math.sqrt on complex, real part

Percentage Accurate: 42.3% → 89.6%

Time: 6.2s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.3% 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: 89.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 12.1%

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

      1. +-commutative 12.1%

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

      2. hypot-def 12.1%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in re around -inf 60.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + 0.25 \cdot \frac{{im}^{4}}{{re}^{3}}}} \]
   5. Step-by-step derivation

      1. +-commutative 60.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + -1 \cdot \frac{{im}^{2}}{re}}} \]

      2. mul-1-neg 60.1%

         \[\leadsto 0.5 \cdot \sqrt{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]

      3. unsub-neg 60.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} - \frac{{im}^{2}}{re}}} \]

      4. *-commutative 60.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25} - \frac{{im}^{2}}{re}} \]

      5. unpow2 60.1%

         \[\leadsto 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{\color{blue}{im \cdot im}}{re}} \]

      6. associate-/l* 61.7%

         \[\leadsto 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \color{blue}{\frac{im}{\frac{re}{im}}}} \]

   6. Simplified 61.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{im}{\frac{re}{im}}}} \]

   7. Taylor expanded in im around 0 63.4%

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

      1. mul-1-neg 63.4%

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

      2. unpow2 63.4%

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

      3. associate-*r/ 64.8%

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

      4. distribute-rgt-neg-in 64.8%

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

      5. distribute-neg-frac 64.8%

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

   9. Simplified 64.8%

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

       1. sqrt-prod 55.3%

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

   11. Applied egg-rr 55.3%

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

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

   1. Initial program 45.1%

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

      1. +-commutative 45.1%

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

      2. hypot-def 90.7%

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

   3. Simplified 90.7%

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

4. Final simplification 86.6%

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

Alternative 2: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -1.56 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -1.56e+65)
   (* 0.5 (sqrt (* (- im) (/ im re))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -1.56e+65) {
		tmp = 0.5 * sqrt((-im * (im / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -1.56e+65) {
		tmp = 0.5 * Math.sqrt((-im * (im / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -1.56e+65:
		tmp = 0.5 * math.sqrt((-im * (im / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -1.56e+65)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) * Float64(im / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.56e+65)
		tmp = 0.5 * sqrt((-im * (im / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -1.56e+65], N[(0.5 * N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.5599999999999999e65

   1. Initial program 10.9%

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

      1. +-commutative 10.9%

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

      2. hypot-def 29.8%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in re around -inf 44.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + 0.25 \cdot \frac{{im}^{4}}{{re}^{3}}}} \]
   5. Step-by-step derivation

      1. +-commutative 44.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + -1 \cdot \frac{{im}^{2}}{re}}} \]

      2. mul-1-neg 44.3%

         \[\leadsto 0.5 \cdot \sqrt{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]

      3. unsub-neg 44.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} - \frac{{im}^{2}}{re}}} \]

      4. *-commutative 44.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25} - \frac{{im}^{2}}{re}} \]

      5. unpow2 44.3%

         \[\leadsto 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{\color{blue}{im \cdot im}}{re}} \]

      6. associate-/l* 44.2%

         \[\leadsto 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \color{blue}{\frac{im}{\frac{re}{im}}}} \]

   6. Simplified 44.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{im}{\frac{re}{im}}}} \]

   7. Taylor expanded in im around 0 56.9%

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

      1. mul-1-neg 56.9%

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

      2. unpow2 56.9%

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

      3. associate-*r/ 61.7%

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

      4. distribute-rgt-neg-in 61.7%

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

      5. distribute-neg-frac 61.7%

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

   9. Simplified 61.7%

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

   if -1.5599999999999999e65 < re

   1. Initial program 47.4%

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

      1. +-commutative 47.4%

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

      2. hypot-def 92.0%

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

   3. Simplified 92.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.56 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3: 70.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} t_0 := 2 \cdot \left(re + im\right)\\ \mathbf{if}\;re \leq -3.6 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 3.55 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot \frac{re}{im} + t_0}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+43} \lor \neg \left(re \leq 1.12 \cdot 10^{+85}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 2.0 (+ re im))))
   (if (<= re -3.6e+62)
     (* 0.5 (sqrt (* (- im) (/ im re))))
     (if (<= re 3.55e-9)
       (* 0.5 (sqrt (+ (* re (/ re im)) t_0)))
       (if (or (<= re 5e+43) (not (<= re 1.12e+85)))
         (* 0.5 (* 2.0 (sqrt re)))
         (* 0.5 (sqrt t_0)))))))

C

im = abs(im);
double code(double re, double im) {
	double t_0 = 2.0 * (re + im);
	double tmp;
	if (re <= -3.6e+62) {
		tmp = 0.5 * sqrt((-im * (im / re)));
	} else if (re <= 3.55e-9) {
		tmp = 0.5 * sqrt(((re * (re / im)) + t_0));
	} else if ((re <= 5e+43) || !(re <= 1.12e+85)) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * (re + im)
    if (re <= (-3.6d+62)) then
        tmp = 0.5d0 * sqrt((-im * (im / re)))
    else if (re <= 3.55d-9) then
        tmp = 0.5d0 * sqrt(((re * (re / im)) + t_0))
    else if ((re <= 5d+43) .or. (.not. (re <= 1.12d+85))) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double t_0 = 2.0 * (re + im);
	double tmp;
	if (re <= -3.6e+62) {
		tmp = 0.5 * Math.sqrt((-im * (im / re)));
	} else if (re <= 3.55e-9) {
		tmp = 0.5 * Math.sqrt(((re * (re / im)) + t_0));
	} else if ((re <= 5e+43) || !(re <= 1.12e+85)) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	t_0 = 2.0 * (re + im)
	tmp = 0
	if re <= -3.6e+62:
		tmp = 0.5 * math.sqrt((-im * (im / re)))
	elif re <= 3.55e-9:
		tmp = 0.5 * math.sqrt(((re * (re / im)) + t_0))
	elif (re <= 5e+43) or not (re <= 1.12e+85):
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt(t_0)
	return tmp

Julia

im = abs(im)
function code(re, im)
	t_0 = Float64(2.0 * Float64(re + im))
	tmp = 0.0
	if (re <= -3.6e+62)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) * Float64(im / re))));
	elseif (re <= 3.55e-9)
		tmp = Float64(0.5 * sqrt(Float64(Float64(re * Float64(re / im)) + t_0)));
	elseif ((re <= 5e+43) || !(re <= 1.12e+85))
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(t_0));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	t_0 = 2.0 * (re + im);
	tmp = 0.0;
	if (re <= -3.6e+62)
		tmp = 0.5 * sqrt((-im * (im / re)));
	elseif (re <= 3.55e-9)
		tmp = 0.5 * sqrt(((re * (re / im)) + t_0));
	elseif ((re <= 5e+43) || ~((re <= 1.12e+85)))
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := Block[{t$95$0 = N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3.6e+62], N[(0.5 * N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.55e-9], N[(0.5 * N[Sqrt[N[(N[(re * N[(re / im), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 5e+43], N[Not[LessEqual[re, 1.12e+85]], $MachinePrecision]], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.6e62

   1. Initial program 10.8%

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

      1. +-commutative 10.8%

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

      2. hypot-def 29.3%

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

   3. Simplified 29.3%

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

   4. Taylor expanded in re around -inf 43.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + 0.25 \cdot \frac{{im}^{4}}{{re}^{3}}}} \]
   5. Step-by-step derivation

      1. +-commutative 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + -1 \cdot \frac{{im}^{2}}{re}}} \]

      2. mul-1-neg 43.4%

         \[\leadsto 0.5 \cdot \sqrt{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]

      3. unsub-neg 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} - \frac{{im}^{2}}{re}}} \]

      4. *-commutative 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25} - \frac{{im}^{2}}{re}} \]

      5. unpow2 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{\color{blue}{im \cdot im}}{re}} \]

      6. associate-/l* 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \color{blue}{\frac{im}{\frac{re}{im}}}} \]

   6. Simplified 43.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{im}{\frac{re}{im}}}} \]

   7. Taylor expanded in im around 0 55.8%

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

      1. mul-1-neg 55.8%

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

      2. unpow2 55.8%

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

      3. associate-*r/ 60.5%

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

      4. distribute-rgt-neg-in 60.5%

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

      5. distribute-neg-frac 60.5%

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

   9. Simplified 60.5%

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

   if -3.6e62 < re < 3.54999999999999994e-9

   1. Initial program 49.2%

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

      1. +-commutative 49.2%

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

      2. hypot-def 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in re around 0 44.6%

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

      1. unpow2 44.6%

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

      2. distribute-lft-out 44.6%

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

   6. Simplified 44.6%

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

      1. associate-/l* 44.5%

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

      2. associate-/r/ 44.5%

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

   8. Applied egg-rr 44.5%

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

   if 3.54999999999999994e-9 < re < 5.0000000000000004e43 or 1.11999999999999993e85 < re

   1. Initial program 42.5%

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

      1. +-commutative 42.5%

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

      2. hypot-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in im around 0 84.3%

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

      1. unpow2 84.3%

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

      2. rem-square-sqrt 86.0%

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

   6. Simplified 86.0%

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

   if 5.0000000000000004e43 < re < 1.11999999999999993e85

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 46.1%

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

      1. distribute-lft-out 46.1%

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

      2. +-commutative 46.1%

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

      3. *-commutative 46.1%

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

      4. +-commutative 46.1%

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

   6. Simplified 46.1%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 3.55 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot \frac{re}{im} + 2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+43} \lor \neg \left(re \leq 1.12 \cdot 10^{+85}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4: 70.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -2.35 \cdot 10^{+63}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{-9} \lor \neg \left(re \leq 1.7 \cdot 10^{+47}\right) \land re \leq 1.4 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -2.35e+63)
   (* 0.5 (sqrt (/ (- im) (/ re im))))
   (if (or (<= re 3.3e-9) (and (not (<= re 1.7e+47)) (<= re 1.4e+84)))
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -2.35e+63) {
		tmp = 0.5 * sqrt((-im / (re / im)));
	} else if ((re <= 3.3e-9) || (!(re <= 1.7e+47) && (re <= 1.4e+84))) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.35d+63)) then
        tmp = 0.5d0 * sqrt((-im / (re / im)))
    else if ((re <= 3.3d-9) .or. (.not. (re <= 1.7d+47)) .and. (re <= 1.4d+84)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -2.35e+63) {
		tmp = 0.5 * Math.sqrt((-im / (re / im)));
	} else if ((re <= 3.3e-9) || (!(re <= 1.7e+47) && (re <= 1.4e+84))) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -2.35e+63:
		tmp = 0.5 * math.sqrt((-im / (re / im)))
	elif (re <= 3.3e-9) or (not (re <= 1.7e+47) and (re <= 1.4e+84)):
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -2.35e+63)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) / Float64(re / im))));
	elseif ((re <= 3.3e-9) || (!(re <= 1.7e+47) && (re <= 1.4e+84)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.35e+63)
		tmp = 0.5 * sqrt((-im / (re / im)));
	elseif ((re <= 3.3e-9) || (~((re <= 1.7e+47)) && (re <= 1.4e+84)))
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -2.35e+63], N[(0.5 * N[Sqrt[N[((-im) / N[(re / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 3.3e-9], And[N[Not[LessEqual[re, 1.7e+47]], $MachinePrecision], LessEqual[re, 1.4e+84]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.3500000000000001e63

   1. Initial program 10.8%

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

      1. +-commutative 10.8%

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

      2. hypot-def 29.3%

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

   3. Simplified 29.3%

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

   4. Taylor expanded in re around -inf 43.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + 0.25 \cdot \frac{{im}^{4}}{{re}^{3}}}} \]
   5. Step-by-step derivation

      1. +-commutative 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + -1 \cdot \frac{{im}^{2}}{re}}} \]

      2. mul-1-neg 43.4%

         \[\leadsto 0.5 \cdot \sqrt{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]

      3. unsub-neg 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} - \frac{{im}^{2}}{re}}} \]

      4. *-commutative 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25} - \frac{{im}^{2}}{re}} \]

      5. unpow2 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{\color{blue}{im \cdot im}}{re}} \]

      6. associate-/l* 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \color{blue}{\frac{im}{\frac{re}{im}}}} \]

   6. Simplified 43.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{im}{\frac{re}{im}}}} \]

   7. Taylor expanded in im around 0 55.8%

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

      1. mul-1-neg 55.8%

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

      2. unpow2 55.8%

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

      3. associate-*r/ 60.5%

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

      4. distribute-rgt-neg-in 60.5%

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

      5. distribute-neg-frac 60.5%

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

   9. Simplified 60.5%

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

   10. Taylor expanded in im around 0 55.8%

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

       1. mul-1-neg 55.8%

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

       2. distribute-frac-neg 55.8%

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

       3. unpow2 55.8%

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

       4. distribute-lft-neg-in 55.8%

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

       5. associate-/l* 60.4%

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

       6. distribute-neg-frac 60.4%

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

   12. Simplified 60.4%

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

   if -2.3500000000000001e63 < re < 3.30000000000000018e-9 or 1.6999999999999999e47 < re < 1.39999999999999991e84

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-def 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in re around 0 45.3%

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

      1. distribute-lft-out 45.3%

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

      2. +-commutative 45.3%

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

      3. *-commutative 45.3%

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

      4. +-commutative 45.3%

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

   6. Simplified 45.3%

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

   if 3.30000000000000018e-9 < re < 1.6999999999999999e47 or 1.39999999999999991e84 < re

   1. Initial program 42.5%

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

      1. +-commutative 42.5%

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

      2. hypot-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in im around 0 84.3%

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

      1. unpow2 84.3%

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

      2. rem-square-sqrt 86.0%

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

   6. Simplified 86.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.35 \cdot 10^{+63}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 3.3 \cdot 10^{-9} \lor \neg \left(re \leq 1.7 \cdot 10^{+47}\right) \land re \leq 1.4 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5: 70.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 3.75 \cdot 10^{-9} \lor \neg \left(re \leq 1.9 \cdot 10^{+46}\right) \land re \leq 1.25 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -3.4e+62)
   (* 0.5 (sqrt (* (- im) (/ im re))))
   (if (or (<= re 3.75e-9) (and (not (<= re 1.9e+46)) (<= re 1.25e+84)))
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -3.4e+62) {
		tmp = 0.5 * sqrt((-im * (im / re)));
	} else if ((re <= 3.75e-9) || (!(re <= 1.9e+46) && (re <= 1.25e+84))) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.4d+62)) then
        tmp = 0.5d0 * sqrt((-im * (im / re)))
    else if ((re <= 3.75d-9) .or. (.not. (re <= 1.9d+46)) .and. (re <= 1.25d+84)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -3.4e+62) {
		tmp = 0.5 * Math.sqrt((-im * (im / re)));
	} else if ((re <= 3.75e-9) || (!(re <= 1.9e+46) && (re <= 1.25e+84))) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -3.4e+62:
		tmp = 0.5 * math.sqrt((-im * (im / re)))
	elif (re <= 3.75e-9) or (not (re <= 1.9e+46) and (re <= 1.25e+84)):
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -3.4e+62)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) * Float64(im / re))));
	elseif ((re <= 3.75e-9) || (!(re <= 1.9e+46) && (re <= 1.25e+84)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.4e+62)
		tmp = 0.5 * sqrt((-im * (im / re)));
	elseif ((re <= 3.75e-9) || (~((re <= 1.9e+46)) && (re <= 1.25e+84)))
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -3.4e+62], N[(0.5 * N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 3.75e-9], And[N[Not[LessEqual[re, 1.9e+46]], $MachinePrecision], LessEqual[re, 1.25e+84]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.40000000000000014e62

   1. Initial program 10.8%

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

      1. +-commutative 10.8%

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

      2. hypot-def 29.3%

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

   3. Simplified 29.3%

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

   4. Taylor expanded in re around -inf 43.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re} + 0.25 \cdot \frac{{im}^{4}}{{re}^{3}}}} \]
   5. Step-by-step derivation

      1. +-commutative 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + -1 \cdot \frac{{im}^{2}}{re}}} \]

      2. mul-1-neg 43.4%

         \[\leadsto 0.5 \cdot \sqrt{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} + \color{blue}{\left(-\frac{{im}^{2}}{re}\right)}} \]

      3. unsub-neg 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{im}^{4}}{{re}^{3}} - \frac{{im}^{2}}{re}}} \]

      4. *-commutative 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25} - \frac{{im}^{2}}{re}} \]

      5. unpow2 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{\color{blue}{im \cdot im}}{re}} \]

      6. associate-/l* 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \color{blue}{\frac{im}{\frac{re}{im}}}} \]

   6. Simplified 43.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{4}}{{re}^{3}} \cdot 0.25 - \frac{im}{\frac{re}{im}}}} \]

   7. Taylor expanded in im around 0 55.8%

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

      1. mul-1-neg 55.8%

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

      2. unpow2 55.8%

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

      3. associate-*r/ 60.5%

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

      4. distribute-rgt-neg-in 60.5%

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

      5. distribute-neg-frac 60.5%

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

   9. Simplified 60.5%

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

   if -3.40000000000000014e62 < re < 3.74999999999999966e-9 or 1.9e46 < re < 1.25e84

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-def 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in re around 0 45.3%

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

      1. distribute-lft-out 45.3%

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

      2. +-commutative 45.3%

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

      3. *-commutative 45.3%

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

      4. +-commutative 45.3%

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

   6. Simplified 45.3%

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

   if 3.74999999999999966e-9 < re < 1.9e46 or 1.25e84 < re

   1. Initial program 42.5%

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

      1. +-commutative 42.5%

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

      2. hypot-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in im around 0 84.3%

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

      1. unpow2 84.3%

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

      2. rem-square-sqrt 86.0%

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

   6. Simplified 86.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 3.75 \cdot 10^{-9} \lor \neg \left(re \leq 1.9 \cdot 10^{+46}\right) \land re \leq 1.25 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 6: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+47} \lor \neg \left(re \leq 1.6 \cdot 10^{+84}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re 3.7e-9)
   (* 0.5 (sqrt (* 2.0 im)))
   (if (or (<= re 1.4e+47) (not (<= re 1.6e+84)))
     (* 0.5 (* 2.0 (sqrt re)))
     (* 0.5 (sqrt (* 2.0 (+ re im)))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= 3.7e-9) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else if ((re <= 1.4e+47) || !(re <= 1.6e+84)) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.7d-9) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else if ((re <= 1.4d+47) .or. (.not. (re <= 1.6d+84))) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= 3.7e-9) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else if ((re <= 1.4e+47) || !(re <= 1.6e+84)) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= 3.7e-9:
		tmp = 0.5 * math.sqrt((2.0 * im))
	elif (re <= 1.4e+47) or not (re <= 1.6e+84):
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= 3.7e-9)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	elseif ((re <= 1.4e+47) || !(re <= 1.6e+84))
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.7e-9)
		tmp = 0.5 * sqrt((2.0 * im));
	elseif ((re <= 1.4e+47) || ~((re <= 1.6e+84)))
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, 3.7e-9], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 1.4e+47], N[Not[LessEqual[re, 1.6e+84]], $MachinePrecision]], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 3.7e-9

   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. +-commutative 39.6%

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

      2. hypot-def 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in re around 0 35.5%

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

      1. *-commutative 35.5%

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

   6. Simplified 35.5%

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

   if 3.7e-9 < re < 1.39999999999999994e47 or 1.60000000000000005e84 < re

   1. Initial program 42.5%

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

      1. +-commutative 42.5%

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

      2. hypot-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in im around 0 84.3%

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

      1. unpow2 84.3%

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

      2. rem-square-sqrt 86.0%

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

   6. Simplified 86.0%

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

   if 1.39999999999999994e47 < re < 1.60000000000000005e84

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 46.1%

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

      1. distribute-lft-out 46.1%

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

      2. +-commutative 46.1%

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

      3. *-commutative 46.1%

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

      4. +-commutative 46.1%

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

   6. Simplified 46.1%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+47} \lor \neg \left(re \leq 1.6 \cdot 10^{+84}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 7: 65.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 2.65 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re 2.65e-9) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* 2.0 (sqrt re)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= 2.65e-9) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 2.65d-9) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= 2.65e-9) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= 2.65e-9:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= 2.65e-9)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.65e-9)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, 2.65e-9], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.65000000000000015e-9

   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. +-commutative 39.6%

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

      2. hypot-def 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in re around 0 35.5%

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

      1. *-commutative 35.5%

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

   6. Simplified 35.5%

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

   if 2.65000000000000015e-9 < re

   1. Initial program 44.9%

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

      1. +-commutative 44.9%

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

      2. hypot-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in im around 0 78.2%

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

      1. unpow2 78.2%

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

      2. rem-square-sqrt 79.7%

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

   6. Simplified 79.7%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.65 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 8: 53.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.3%

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

   1. +-commutative 41.3%

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

   2. hypot-def 81.5%

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

3. Simplified 81.5%

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

4. Taylor expanded in re around 0 28.4%

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

   1. *-commutative 28.4%

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

6. Simplified 28.4%

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

7. Final simplification 28.4%

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

Developer target: 48.9% 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 2023189 
(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)))))