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

Percentage Accurate: 41.0% → 88.1%

Time: 4.9s

Alternatives: 4

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: 41.0% 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: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.9 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.9e-20)
   (sqrt (* 0.5 (- (hypot re im) re)))
   (* (pow re -0.5) (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.9e-20) {
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	} else {
		tmp = pow(re, -0.5) * (0.5 * im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.9e-20) {
		tmp = Math.sqrt((0.5 * (Math.hypot(re, im) - re)));
	} else {
		tmp = Math.pow(re, -0.5) * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.9e-20:
		tmp = math.sqrt((0.5 * (math.hypot(re, im) - re)))
	else:
		tmp = math.pow(re, -0.5) * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.9e-20)
		tmp = sqrt(Float64(0.5 * Float64(hypot(re, im) - re)));
	else
		tmp = Float64((re ^ -0.5) * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.9e-20)
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	else
		tmp = (re ^ -0.5) * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.9e-20], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.9000000000000002e-20

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

      2. hypot-udef 91.9%

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

      3. *-commutative 91.9%

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

      4. add-sqr-sqrt 91.1%

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

      5. sqrt-unprod 91.9%

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

      6. *-commutative 91.9%

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

      7. *-commutative 91.9%

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

      8. swap-sqr 91.9%

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

      9. add-sqr-sqrt 91.9%

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

      10. metadata-eval 91.9%

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

   3. Applied egg-rr 91.9%

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

      1. *-commutative 91.9%

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

      2. associate-*r* 91.9%

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

      3. metadata-eval 91.9%

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

   5. Simplified 91.9%

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

   if 4.9000000000000002e-20 < re

   1. Initial program 16.7%

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

   2. Taylor expanded in im around 0 89.4%

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

      1. associate-*r* 89.4%

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

      2. *-commutative 89.4%

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

   4. Simplified 89.4%

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

      1. add-log-exp 33.0%

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

      2. *-un-lft-identity 33.0%

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

      3. log-prod 33.0%

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

      4. metadata-eval 33.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \]

      5. add-log-exp 89.4%

         \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \]

      6. associate-*l* 89.4%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \left(\left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      7. *-commutative 89.4%

         \[\leadsto 0 + 0.5 \cdot \left(\color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1}{re}}\right) \]

      8. associate-*r* 89.4%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      9. associate-*l* 89.4%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      10. associate-*r* 89.4%

          \[\leadsto 0 + \color{blue}{\left(0.5 \cdot im\right) \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      11. sqrt-unprod 89.9%

          \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \left(\sqrt{0.5} \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{re}}}\right) \]

      12. sqrt-unprod 89.9%

          \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \color{blue}{\sqrt{0.5 \cdot \left(2 \cdot \frac{1}{re}\right)}} \]

      13. un-div-inv 89.9%

          \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \sqrt{0.5 \cdot \color{blue}{\frac{2}{re}}} \]

   6. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{0 + \left(0.5 \cdot im\right) \cdot \sqrt{0.5 \cdot \frac{2}{re}}} \]
   7. Step-by-step derivation

      1. +-lft-identity 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-*r/ 89.9%

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

      4. metadata-eval 89.9%

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

      5. unpow1/2 89.9%

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

      6. unpow-1 89.9%

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

      7. exp-to-pow 85.1%

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

      8. exp-prod 85.1%

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

      9. associate-*l* 85.1%

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

      10. metadata-eval 85.1%

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

      11. exp-to-pow 90.0%

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

   8. Simplified 90.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.9 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 2: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 0.00075:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.5e+20)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 0.00075)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (pow re -0.5) (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.5e+20) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 0.00075) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = pow(re, -0.5) * (0.5 * 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 <= (-1.5d+20)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 0.00075d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (re ** (-0.5d0)) * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.5e+20) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 0.00075) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = Math.pow(re, -0.5) * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.5e+20:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 0.00075:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = math.pow(re, -0.5) * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.5e+20)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 0.00075)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64((re ^ -0.5) * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.5e+20)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 0.00075)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (re ^ -0.5) * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.5e+20], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.00075], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.5e20

   1. Initial program 35.6%

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

   2. Taylor expanded in re around -inf 79.3%

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

      1. *-commutative 79.3%

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

   4. Simplified 79.3%

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

   if -1.5e20 < re < 7.5000000000000002e-4

   1. Initial program 53.6%

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

   2. Taylor expanded in re around 0 77.7%

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

   if 7.5000000000000002e-4 < re

   1. Initial program 16.7%

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

   2. Taylor expanded in im around 0 89.4%

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

      1. associate-*r* 89.4%

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

      2. *-commutative 89.4%

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

   4. Simplified 89.4%

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

      1. add-log-exp 33.0%

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

      2. *-un-lft-identity 33.0%

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

      3. log-prod 33.0%

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

      4. metadata-eval 33.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \]

      5. add-log-exp 89.4%

         \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \]

      6. associate-*l* 89.4%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \left(\left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      7. *-commutative 89.4%

         \[\leadsto 0 + 0.5 \cdot \left(\color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1}{re}}\right) \]

      8. associate-*r* 89.4%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      9. associate-*l* 89.4%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      10. associate-*r* 89.4%

          \[\leadsto 0 + \color{blue}{\left(0.5 \cdot im\right) \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      11. sqrt-unprod 89.9%

          \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \left(\sqrt{0.5} \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{re}}}\right) \]

      12. sqrt-unprod 89.9%

          \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \color{blue}{\sqrt{0.5 \cdot \left(2 \cdot \frac{1}{re}\right)}} \]

      13. un-div-inv 89.9%

          \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \sqrt{0.5 \cdot \color{blue}{\frac{2}{re}}} \]

   6. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{0 + \left(0.5 \cdot im\right) \cdot \sqrt{0.5 \cdot \frac{2}{re}}} \]
   7. Step-by-step derivation

      1. +-lft-identity 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-*r/ 89.9%

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

      4. metadata-eval 89.9%

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

      5. unpow1/2 89.9%

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

      6. unpow-1 89.9%

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

      7. exp-to-pow 85.1%

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

      8. exp-prod 85.1%

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

      9. associate-*l* 85.1%

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

      10. metadata-eval 85.1%

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

      11. exp-to-pow 90.0%

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

   8. Simplified 90.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 0.00075:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 3: 53.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e-310)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* (pow re -0.5) (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = pow(re, -0.5) * (0.5 * 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 <= (-5d-310)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else
        tmp = (re ** (-0.5d0)) * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = Math.pow(re, -0.5) * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = math.pow(re, -0.5) * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	else
		tmp = Float64((re ^ -0.5) * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e-310)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = (re ^ -0.5) * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 53.4%

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

   2. Taylor expanded in re around -inf 47.1%

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

      1. *-commutative 47.1%

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

   4. Simplified 47.1%

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

   if -4.999999999999985e-310 < re

   1. Initial program 27.3%

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

   2. Taylor expanded in im around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. *-commutative 60.3%

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

   4. Simplified 60.3%

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

      1. add-log-exp 18.3%

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

      2. *-un-lft-identity 18.3%

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

      3. log-prod 18.3%

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

      4. metadata-eval 18.3%

         \[\leadsto \color{blue}{0} + \log \left(e^{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \]

      5. add-log-exp 60.3%

         \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \]

      6. associate-*l* 60.3%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \left(\left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

      7. *-commutative 60.3%

         \[\leadsto 0 + 0.5 \cdot \left(\color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1}{re}}\right) \]

      8. associate-*r* 60.4%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      9. associate-*l* 60.3%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      10. associate-*r* 60.4%

          \[\leadsto 0 + \color{blue}{\left(0.5 \cdot im\right) \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

      11. sqrt-unprod 60.6%

          \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \left(\sqrt{0.5} \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{re}}}\right) \]

      12. sqrt-unprod 60.7%

          \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \color{blue}{\sqrt{0.5 \cdot \left(2 \cdot \frac{1}{re}\right)}} \]

      13. un-div-inv 60.7%

          \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \sqrt{0.5 \cdot \color{blue}{\frac{2}{re}}} \]

   6. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{0 + \left(0.5 \cdot im\right) \cdot \sqrt{0.5 \cdot \frac{2}{re}}} \]
   7. Step-by-step derivation

      1. +-lft-identity 60.7%

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

      2. *-commutative 60.7%

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

      3. associate-*r/ 60.7%

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

      4. metadata-eval 60.7%

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

      5. unpow1/2 60.7%

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

      6. unpow-1 60.7%

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

      7. exp-to-pow 57.5%

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

      8. exp-prod 57.5%

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

      9. associate-*l* 57.5%

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

      10. metadata-eval 57.5%

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

      11. exp-to-pow 60.8%

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

   8. Simplified 60.8%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 4: 26.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* (pow re -0.5) (* 0.5 im)))

C

double code(double re, double im) {
	return pow(re, -0.5) * (0.5 * im);
}

Fortran

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

Java

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

Python

def code(re, im):
	return math.pow(re, -0.5) * (0.5 * im)

Julia

function code(re, im)
	return Float64((re ^ -0.5) * Float64(0.5 * im))
end

MATLAB

function tmp = code(re, im)
	tmp = (re ^ -0.5) * (0.5 * im);
end

Wolfram

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

Derivation

1. Initial program 41.5%

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

2. Taylor expanded in im around 0 27.6%

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

   1. associate-*r* 27.6%

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

   2. *-commutative 27.6%

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

4. Simplified 27.6%

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

   1. add-log-exp 8.4%

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

   2. *-un-lft-identity 8.4%

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

   3. log-prod 8.4%

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

   4. metadata-eval 8.4%

      \[\leadsto \color{blue}{0} + \log \left(e^{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \]

   5. add-log-exp 27.6%

      \[\leadsto 0 + \color{blue}{\left(0.5 \cdot \left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right)\right) \cdot \sqrt{\frac{1}{re}}} \]

   6. associate-*l* 27.6%

      \[\leadsto 0 + \color{blue}{0.5 \cdot \left(\left(\sqrt{2} \cdot \left(im \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]

   7. *-commutative 27.6%

      \[\leadsto 0 + 0.5 \cdot \left(\color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1}{re}}\right) \]

   8. associate-*r* 27.6%

      \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

   9. associate-*l* 27.6%

      \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(im \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

   10. associate-*r* 27.6%

       \[\leadsto 0 + \color{blue}{\left(0.5 \cdot im\right) \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

   11. sqrt-unprod 27.7%

       \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \left(\sqrt{0.5} \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{re}}}\right) \]

   12. sqrt-unprod 27.7%

       \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \color{blue}{\sqrt{0.5 \cdot \left(2 \cdot \frac{1}{re}\right)}} \]

   13. un-div-inv 27.7%

       \[\leadsto 0 + \left(0.5 \cdot im\right) \cdot \sqrt{0.5 \cdot \color{blue}{\frac{2}{re}}} \]

6. Applied egg-rr 27.7%

   \[\leadsto \color{blue}{0 + \left(0.5 \cdot im\right) \cdot \sqrt{0.5 \cdot \frac{2}{re}}} \]
7. Step-by-step derivation

   1. +-lft-identity 27.7%

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

   2. *-commutative 27.7%

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

   3. associate-*r/ 27.7%

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

   4. metadata-eval 27.7%

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

   5. unpow1/2 27.7%

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

   6. unpow-1 27.7%

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

   7. exp-to-pow 26.3%

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

   8. exp-prod 26.3%

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

   9. associate-*l* 26.3%

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

   10. metadata-eval 26.3%

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

   11. exp-to-pow 27.8%

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

8. Simplified 27.8%

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

9. Final simplification 27.8%

   \[\leadsto {re}^{-0.5} \cdot \left(0.5 \cdot im\right) \]

Reproduce

herbie shell --seed 2023264 
(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)))))