math.sqrt on complex, real part

Percentage Accurate: 41.1% → 84.1%

Time: 6.9s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.1% 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: 84.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im im)))) 0.0)
   (* 0.5 (sqrt (* im (/ im (- re)))))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(im * N[(im / (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.2%

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

      1. +-commutative 3.2%

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

      2. hypot-def 16.5%

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

   3. Simplified 16.5%

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

   4. Taylor expanded in re around -inf 31.0%

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

      1. associate-*r/ 31.0%

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

      2. neg-mul-1 31.0%

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

      3. unpow2 31.0%

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

      4. distribute-rgt-neg-in 31.0%

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

   6. Simplified 31.0%

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

   7. Taylor expanded in im around 0 31.0%

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

      1. *-commutative 31.0%

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

      2. unpow2 31.0%

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

      3. metadata-eval 31.0%

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

      4. times-frac 31.0%

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

      5. *-commutative 31.0%

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

      6. neg-mul-1 31.0%

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

      7. associate-*r/ 30.9%

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

      8. associate-*l* 56.4%

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

      9. associate-*r/ 56.4%

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

      10. *-rgt-identity 56.4%

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

   9. Simplified 56.4%

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

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

   1. Initial program 47.5%

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

      1. +-commutative 47.5%

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

      2. hypot-def 92.4%

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

   3. Simplified 92.4%

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

      1. add-sqr-sqrt 91.8%

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

      2. sqrt-unprod 92.4%

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

      3. *-commutative 92.4%

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

      4. *-commutative 92.4%

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

      5. swap-sqr 92.4%

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

      6. add-sqr-sqrt 92.4%

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

      7. *-commutative 92.4%

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

      8. metadata-eval 92.4%

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

   5. Applied egg-rr 92.4%

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

      1. associate-*l* 92.4%

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

      2. metadata-eval 92.4%

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

   7. Simplified 92.4%

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

4. Final simplification 87.8%

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

Alternative 2: 59.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.8 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -1.45 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-277}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;im \leq 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{-197}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5.8e-118)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im -1.45e-237)
     (sqrt re)
     (if (<= im -1.7e-277)
       (* 0.5 (sqrt (* im (/ im (- re)))))
       (if (<= im 1e-305)
         (sqrt re)
         (if (<= im 2.55e-197)
           (* 0.5 (/ im (sqrt (- re))))
           (if (<= im 1.6e-112)
             (sqrt re)
             (* 0.5 (sqrt (* 2.0 (+ re im)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -5.8e-118) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= -1.45e-237) {
		tmp = sqrt(re);
	} else if (im <= -1.7e-277) {
		tmp = 0.5 * sqrt((im * (im / -re)));
	} else if (im <= 1e-305) {
		tmp = sqrt(re);
	} else if (im <= 2.55e-197) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (im <= 1.6e-112) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-5.8d-118)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= (-1.45d-237)) then
        tmp = sqrt(re)
    else if (im <= (-1.7d-277)) then
        tmp = 0.5d0 * sqrt((im * (im / -re)))
    else if (im <= 1d-305) then
        tmp = sqrt(re)
    else if (im <= 2.55d-197) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if (im <= 1.6d-112) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -5.8e-118) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= -1.45e-237) {
		tmp = Math.sqrt(re);
	} else if (im <= -1.7e-277) {
		tmp = 0.5 * Math.sqrt((im * (im / -re)));
	} else if (im <= 1e-305) {
		tmp = Math.sqrt(re);
	} else if (im <= 2.55e-197) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (im <= 1.6e-112) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -5.8e-118:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= -1.45e-237:
		tmp = math.sqrt(re)
	elif im <= -1.7e-277:
		tmp = 0.5 * math.sqrt((im * (im / -re)))
	elif im <= 1e-305:
		tmp = math.sqrt(re)
	elif im <= 2.55e-197:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif im <= 1.6e-112:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -5.8e-118)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= -1.45e-237)
		tmp = sqrt(re);
	elseif (im <= -1.7e-277)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(im / Float64(-re)))));
	elseif (im <= 1e-305)
		tmp = sqrt(re);
	elseif (im <= 2.55e-197)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (im <= 1.6e-112)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -5.8e-118)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= -1.45e-237)
		tmp = sqrt(re);
	elseif (im <= -1.7e-277)
		tmp = 0.5 * sqrt((im * (im / -re)));
	elseif (im <= 1e-305)
		tmp = sqrt(re);
	elseif (im <= 2.55e-197)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (im <= 1.6e-112)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -5.8e-118], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -1.45e-237], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, -1.7e-277], N[(0.5 * N[Sqrt[N[(im * N[(im / (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e-305], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 2.55e-197], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.6e-112], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -5.79999999999999961e-118

   1. Initial program 43.2%

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

      1. +-commutative 43.2%

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

      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%

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

   4. Taylor expanded in im around -inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. sub-neg 77.5%

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

   6. Simplified 77.5%

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

   if -5.79999999999999961e-118 < im < -1.45000000000000005e-237 or -1.69999999999999991e-277 < im < 9.99999999999999996e-306 or 2.5500000000000001e-197 < im < 1.59999999999999997e-112

   1. Initial program 47.6%

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

      1. +-commutative 47.6%

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

      2. hypot-def 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in im around 0 60.2%

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

      1. associate-*r* 60.2%

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

      2. unpow2 60.2%

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

      3. rem-square-sqrt 61.4%

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

      4. metadata-eval 61.4%

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

      5. *-lft-identity 61.4%

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

   6. Simplified 61.4%

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

   if -1.45000000000000005e-237 < im < -1.69999999999999991e-277

   1. Initial program 14.2%

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

      1. +-commutative 14.2%

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

      2. hypot-def 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in re around -inf 26.1%

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

      1. associate-*r/ 26.1%

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

      2. neg-mul-1 26.1%

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

      3. unpow2 26.1%

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

      4. distribute-rgt-neg-in 26.1%

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

   6. Simplified 26.1%

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

   7. Taylor expanded in im around 0 26.1%

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

      1. *-commutative 26.1%

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

      2. unpow2 26.1%

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

      3. metadata-eval 26.1%

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

      4. times-frac 26.1%

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

      5. *-commutative 26.1%

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

      6. neg-mul-1 26.1%

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

      7. associate-*r/ 26.1%

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

      8. associate-*l* 57.4%

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

      9. associate-*r/ 57.2%

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

      10. *-rgt-identity 57.2%

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

   9. Simplified 57.2%

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

   if 9.99999999999999996e-306 < im < 2.5500000000000001e-197

   1. Initial program 12.2%

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

      1. +-commutative 12.2%

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

      2. hypot-def 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in re around -inf 26.6%

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

      1. associate-*r/ 26.6%

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

      2. neg-mul-1 26.6%

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

      3. unpow2 26.6%

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

      4. distribute-rgt-neg-in 26.6%

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

   6. Simplified 26.6%

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

      1. frac-2neg 26.6%

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

      2. sqrt-div 25.8%

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

      3. distribute-rgt-neg-out 25.8%

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

      4. remove-double-neg 25.8%

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

      5. sqrt-unprod 68.6%

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

      6. add-sqr-sqrt 68.8%

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

   8. Applied egg-rr 68.8%

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

   if 1.59999999999999997e-112 < im

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-def 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in re around 0 73.7%

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

      1. distribute-lft-out 73.7%

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

      2. +-commutative 73.7%

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

      3. *-commutative 73.7%

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

      4. +-commutative 73.7%

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

   6. Simplified 73.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right) \cdot 2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.8 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -1.45 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-277}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;im \leq 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{-197}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.65 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 8.4 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.35e-117)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im 2.65e-306)
     (sqrt re)
     (if (<= im 9.5e-190)
       (* 0.5 (/ im (sqrt (- re))))
       (if (<= im 8.4e-113) (sqrt re) (* 0.5 (sqrt (* 2.0 (+ re im)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.35e-117) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= 2.65e-306) {
		tmp = sqrt(re);
	} else if (im <= 9.5e-190) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (im <= 8.4e-113) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.35d-117)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= 2.65d-306) then
        tmp = sqrt(re)
    else if (im <= 9.5d-190) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if (im <= 8.4d-113) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.35e-117) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= 2.65e-306) {
		tmp = Math.sqrt(re);
	} else if (im <= 9.5e-190) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (im <= 8.4e-113) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.35e-117:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= 2.65e-306:
		tmp = math.sqrt(re)
	elif im <= 9.5e-190:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif im <= 8.4e-113:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.35e-117)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= 2.65e-306)
		tmp = sqrt(re);
	elseif (im <= 9.5e-190)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (im <= 8.4e-113)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.35e-117)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= 2.65e-306)
		tmp = sqrt(re);
	elseif (im <= 9.5e-190)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (im <= 8.4e-113)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.35e-117], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.65e-306], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 9.5e-190], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.4e-113], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.35000000000000001e-117

   1. Initial program 43.2%

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

      1. +-commutative 43.2%

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

      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%

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

   4. Taylor expanded in im around -inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. sub-neg 77.5%

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

   6. Simplified 77.5%

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

   if -1.35000000000000001e-117 < im < 2.6499999999999999e-306 or 9.50000000000000055e-190 < im < 8.39999999999999999e-113

   1. Initial program 42.8%

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

      1. +-commutative 42.8%

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

      2. hypot-def 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in im around 0 53.2%

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

      1. associate-*r* 53.2%

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

      2. unpow2 53.2%

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

      3. rem-square-sqrt 54.3%

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

      4. metadata-eval 54.3%

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

      5. *-lft-identity 54.3%

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

   6. Simplified 54.3%

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

   if 2.6499999999999999e-306 < im < 9.50000000000000055e-190

   1. Initial program 12.2%

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

      1. +-commutative 12.2%

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

      2. hypot-def 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in re around -inf 26.6%

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

      1. associate-*r/ 26.6%

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

      2. neg-mul-1 26.6%

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

      3. unpow2 26.6%

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

      4. distribute-rgt-neg-in 26.6%

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

   6. Simplified 26.6%

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

      1. frac-2neg 26.6%

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

      2. sqrt-div 25.8%

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

      3. distribute-rgt-neg-out 25.8%

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

      4. remove-double-neg 25.8%

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

      5. sqrt-unprod 68.6%

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

      6. add-sqr-sqrt 68.8%

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

   8. Applied egg-rr 68.8%

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

   if 8.39999999999999999e-113 < im

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-def 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in re around 0 73.7%

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

      1. distribute-lft-out 73.7%

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

      2. +-commutative 73.7%

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

      3. *-commutative 73.7%

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

      4. +-commutative 73.7%

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

   6. Simplified 73.7%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.65 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 8.4 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4: 58.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.75 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 7.8 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-196}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.75e-118)
   (* 0.5 (sqrt (* im -2.0)))
   (if (<= im 7.8e-306)
     (sqrt re)
     (if (<= im 5.8e-196)
       (* 0.5 (/ im (sqrt (- re))))
       (if (<= im 7.2e-113) (sqrt re) (* 0.5 (sqrt (* im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.75e-118) {
		tmp = 0.5 * sqrt((im * -2.0));
	} else if (im <= 7.8e-306) {
		tmp = sqrt(re);
	} else if (im <= 5.8e-196) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (im <= 7.2e-113) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.75d-118)) then
        tmp = 0.5d0 * sqrt((im * (-2.0d0)))
    else if (im <= 7.8d-306) then
        tmp = sqrt(re)
    else if (im <= 5.8d-196) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if (im <= 7.2d-113) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.75e-118) {
		tmp = 0.5 * Math.sqrt((im * -2.0));
	} else if (im <= 7.8e-306) {
		tmp = Math.sqrt(re);
	} else if (im <= 5.8e-196) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (im <= 7.2e-113) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.75e-118:
		tmp = 0.5 * math.sqrt((im * -2.0))
	elif im <= 7.8e-306:
		tmp = math.sqrt(re)
	elif im <= 5.8e-196:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif im <= 7.2e-113:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.75e-118)
		tmp = Float64(0.5 * sqrt(Float64(im * -2.0)));
	elseif (im <= 7.8e-306)
		tmp = sqrt(re);
	elseif (im <= 5.8e-196)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (im <= 7.2e-113)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.75e-118)
		tmp = 0.5 * sqrt((im * -2.0));
	elseif (im <= 7.8e-306)
		tmp = sqrt(re);
	elseif (im <= 5.8e-196)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (im <= 7.2e-113)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.75e-118], N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.8e-306], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 5.8e-196], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e-113], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.75e-118

   1. Initial program 43.2%

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

      1. +-commutative 43.2%

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

      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%

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

   4. Taylor expanded in im around -inf 75.9%

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

      1. *-commutative 75.9%

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

   6. Simplified 75.9%

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

   if -1.75e-118 < im < 7.799999999999999e-306 or 5.79999999999999974e-196 < im < 7.1999999999999995e-113

   1. Initial program 42.8%

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

      1. +-commutative 42.8%

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

      2. hypot-def 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in im around 0 53.2%

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

      1. associate-*r* 53.2%

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

      2. unpow2 53.2%

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

      3. rem-square-sqrt 54.3%

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

      4. metadata-eval 54.3%

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

      5. *-lft-identity 54.3%

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

   6. Simplified 54.3%

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

   if 7.799999999999999e-306 < im < 5.79999999999999974e-196

   1. Initial program 12.2%

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

      1. +-commutative 12.2%

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

      2. hypot-def 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in re around -inf 26.6%

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

      1. associate-*r/ 26.6%

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

      2. neg-mul-1 26.6%

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

      3. unpow2 26.6%

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

      4. distribute-rgt-neg-in 26.6%

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

   6. Simplified 26.6%

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

      1. frac-2neg 26.6%

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

      2. sqrt-div 25.8%

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

      3. distribute-rgt-neg-out 25.8%

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

      4. remove-double-neg 25.8%

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

      5. sqrt-unprod 68.6%

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

      6. add-sqr-sqrt 68.8%

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

   8. Applied egg-rr 68.8%

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

   if 7.1999999999999995e-113 < im

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-def 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in re around 0 72.7%

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

      1. *-commutative 72.7%

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

   6. Simplified 72.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.75 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 7.8 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-196}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5: 59.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.8 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.8e-118)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im 2.6e-305)
     (sqrt re)
     (if (<= im 2.7e-187)
       (* 0.5 (/ im (sqrt (- re))))
       (if (<= im 1.5e-111) (sqrt re) (* 0.5 (sqrt (* im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.8e-118) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= 2.6e-305) {
		tmp = sqrt(re);
	} else if (im <= 2.7e-187) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (im <= 1.5e-111) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.8d-118)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= 2.6d-305) then
        tmp = sqrt(re)
    else if (im <= 2.7d-187) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if (im <= 1.5d-111) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.8e-118) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= 2.6e-305) {
		tmp = Math.sqrt(re);
	} else if (im <= 2.7e-187) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (im <= 1.5e-111) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.8e-118:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= 2.6e-305:
		tmp = math.sqrt(re)
	elif im <= 2.7e-187:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif im <= 1.5e-111:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.8e-118)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= 2.6e-305)
		tmp = sqrt(re);
	elseif (im <= 2.7e-187)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (im <= 1.5e-111)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.8e-118)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= 2.6e-305)
		tmp = sqrt(re);
	elseif (im <= 2.7e-187)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (im <= 1.5e-111)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.8e-118], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e-305], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 2.7e-187], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.5e-111], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.8000000000000001e-118

   1. Initial program 43.2%

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

      1. +-commutative 43.2%

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

      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%

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

   4. Taylor expanded in im around -inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. sub-neg 77.5%

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

   6. Simplified 77.5%

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

   if -1.8000000000000001e-118 < im < 2.6000000000000002e-305 or 2.7000000000000001e-187 < im < 1.50000000000000004e-111

   1. Initial program 42.8%

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

      1. +-commutative 42.8%

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

      2. hypot-def 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in im around 0 53.2%

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

      1. associate-*r* 53.2%

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

      2. unpow2 53.2%

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

      3. rem-square-sqrt 54.3%

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

      4. metadata-eval 54.3%

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

      5. *-lft-identity 54.3%

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

   6. Simplified 54.3%

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

   if 2.6000000000000002e-305 < im < 2.7000000000000001e-187

   1. Initial program 12.2%

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

      1. +-commutative 12.2%

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

      2. hypot-def 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in re around -inf 26.6%

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

      1. associate-*r/ 26.6%

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

      2. neg-mul-1 26.6%

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

      3. unpow2 26.6%

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

      4. distribute-rgt-neg-in 26.6%

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

   6. Simplified 26.6%

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

      1. frac-2neg 26.6%

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

      2. sqrt-div 25.8%

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

      3. distribute-rgt-neg-out 25.8%

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

      4. remove-double-neg 25.8%

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

      5. sqrt-unprod 68.6%

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

      6. add-sqr-sqrt 68.8%

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

   8. Applied egg-rr 68.8%

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

   if 1.50000000000000004e-111 < im

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-def 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in re around 0 72.7%

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

      1. *-commutative 72.7%

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

   6. Simplified 72.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.8 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6: 59.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -2e-118)
   (* 0.5 (sqrt (* im -2.0)))
   (if (<= im 3.5e-110) (sqrt re) (* 0.5 (sqrt (* im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2e-118) {
		tmp = 0.5 * sqrt((im * -2.0));
	} else if (im <= 3.5e-110) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-2d-118)) then
        tmp = 0.5d0 * sqrt((im * (-2.0d0)))
    else if (im <= 3.5d-110) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -2e-118) {
		tmp = 0.5 * Math.sqrt((im * -2.0));
	} else if (im <= 3.5e-110) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2e-118:
		tmp = 0.5 * math.sqrt((im * -2.0))
	elif im <= 3.5e-110:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2e-118)
		tmp = Float64(0.5 * sqrt(Float64(im * -2.0)));
	elseif (im <= 3.5e-110)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2e-118)
		tmp = 0.5 * sqrt((im * -2.0));
	elseif (im <= 3.5e-110)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2e-118], N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.5e-110], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.99999999999999997e-118

   1. Initial program 43.2%

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

      1. +-commutative 43.2%

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

      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%

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

   4. Taylor expanded in im around -inf 75.9%

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

      1. *-commutative 75.9%

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

   6. Simplified 75.9%

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

   if -1.99999999999999997e-118 < im < 3.49999999999999974e-110

   1. Initial program 34.8%

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

      1. +-commutative 34.8%

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

      2. hypot-def 71.1%

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

   3. Simplified 71.1%

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

   4. Taylor expanded in im around 0 43.0%

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

      1. associate-*r* 43.0%

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

      2. unpow2 43.0%

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

      3. rem-square-sqrt 43.9%

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

      4. metadata-eval 43.9%

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

      5. *-lft-identity 43.9%

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

   6. Simplified 43.9%

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

   if 3.49999999999999974e-110 < im

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-def 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in re around 0 72.7%

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

      1. *-commutative 72.7%

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

   6. Simplified 72.7%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7: 42.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.32e-118) (* 0.5 (sqrt (* im -2.0))) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.32e-118) {
		tmp = 0.5 * sqrt((im * -2.0));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.32d-118)) then
        tmp = 0.5d0 * sqrt((im * (-2.0d0)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.32e-118) {
		tmp = 0.5 * Math.sqrt((im * -2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.32e-118:
		tmp = 0.5 * math.sqrt((im * -2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.32e-118)
		tmp = Float64(0.5 * sqrt(Float64(im * -2.0)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.32e-118)
		tmp = 0.5 * sqrt((im * -2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.32e-118], N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.32000000000000003e-118

   1. Initial program 43.2%

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

      1. +-commutative 43.2%

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

      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%

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

   4. Taylor expanded in im around -inf 75.9%

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

      1. *-commutative 75.9%

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

   6. Simplified 75.9%

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

   if -1.32000000000000003e-118 < im

   1. Initial program 41.0%

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

      1. +-commutative 41.0%

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

      2. hypot-def 78.1%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in im around 0 29.9%

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

      1. associate-*r* 29.9%

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

      2. unpow2 29.9%

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

      3. rem-square-sqrt 30.5%

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

      4. metadata-eval 30.5%

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

      5. *-lft-identity 30.5%

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

   6. Simplified 30.5%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 8: 26.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt re))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 41.8%

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

   1. +-commutative 41.8%

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

   2. hypot-def 82.7%

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

3. Simplified 82.7%

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

4. Taylor expanded in im around 0 25.5%

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

   1. associate-*r* 25.5%

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

   2. unpow2 25.5%

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

   3. rem-square-sqrt 26.0%

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

   4. metadata-eval 26.0%

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

   5. *-lft-identity 26.0%

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

6. Simplified 26.0%

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

7. Final simplification 26.0%

   \[\leadsto \sqrt{re} \]

Developer target: 47.7% 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 2023174 
(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)))))