math.sqrt on complex, real part

Percentage Accurate: 41.1% → 89.3%

Time: 15.9s

Alternatives: 8

Speedup: 1.9×

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: 41.1% 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: 89.3% accurate, 0.3× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im_m im_m)))))) 0.0)
   (* 0.5 (pow (pow (exp 0.25) (+ (log (/ -1.0 re)) (* 2.0 (log im_m)))) 2.0))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

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

Java

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

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im_m * im_m)))))) <= 0.0:
		tmp = 0.5 * math.pow(math.pow(math.exp(0.25), (math.log((-1.0 / re)) + (2.0 * math.log(im_m)))), 2.0)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im_m))))
	return tmp

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] + N[(2.0 * N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 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 12.9%

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

      1. sqr-neg 12.9%

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

      2. +-commutative 12.9%

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

      3. sqr-neg 12.9%

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

      4. +-commutative 12.9%

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

      5. distribute-rgt-in 12.9%

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

      6. cancel-sign-sub 12.9%

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

      7. distribute-rgt-out-- 12.9%

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

      8. sub-neg 12.9%

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

      9. remove-double-neg 12.9%

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

      10. +-commutative 12.9%

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

      11. hypot-define 12.9%

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

   3. Simplified 12.9%

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

      1. *-commutative 12.9%

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

      2. hypot-define 12.9%

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

      3. +-commutative 12.9%

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

      4. *-commutative 12.9%

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

      5. add-sqr-sqrt 12.9%

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

      6. pow2 12.9%

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

      7. pow1/2 12.9%

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

      8. sqrt-pow1 12.9%

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

      9. +-commutative 12.9%

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

      10. hypot-define 12.9%

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

      11. metadata-eval 12.9%

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

   6. Applied egg-rr 12.9%

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

   7. Taylor expanded in re around -inf 57.0%

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

      1. exp-prod 55.0%

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

      2. log-pow 34.8%

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

   9. Simplified 34.8%

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

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

   1. Initial program 50.2%

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

      1. sqr-neg 50.2%

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

      2. +-commutative 50.2%

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

      3. sqr-neg 50.2%

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

      4. +-commutative 50.2%

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

      5. distribute-rgt-in 50.2%

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

      6. cancel-sign-sub 50.2%

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

      7. distribute-rgt-out-- 50.2%

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

      8. sub-neg 50.2%

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

      9. remove-double-neg 50.2%

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

      10. +-commutative 50.2%

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

      11. hypot-define 89.8%

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

   3. Simplified 89.8%

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

4. Final simplification 81.6%

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

Alternative 2: 89.3% accurate, 0.3× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im_m im_m)))))) 0.0)
   (* 0.5 (pow (exp 0.25) (* 2.0 (+ (log (/ -1.0 re)) (* 2.0 (log im_m))))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

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

Java

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

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im_m * im_m)))))) <= 0.0:
		tmp = 0.5 * math.pow(math.exp(0.25), (2.0 * (math.log((-1.0 / re)) + (2.0 * math.log(im_m)))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im_m))))
	return tmp

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Power[N[Exp[0.25], $MachinePrecision], N[(2.0 * N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] + N[(2.0 * N[Log[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 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 12.9%

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

      1. sqr-neg 12.9%

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

      2. +-commutative 12.9%

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

      3. sqr-neg 12.9%

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

      4. +-commutative 12.9%

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

      5. distribute-rgt-in 12.9%

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

      6. cancel-sign-sub 12.9%

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

      7. distribute-rgt-out-- 12.9%

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

      8. sub-neg 12.9%

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

      9. remove-double-neg 12.9%

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

      10. +-commutative 12.9%

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

      11. hypot-define 12.9%

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

   3. Simplified 12.9%

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

      1. *-commutative 12.9%

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

      2. hypot-define 12.9%

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

      3. +-commutative 12.9%

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

      4. *-commutative 12.9%

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

      5. add-sqr-sqrt 12.9%

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

      6. pow2 12.9%

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

      7. pow1/2 12.9%

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

      8. sqrt-pow1 12.9%

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

      9. +-commutative 12.9%

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

      10. hypot-define 12.9%

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

      11. metadata-eval 12.9%

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

   6. Applied egg-rr 12.9%

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

   7. Taylor expanded in re around -inf 57.0%

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

      1. unpow2 57.0%

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

      2. exp-prod 55.8%

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

      3. exp-prod 55.0%

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

      4. pow-sqr 55.0%

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

      5. log-pow 34.8%

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

   9. Simplified 34.8%

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

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

   1. Initial program 50.2%

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

      1. sqr-neg 50.2%

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

      2. +-commutative 50.2%

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

      3. sqr-neg 50.2%

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

      4. +-commutative 50.2%

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

      5. distribute-rgt-in 50.2%

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

      6. cancel-sign-sub 50.2%

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

      7. distribute-rgt-out-- 50.2%

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

      8. sub-neg 50.2%

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

      9. remove-double-neg 50.2%

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

      10. +-commutative 50.2%

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

      11. hypot-define 89.8%

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

   3. Simplified 89.8%

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

4. Final simplification 81.6%

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

Alternative 3: 84.9% accurate, 0.5× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im_m im_m)))))) 0.0)
   (* 0.5 (sqrt (/ (pow im_m 2.0) (- re))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(N[Power[im$95$m, 2.0], $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 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 12.9%

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

      1. sqr-neg 12.9%

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

      2. +-commutative 12.9%

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

      3. sqr-neg 12.9%

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

      4. +-commutative 12.9%

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

      5. distribute-rgt-in 12.9%

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

      6. cancel-sign-sub 12.9%

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

      7. distribute-rgt-out-- 12.9%

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

      8. sub-neg 12.9%

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

      9. remove-double-neg 12.9%

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

      10. +-commutative 12.9%

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

      11. hypot-define 12.9%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in re around -inf 61.0%

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

      1. mul-1-neg 61.0%

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

      2. distribute-neg-frac2 61.0%

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

   7. Simplified 61.0%

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

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

   1. Initial program 50.2%

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

      1. sqr-neg 50.2%

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

      2. +-commutative 50.2%

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

      3. sqr-neg 50.2%

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

      4. +-commutative 50.2%

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

      5. distribute-rgt-in 50.2%

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

      6. cancel-sign-sub 50.2%

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

      7. distribute-rgt-out-- 50.2%

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

      8. sub-neg 50.2%

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

      9. remove-double-neg 50.2%

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

      10. +-commutative 50.2%

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

      11. hypot-define 89.8%

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

   3. Simplified 89.8%

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

4. Final simplification 85.5%

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

Alternative 4: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.55 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im\_m}^{2}}{-re}}\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im\_m}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+14} \lor \neg \left(re \leq 2.65 \cdot 10^{+39}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.55e+14)
   (* 0.5 (sqrt (/ (pow im_m 2.0) (- re))))
   (if (<= re 8.4e-78)
     (* 0.5 (sqrt (* 2.0 im_m)))
     (if (or (<= re 8.2e+14) (not (<= re 2.65e+39)))
       (* 0.5 (* 2.0 (sqrt re)))
       (* 0.5 (sqrt (* 2.0 (+ re im_m))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.55e+14) {
		tmp = 0.5 * sqrt((pow(im_m, 2.0) / -re));
	} else if (re <= 8.4e-78) {
		tmp = 0.5 * sqrt((2.0 * im_m));
	} else if ((re <= 8.2e+14) || !(re <= 2.65e+39)) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-1.55d+14)) then
        tmp = 0.5d0 * sqrt(((im_m ** 2.0d0) / -re))
    else if (re <= 8.4d-78) then
        tmp = 0.5d0 * sqrt((2.0d0 * im_m))
    else if ((re <= 8.2d+14) .or. (.not. (re <= 2.65d+39))) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -1.55e+14) {
		tmp = 0.5 * Math.sqrt((Math.pow(im_m, 2.0) / -re));
	} else if (re <= 8.4e-78) {
		tmp = 0.5 * Math.sqrt((2.0 * im_m));
	} else if ((re <= 8.2e+14) || !(re <= 2.65e+39)) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.55e+14:
		tmp = 0.5 * math.sqrt((math.pow(im_m, 2.0) / -re))
	elif re <= 8.4e-78:
		tmp = 0.5 * math.sqrt((2.0 * im_m))
	elif (re <= 8.2e+14) or not (re <= 2.65e+39):
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.55e+14)
		tmp = Float64(0.5 * sqrt(Float64((im_m ^ 2.0) / Float64(-re))));
	elseif (re <= 8.4e-78)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im_m)));
	elseif ((re <= 8.2e+14) || !(re <= 2.65e+39))
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -1.55e+14)
		tmp = 0.5 * sqrt(((im_m ^ 2.0) / -re));
	elseif (re <= 8.4e-78)
		tmp = 0.5 * sqrt((2.0 * im_m));
	elseif ((re <= 8.2e+14) || ~((re <= 2.65e+39)))
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.55e+14], N[(0.5 * N[Sqrt[N[(N[Power[im$95$m, 2.0], $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.4e-78], N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 8.2e+14], N[Not[LessEqual[re, 2.65e+39]], $MachinePrecision]], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.55e14

   1. Initial program 9.6%

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

      1. sqr-neg 9.6%

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

      2. +-commutative 9.6%

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

      3. sqr-neg 9.6%

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

      4. +-commutative 9.6%

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

      5. distribute-rgt-in 9.6%

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

      6. cancel-sign-sub 9.6%

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

      7. distribute-rgt-out-- 9.6%

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

      8. sub-neg 9.6%

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

      9. remove-double-neg 9.6%

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

      10. +-commutative 9.6%

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

      11. hypot-define 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in re around -inf 61.7%

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

      1. mul-1-neg 61.7%

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

      2. distribute-neg-frac2 61.7%

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

   7. Simplified 61.7%

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

   if -1.55e14 < re < 8.4000000000000002e-78

   1. Initial program 48.1%

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

      1. sqr-neg 48.1%

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

      2. +-commutative 48.1%

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

      3. sqr-neg 48.1%

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

      4. +-commutative 48.1%

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

      5. distribute-rgt-in 48.1%

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

      6. cancel-sign-sub 48.1%

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

      7. distribute-rgt-out-- 48.1%

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

      8. sub-neg 48.1%

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

      9. remove-double-neg 48.1%

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

      10. +-commutative 48.1%

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

      11. hypot-define 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in re around 0 38.5%

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

      1. *-commutative 38.5%

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

   7. Simplified 38.5%

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

   if 8.4000000000000002e-78 < re < 8.2e14 or 2.64999999999999989e39 < re

   1. Initial program 63.6%

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

      1. sqr-neg 63.6%

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

      2. +-commutative 63.6%

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

      3. sqr-neg 63.6%

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

      4. +-commutative 63.6%

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

      5. distribute-rgt-in 63.6%

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

      6. cancel-sign-sub 63.6%

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

      7. distribute-rgt-out-- 63.6%

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

      8. sub-neg 63.6%

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

      9. remove-double-neg 63.6%

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

      10. +-commutative 63.6%

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

      11. hypot-define 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. Add Preprocessing

   5. Taylor expanded in im around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow2 73.6%

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

      3. rem-square-sqrt 75.0%

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

   7. Simplified 75.0%

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

   if 8.2e14 < re < 2.64999999999999989e39

   1. Initial program 45.3%

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

      1. sqr-neg 45.3%

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

      2. +-commutative 45.3%

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

      3. sqr-neg 45.3%

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

      4. +-commutative 45.3%

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

      5. distribute-rgt-in 45.3%

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

      6. cancel-sign-sub 45.3%

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

      7. distribute-rgt-out-- 45.3%

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

      8. sub-neg 45.3%

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

      9. remove-double-neg 45.3%

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

      10. +-commutative 45.3%

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

      11. hypot-define 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. Add Preprocessing

   5. Taylor expanded in re around 0 45.7%

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

      1. distribute-lft-out 45.7%

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

      2. *-commutative 45.7%

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

   7. Simplified 45.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right) \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.55 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{-re}}\\ \mathbf{elif}\;re \leq 8.4 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{+14} \lor \neg \left(re \leq 2.65 \cdot 10^{+39}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.7 \cdot 10^{+197}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im\_m}\\ \mathbf{elif}\;re \leq 5.7 \cdot 10^{+16} \lor \neg \left(re \leq 2.8 \cdot 10^{+39}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.7e+197)
   (* 0.5 (sqrt 0.0))
   (if (<= re 6e-78)
     (* 0.5 (sqrt (* 2.0 im_m)))
     (if (or (<= re 5.7e+16) (not (<= re 2.8e+39)))
       (* 0.5 (* 2.0 (sqrt re)))
       (* 0.5 (sqrt (* 2.0 (+ re im_m))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.7e+197) {
		tmp = 0.5 * sqrt(0.0);
	} else if (re <= 6e-78) {
		tmp = 0.5 * sqrt((2.0 * im_m));
	} else if ((re <= 5.7e+16) || !(re <= 2.8e+39)) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-3.7d+197)) then
        tmp = 0.5d0 * sqrt(0.0d0)
    else if (re <= 6d-78) then
        tmp = 0.5d0 * sqrt((2.0d0 * im_m))
    else if ((re <= 5.7d+16) .or. (.not. (re <= 2.8d+39))) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3.7e+197) {
		tmp = 0.5 * Math.sqrt(0.0);
	} else if (re <= 6e-78) {
		tmp = 0.5 * Math.sqrt((2.0 * im_m));
	} else if ((re <= 5.7e+16) || !(re <= 2.8e+39)) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.7e+197:
		tmp = 0.5 * math.sqrt(0.0)
	elif re <= 6e-78:
		tmp = 0.5 * math.sqrt((2.0 * im_m))
	elif (re <= 5.7e+16) or not (re <= 2.8e+39):
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.7e+197)
		tmp = Float64(0.5 * sqrt(0.0));
	elseif (re <= 6e-78)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im_m)));
	elseif ((re <= 5.7e+16) || !(re <= 2.8e+39))
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3.7e+197)
		tmp = 0.5 * sqrt(0.0);
	elseif (re <= 6e-78)
		tmp = 0.5 * sqrt((2.0 * im_m));
	elseif ((re <= 5.7e+16) || ~((re <= 2.8e+39)))
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.7e+197], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6e-78], N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 5.7e+16], N[Not[LessEqual[re, 2.8e+39]], $MachinePrecision]], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.7000000000000003e197

   1. Initial program 2.7%

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

      1. sqr-neg 2.7%

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

      2. +-commutative 2.7%

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

      3. sqr-neg 2.7%

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

      4. +-commutative 2.7%

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

      5. distribute-rgt-in 2.7%

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

      6. cancel-sign-sub 2.7%

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

      7. distribute-rgt-out-- 2.7%

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

      8. sub-neg 2.7%

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

      9. remove-double-neg 2.7%

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

      10. +-commutative 2.7%

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

      11. hypot-define 40.6%

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

   3. Simplified 40.6%

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

      1. hypot-define 2.7%

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

      2. +-commutative 2.7%

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

      3. distribute-rgt-in 2.7%

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

      4. add-sqr-sqrt 2.7%

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

      5. *-commutative 2.7%

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

      6. fma-define 2.7%

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

      7. *-commutative 2.7%

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

      8. hypot-define 2.7%

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

      9. *-commutative 2.7%

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

      10. hypot-define 11.9%

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

      11. *-commutative 11.9%

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

   6. Applied egg-rr 11.9%

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

   7. Taylor expanded in re around -inf 5.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \left(re \cdot \left({\left(\sqrt{2}\right)}^{2} - 2\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 5.2%

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

      2. neg-mul-1 5.2%

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

      3. unpow2 5.2%

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

      4. rem-square-sqrt 31.3%

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

      5. metadata-eval 31.3%

         \[\leadsto 0.5 \cdot \sqrt{\left(-re\right) \cdot \color{blue}{0}} \]

      6. mul0-rgt 31.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

   9. Simplified 31.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

   if -3.7000000000000003e197 < re < 5.99999999999999975e-78

   1. Initial program 40.6%

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

      1. sqr-neg 40.6%

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

      2. +-commutative 40.6%

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

      3. sqr-neg 40.6%

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

      4. +-commutative 40.6%

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

      5. distribute-rgt-in 40.6%

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

      6. cancel-sign-sub 40.6%

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

      7. distribute-rgt-out-- 40.6%

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

      8. sub-neg 40.6%

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

      9. remove-double-neg 40.6%

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

      10. +-commutative 40.6%

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

      11. hypot-define 71.2%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in re around 0 33.2%

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

      1. *-commutative 33.2%

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

   7. Simplified 33.2%

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

   if 5.99999999999999975e-78 < re < 5.7e16 or 2.80000000000000001e39 < re

   1. Initial program 63.6%

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

      1. sqr-neg 63.6%

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

      2. +-commutative 63.6%

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

      3. sqr-neg 63.6%

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

      4. +-commutative 63.6%

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

      5. distribute-rgt-in 63.6%

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

      6. cancel-sign-sub 63.6%

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

      7. distribute-rgt-out-- 63.6%

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

      8. sub-neg 63.6%

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

      9. remove-double-neg 63.6%

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

      10. +-commutative 63.6%

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

      11. hypot-define 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. Add Preprocessing

   5. Taylor expanded in im around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow2 73.6%

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

      3. rem-square-sqrt 75.0%

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

   7. Simplified 75.0%

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

   if 5.7e16 < re < 2.80000000000000001e39

   1. Initial program 45.3%

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

      1. sqr-neg 45.3%

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

      2. +-commutative 45.3%

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

      3. sqr-neg 45.3%

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

      4. +-commutative 45.3%

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

      5. distribute-rgt-in 45.3%

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

      6. cancel-sign-sub 45.3%

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

      7. distribute-rgt-out-- 45.3%

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

      8. sub-neg 45.3%

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

      9. remove-double-neg 45.3%

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

      10. +-commutative 45.3%

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

      11. hypot-define 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. Add Preprocessing

   5. Taylor expanded in re around 0 45.7%

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

      1. distribute-lft-out 45.7%

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

      2. *-commutative 45.7%

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

   7. Simplified 45.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right) \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.7 \cdot 10^{+197}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{-78}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 5.7 \cdot 10^{+16} \lor \neg \left(re \leq 2.8 \cdot 10^{+39}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+199}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{-78} \lor \neg \left(re \leq 5.5 \cdot 10^{+18}\right) \land re \leq 1.18 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5e+199)
   (* 0.5 (sqrt 0.0))
   (if (or (<= re 6.2e-78) (and (not (<= re 5.5e+18)) (<= re 1.18e+39)))
     (* 0.5 (sqrt (* 2.0 im_m)))
     (* 0.5 (* 2.0 (sqrt re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5e+199) {
		tmp = 0.5 * sqrt(0.0);
	} else if ((re <= 6.2e-78) || (!(re <= 5.5e+18) && (re <= 1.18e+39))) {
		tmp = 0.5 * sqrt((2.0 * im_m));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-5d+199)) then
        tmp = 0.5d0 * sqrt(0.0d0)
    else if ((re <= 6.2d-78) .or. (.not. (re <= 5.5d+18)) .and. (re <= 1.18d+39)) then
        tmp = 0.5d0 * sqrt((2.0d0 * im_m))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -5e+199) {
		tmp = 0.5 * Math.sqrt(0.0);
	} else if ((re <= 6.2e-78) || (!(re <= 5.5e+18) && (re <= 1.18e+39))) {
		tmp = 0.5 * Math.sqrt((2.0 * im_m));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -5e+199:
		tmp = 0.5 * math.sqrt(0.0)
	elif (re <= 6.2e-78) or (not (re <= 5.5e+18) and (re <= 1.18e+39)):
		tmp = 0.5 * math.sqrt((2.0 * im_m))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -5e+199)
		tmp = Float64(0.5 * sqrt(0.0));
	elseif ((re <= 6.2e-78) || (!(re <= 5.5e+18) && (re <= 1.18e+39)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im_m)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -5e+199)
		tmp = 0.5 * sqrt(0.0);
	elseif ((re <= 6.2e-78) || (~((re <= 5.5e+18)) && (re <= 1.18e+39)))
		tmp = 0.5 * sqrt((2.0 * im_m));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -5e+199], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 6.2e-78], And[N[Not[LessEqual[re, 5.5e+18]], $MachinePrecision], LessEqual[re, 1.18e+39]]], N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $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 < -4.9999999999999998e199

   1. Initial program 2.7%

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

      1. sqr-neg 2.7%

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

      2. +-commutative 2.7%

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

      3. sqr-neg 2.7%

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

      4. +-commutative 2.7%

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

      5. distribute-rgt-in 2.7%

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

      6. cancel-sign-sub 2.7%

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

      7. distribute-rgt-out-- 2.7%

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

      8. sub-neg 2.7%

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

      9. remove-double-neg 2.7%

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

      10. +-commutative 2.7%

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

      11. hypot-define 40.6%

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

   3. Simplified 40.6%

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

      1. hypot-define 2.7%

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

      2. +-commutative 2.7%

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

      3. distribute-rgt-in 2.7%

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

      4. add-sqr-sqrt 2.7%

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

      5. *-commutative 2.7%

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

      6. fma-define 2.7%

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

      7. *-commutative 2.7%

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

      8. hypot-define 2.7%

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

      9. *-commutative 2.7%

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

      10. hypot-define 11.9%

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

      11. *-commutative 11.9%

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

   6. Applied egg-rr 11.9%

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

   7. Taylor expanded in re around -inf 5.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \left(re \cdot \left({\left(\sqrt{2}\right)}^{2} - 2\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 5.2%

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

      2. neg-mul-1 5.2%

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

      3. unpow2 5.2%

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

      4. rem-square-sqrt 31.3%

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

      5. metadata-eval 31.3%

         \[\leadsto 0.5 \cdot \sqrt{\left(-re\right) \cdot \color{blue}{0}} \]

      6. mul0-rgt 31.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

   9. Simplified 31.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

   if -4.9999999999999998e199 < re < 6.20000000000000035e-78 or 5.5e18 < re < 1.17999999999999996e39

   1. Initial program 40.8%

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

      1. sqr-neg 40.8%

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

      2. +-commutative 40.8%

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

      3. sqr-neg 40.8%

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

      4. +-commutative 40.8%

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

      5. distribute-rgt-in 40.8%

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

      6. cancel-sign-sub 40.8%

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

      7. distribute-rgt-out-- 40.8%

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

      8. sub-neg 40.8%

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

      9. remove-double-neg 40.8%

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

      10. +-commutative 40.8%

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

      11. hypot-define 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in re around 0 33.6%

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

      1. *-commutative 33.6%

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

   7. Simplified 33.6%

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

   if 6.20000000000000035e-78 < re < 5.5e18 or 1.17999999999999996e39 < re

   1. Initial program 63.6%

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

      1. sqr-neg 63.6%

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

      2. +-commutative 63.6%

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

      3. sqr-neg 63.6%

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

      4. +-commutative 63.6%

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

      5. distribute-rgt-in 63.6%

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

      6. cancel-sign-sub 63.6%

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

      7. distribute-rgt-out-- 63.6%

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

      8. sub-neg 63.6%

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

      9. remove-double-neg 63.6%

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

      10. +-commutative 63.6%

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

      11. hypot-define 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. Add Preprocessing

   5. Taylor expanded in im around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow2 73.6%

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

      3. rem-square-sqrt 75.0%

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

   7. Simplified 75.0%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+199}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{elif}\;re \leq 6.2 \cdot 10^{-78} \lor \neg \left(re \leq 5.5 \cdot 10^{+18}\right) \land re \leq 1.18 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 3.8 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im\_m}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 3.8e-265) (* 0.5 (sqrt 0.0)) (* 0.5 (sqrt (* 2.0 im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.8e-265) {
		tmp = 0.5 * sqrt(0.0);
	} else {
		tmp = 0.5 * sqrt((2.0 * im_m));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.8d-265) then
        tmp = 0.5d0 * sqrt(0.0d0)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im_m))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.8e-265) {
		tmp = 0.5 * Math.sqrt(0.0);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im_m));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 3.8e-265:
		tmp = 0.5 * math.sqrt(0.0)
	else:
		tmp = 0.5 * math.sqrt((2.0 * im_m))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 3.8e-265)
		tmp = Float64(0.5 * sqrt(0.0));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im_m)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 3.8e-265)
		tmp = 0.5 * sqrt(0.0);
	else
		tmp = 0.5 * sqrt((2.0 * im_m));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 3.8e-265], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7999999999999998e-265

   1. Initial program 39.6%

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

      1. sqr-neg 39.6%

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

      2. +-commutative 39.6%

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

      3. sqr-neg 39.6%

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

      4. +-commutative 39.6%

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

      5. distribute-rgt-in 39.6%

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

      6. cancel-sign-sub 39.6%

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

      7. distribute-rgt-out-- 39.6%

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

      8. sub-neg 39.6%

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

      9. remove-double-neg 39.6%

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

      10. +-commutative 39.6%

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

      11. hypot-define 75.5%

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

   3. Simplified 75.5%

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

      1. hypot-define 39.6%

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

      2. +-commutative 39.6%

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

      3. distribute-rgt-in 39.6%

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

      4. add-sqr-sqrt 38.6%

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

      5. *-commutative 38.6%

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

      6. fma-define 37.2%

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

      7. *-commutative 37.2%

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

      8. hypot-define 37.3%

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

      9. *-commutative 37.3%

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

      10. hypot-define 70.3%

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

      11. *-commutative 70.3%

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

   6. Applied egg-rr 70.3%

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

   7. Taylor expanded in re around -inf 3.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \left(re \cdot \left({\left(\sqrt{2}\right)}^{2} - 2\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 3.4%

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

      2. neg-mul-1 3.4%

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

      3. unpow2 3.4%

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

      4. rem-square-sqrt 8.1%

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

      5. metadata-eval 8.1%

         \[\leadsto 0.5 \cdot \sqrt{\left(-re\right) \cdot \color{blue}{0}} \]

      6. mul0-rgt 8.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

   9. Simplified 8.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

   if 3.7999999999999998e-265 < im

   1. Initial program 51.3%

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

      1. sqr-neg 51.3%

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

      2. +-commutative 51.3%

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

      3. sqr-neg 51.3%

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

      4. +-commutative 51.3%

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

      5. distribute-rgt-in 51.3%

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

      6. cancel-sign-sub 51.3%

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

      7. distribute-rgt-out-- 51.3%

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

      8. sub-neg 51.3%

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

      9. remove-double-neg 51.3%

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

      10. +-commutative 51.3%

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

      11. hypot-define 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in re around 0 55.2%

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

      1. *-commutative 55.2%

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

   7. Simplified 55.2%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 6.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.5 \cdot \sqrt{0} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* 0.5 (sqrt 0.0)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.5 * sqrt(0.0);
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.5d0 * sqrt(0.0d0)
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.5 * Math.sqrt(0.0);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.5 * math.sqrt(0.0)

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(0.5 * sqrt(0.0))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.5 * sqrt(0.0);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.7%

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

   1. sqr-neg 44.7%

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

   2. +-commutative 44.7%

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

   3. sqr-neg 44.7%

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

   4. +-commutative 44.7%

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

   5. distribute-rgt-in 44.7%

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

   6. cancel-sign-sub 44.7%

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

   7. distribute-rgt-out-- 44.7%

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

   8. sub-neg 44.7%

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

   9. remove-double-neg 44.7%

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

   10. +-commutative 44.7%

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

   11. hypot-define 78.4%

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

3. Simplified 78.4%

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

   1. hypot-define 44.7%

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

   2. +-commutative 44.7%

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

   3. distribute-rgt-in 44.7%

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

   4. add-sqr-sqrt 44.1%

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

   5. *-commutative 44.1%

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

   6. fma-define 43.2%

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

   7. *-commutative 43.2%

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

   8. hypot-define 43.2%

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

   9. *-commutative 43.2%

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

   10. hypot-define 74.5%

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

   11. *-commutative 74.5%

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

6. Applied egg-rr 74.5%

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

7. Taylor expanded in re around -inf 3.1%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \left(re \cdot \left({\left(\sqrt{2}\right)}^{2} - 2\right)\right)}} \]
8. Step-by-step derivation

   1. associate-*r* 3.1%

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

   2. neg-mul-1 3.1%

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

   3. unpow2 3.1%

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

   4. rem-square-sqrt 6.8%

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

   5. metadata-eval 6.8%

      \[\leadsto 0.5 \cdot \sqrt{\left(-re\right) \cdot \color{blue}{0}} \]

   6. mul0-rgt 6.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

9. Simplified 6.8%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

10. Final simplification 6.8%

    \[\leadsto 0.5 \cdot \sqrt{0} \]
11. Add Preprocessing

Developer target: 47.6% 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 2024040 
(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)))))