-out-- 40.7%

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

   7. sub-neg 40.7%

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

   8. remove-double-neg 40.7%

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

   9. hypot-def 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in re around 0 23.7%

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

   1. *-commutative 23.7%

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

6. Simplified 23.7%

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

7. Final simplification 23.7%

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

Developer target: 49.3% 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 2023279 
(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)))))