math.sqrt on complex, imaginary part, im greater than 0 branch

Percentage Accurate: 41.8% → 88.5%

Time: 5.2s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.02 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.02e+18)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.02e+18) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.02e+18) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.02e+18:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.02e+18)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.02e+18)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.02e+18], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.02e18

   1. Initial program 48.0%

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

      1. hypot-def 92.9%

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

   3. Simplified 92.9%

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

   if 1.02e18 < re

   1. Initial program 8.1%

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

   2. Taylor expanded in re around inf 52.1%

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

      1. unpow2 52.1%

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

   4. Simplified 52.1%

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

      1. add-log-exp 18.3%

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

      2. *-un-lft-identity 18.3%

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

      3. log-prod 18.3%

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

      4. metadata-eval 18.3%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{im \cdot im}{re}}}\right)\right) \]

      5. add-log-exp 52.1%

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

      6. sqrt-div 66.0%

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

      7. sqrt-prod 86.2%

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

      8. add-sqr-sqrt 86.6%

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

   6. Applied egg-rr 86.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 86.6%

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

   8. Simplified 86.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.02 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 2: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{if}\;re \leq -7.8 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.55 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \mathbf{elif}\;re \leq -3.3 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 3.7 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))))
   (if (<= re -7.8e+90)
     t_0
     (if (<= re -1.55e+77)
       (* 0.5 (* (sqrt 2.0) (sqrt im)))
       (if (<= re -3.3e+15)
         t_0
         (if (<= re 3.7e+14)
           (* 0.5 (sqrt (* 2.0 (- im re))))
           (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double tmp;
	if (re <= -7.8e+90) {
		tmp = t_0;
	} else if (re <= -1.55e+77) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(im));
	} else if (re <= -3.3e+15) {
		tmp = t_0;
	} else if (re <= 3.7e+14) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / sqrt(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 = 0.5d0 * sqrt((re * (-4.0d0)))
    if (re <= (-7.8d+90)) then
        tmp = t_0
    else if (re <= (-1.55d+77)) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(im))
    else if (re <= (-3.3d+15)) then
        tmp = t_0
    else if (re <= 3.7d+14) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double tmp;
	if (re <= -7.8e+90) {
		tmp = t_0;
	} else if (re <= -1.55e+77) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(im));
	} else if (re <= -3.3e+15) {
		tmp = t_0;
	} else if (re <= 3.7e+14) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	tmp = 0
	if re <= -7.8e+90:
		tmp = t_0
	elif re <= -1.55e+77:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(im))
	elif re <= -3.3e+15:
		tmp = t_0
	elif re <= 3.7e+14:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	tmp = 0.0
	if (re <= -7.8e+90)
		tmp = t_0;
	elseif (re <= -1.55e+77)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(im)));
	elseif (re <= -3.3e+15)
		tmp = t_0;
	elseif (re <= 3.7e+14)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	tmp = 0.0;
	if (re <= -7.8e+90)
		tmp = t_0;
	elseif (re <= -1.55e+77)
		tmp = 0.5 * (sqrt(2.0) * sqrt(im));
	elseif (re <= -3.3e+15)
		tmp = t_0;
	elseif (re <= 3.7e+14)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -7.8e+90], t$95$0, If[LessEqual[re, -1.55e+77], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -3.3e+15], t$95$0, If[LessEqual[re, 3.7e+14], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.8000000000000004e90 or -1.54999999999999999e77 < re < -3.3e15

   1. Initial program 40.2%

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

   2. Taylor expanded in re around -inf 90.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 90.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   4. Simplified 90.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -7.8000000000000004e90 < re < -1.54999999999999999e77

   1. Initial program 20.2%

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

   2. Taylor expanded in re around 0 99.2%

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

   if -3.3e15 < re < 3.7e14

   1. Initial program 53.5%

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

   2. Taylor expanded in re around 0 80.5%

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

   if 3.7e14 < re

   1. Initial program 8.1%

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

   2. Taylor expanded in re around inf 50.9%

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

      1. unpow2 50.9%

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

   4. Simplified 50.9%

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

      1. add-log-exp 18.0%

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

      2. *-un-lft-identity 18.0%

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

      3. log-prod 18.0%

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

      4. metadata-eval 18.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{im \cdot im}{re}}}\right)\right) \]

      5. add-log-exp 50.9%

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

      6. sqrt-div 64.4%

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

      7. sqrt-prod 85.2%

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

      8. add-sqr-sqrt 85.6%

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

   6. Applied egg-rr 85.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 85.6%

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

   8. Simplified 85.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.8 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.55 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \mathbf{elif}\;re \leq -3.3 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.7 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -2.8 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -3.6 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0))))
        (t_1 (* 0.5 (sqrt (* 2.0 (- im re))))))
   (if (<= re -2.8e+91)
     t_0
     (if (<= re -4.2e+75)
       t_1
       (if (<= re -3.6e+16)
         t_0
         (if (<= re 2.4e+14) t_1 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -2.8e+91) {
		tmp = t_0;
	} else if (re <= -4.2e+75) {
		tmp = t_1;
	} else if (re <= -3.6e+16) {
		tmp = t_0;
	} else if (re <= 2.4e+14) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / sqrt(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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    if (re <= (-2.8d+91)) then
        tmp = t_0
    else if (re <= (-4.2d+75)) then
        tmp = t_1
    else if (re <= (-3.6d+16)) then
        tmp = t_0
    else if (re <= 2.4d+14) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -2.8e+91) {
		tmp = t_0;
	} else if (re <= -4.2e+75) {
		tmp = t_1;
	} else if (re <= -3.6e+16) {
		tmp = t_0;
	} else if (re <= 2.4e+14) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((2.0 * (im - re)))
	tmp = 0
	if re <= -2.8e+91:
		tmp = t_0
	elif re <= -4.2e+75:
		tmp = t_1
	elif re <= -3.6e+16:
		tmp = t_0
	elif re <= 2.4e+14:
		tmp = t_1
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	tmp = 0.0
	if (re <= -2.8e+91)
		tmp = t_0;
	elseif (re <= -4.2e+75)
		tmp = t_1;
	elseif (re <= -3.6e+16)
		tmp = t_0;
	elseif (re <= 2.4e+14)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((2.0 * (im - re)));
	tmp = 0.0;
	if (re <= -2.8e+91)
		tmp = t_0;
	elseif (re <= -4.2e+75)
		tmp = t_1;
	elseif (re <= -3.6e+16)
		tmp = t_0;
	elseif (re <= 2.4e+14)
		tmp = t_1;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.8e+91], t$95$0, If[LessEqual[re, -4.2e+75], t$95$1, If[LessEqual[re, -3.6e+16], t$95$0, If[LessEqual[re, 2.4e+14], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.7999999999999999e91 or -4.19999999999999997e75 < re < -3.6e16

   1. Initial program 40.2%

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

   2. Taylor expanded in re around -inf 90.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 90.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   4. Simplified 90.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -2.7999999999999999e91 < re < -4.19999999999999997e75 or -3.6e16 < re < 2.4e14

   1. Initial program 52.0%

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

   2. Taylor expanded in re around 0 80.7%

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

   if 2.4e14 < re

   1. Initial program 8.1%

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

   2. Taylor expanded in re around inf 50.9%

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

      1. unpow2 50.9%

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

   4. Simplified 50.9%

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

      1. add-log-exp 18.0%

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

      2. *-un-lft-identity 18.0%

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

      3. log-prod 18.0%

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

      4. metadata-eval 18.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{im \cdot im}{re}}}\right)\right) \]

      5. add-log-exp 50.9%

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

      6. sqrt-div 64.4%

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

      7. sqrt-prod 85.2%

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

      8. add-sqr-sqrt 85.6%

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

   6. Applied egg-rr 85.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 85.6%

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

   8. Simplified 85.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+91}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -3.6 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -7.8 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.55 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -6.6 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (* 0.5 (sqrt (* 2.0 im)))))
   (if (<= re -7.8e+90)
     t_0
     (if (<= re -1.55e+77)
       t_1
       (if (<= re -6.6e+18)
         t_0
         (if (<= re 2.35e+17) t_1 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((re * -4.0));
	double t_1 = 0.5 * sqrt((2.0 * im));
	double tmp;
	if (re <= -7.8e+90) {
		tmp = t_0;
	} else if (re <= -1.55e+77) {
		tmp = t_1;
	} else if (re <= -6.6e+18) {
		tmp = t_0;
	} else if (re <= 2.35e+17) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / sqrt(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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((re * (-4.0d0)))
    t_1 = 0.5d0 * sqrt((2.0d0 * im))
    if (re <= (-7.8d+90)) then
        tmp = t_0
    else if (re <= (-1.55d+77)) then
        tmp = t_1
    else if (re <= (-6.6d+18)) then
        tmp = t_0
    else if (re <= 2.35d+17) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((re * -4.0));
	double t_1 = 0.5 * Math.sqrt((2.0 * im));
	double tmp;
	if (re <= -7.8e+90) {
		tmp = t_0;
	} else if (re <= -1.55e+77) {
		tmp = t_1;
	} else if (re <= -6.6e+18) {
		tmp = t_0;
	} else if (re <= 2.35e+17) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((re * -4.0))
	t_1 = 0.5 * math.sqrt((2.0 * im))
	tmp = 0
	if re <= -7.8e+90:
		tmp = t_0
	elif re <= -1.55e+77:
		tmp = t_1
	elif re <= -6.6e+18:
		tmp = t_0
	elif re <= 2.35e+17:
		tmp = t_1
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(re * -4.0)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * im)))
	tmp = 0.0
	if (re <= -7.8e+90)
		tmp = t_0;
	elseif (re <= -1.55e+77)
		tmp = t_1;
	elseif (re <= -6.6e+18)
		tmp = t_0;
	elseif (re <= 2.35e+17)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((re * -4.0));
	t_1 = 0.5 * sqrt((2.0 * im));
	tmp = 0.0;
	if (re <= -7.8e+90)
		tmp = t_0;
	elseif (re <= -1.55e+77)
		tmp = t_1;
	elseif (re <= -6.6e+18)
		tmp = t_0;
	elseif (re <= 2.35e+17)
		tmp = t_1;
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -7.8e+90], t$95$0, If[LessEqual[re, -1.55e+77], t$95$1, If[LessEqual[re, -6.6e+18], t$95$0, If[LessEqual[re, 2.35e+17], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.8000000000000004e90 or -1.54999999999999999e77 < re < -6.6e18

   1. Initial program 40.2%

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

   2. Taylor expanded in re around -inf 90.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 90.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   4. Simplified 90.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -7.8000000000000004e90 < re < -1.54999999999999999e77 or -6.6e18 < re < 2.35e17

   1. Initial program 52.0%

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

   2. Taylor expanded in re around 0 79.6%

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

      1. *-commutative 79.6%

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

   4. Simplified 79.6%

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

   if 2.35e17 < re

   1. Initial program 8.1%

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

   2. Taylor expanded in re around inf 50.9%

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

      1. unpow2 50.9%

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

   4. Simplified 50.9%

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

      1. add-log-exp 18.0%

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

      2. *-un-lft-identity 18.0%

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

      3. log-prod 18.0%

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

      4. metadata-eval 18.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{\frac{im \cdot im}{re}}}\right)\right) \]

      5. add-log-exp 50.9%

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

      6. sqrt-div 64.4%

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

      7. sqrt-prod 85.2%

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

      8. add-sqr-sqrt 85.6%

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

   6. Applied egg-rr 85.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 85.6%

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

   8. Simplified 85.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.8 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.55 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq -6.6 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{+91} \lor \neg \left(re \leq -5.2 \cdot 10^{+76}\right) \land re \leq -1.1 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -1.3e+91) (and (not (<= re -5.2e+76)) (<= re -1.1e+18)))
   (* 0.5 (sqrt (* re -4.0)))
   (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -1.3e+91) || (!(re <= -5.2e+76) && (re <= -1.1e+18))) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-1.3d+91)) .or. (.not. (re <= (-5.2d+76))) .and. (re <= (-1.1d+18))) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -1.3e+91) || (!(re <= -5.2e+76) && (re <= -1.1e+18))) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -1.3e+91) or (not (re <= -5.2e+76) and (re <= -1.1e+18)):
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -1.3e+91) || (!(re <= -5.2e+76) && (re <= -1.1e+18)))
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -1.3e+91) || (~((re <= -5.2e+76)) && (re <= -1.1e+18)))
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -1.3e+91], And[N[Not[LessEqual[re, -5.2e+76]], $MachinePrecision], LessEqual[re, -1.1e+18]]], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.3e91 or -5.1999999999999999e76 < re < -1.1e18

   1. Initial program 40.2%

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

   2. Taylor expanded in re around -inf 90.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   3. Step-by-step derivation

      1. *-commutative 90.3%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   4. Simplified 90.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.3e91 < re < -5.1999999999999999e76 or -1.1e18 < re

   1. Initial program 36.4%

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

   2. Taylor expanded in re around 0 58.2%

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

      1. *-commutative 58.2%

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

   4. Simplified 58.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{+91} \lor \neg \left(re \leq -5.2 \cdot 10^{+76}\right) \land re \leq -1.1 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 6: 52.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

2. Taylor expanded in re around 0 48.8%

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

   1. *-commutative 48.8%

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

4. Simplified 48.8%

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

5. Final simplification 48.8%

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

Reproduce

herbie shell --seed 2023200 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))