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

Percentage Accurate: 41.4% → 87.9%

Time: 7.0s

Alternatives: 5

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: 41.4% 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: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.35:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.35)
   (* 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 <= 0.35) {
		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 <= 0.35) {
		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 <= 0.35:
		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 <= 0.35)
		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 <= 0.35)
		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, 0.35], 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 < 0.34999999999999998

   1. Initial program 59.0%

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

      1. hypot-def 94.9%

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

   3. Simplified 94.9%

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

   if 0.34999999999999998 < re

   1. Initial program 10.9%

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

   2. Taylor expanded in re around inf 65.2%

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

      1. unpow2 65.2%

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

   4. Simplified 65.2%

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

   5. Taylor expanded in im around 0 85.2%

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

      1. *-commutative 85.2%

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

      2. rem-exp-log 80.2%

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

      3. unpow1/2 80.2%

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

      4. rem-exp-log 79.5%

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

      5. exp-neg 79.5%

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

      6. exp-prod 79.5%

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

      7. distribute-lft-neg-out 79.5%

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

      8. distribute-rgt-neg-in 79.5%

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

      9. metadata-eval 79.5%

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

      10. *-commutative 79.5%

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

      11. log-pow 79.5%

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

      12. exp-to-pow 79.5%

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

      13. metadata-eval 79.5%

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

      14. distribute-rgt-neg-in 79.5%

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

      15. exp-neg 79.5%

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

      16. log-rec 79.5%

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

      17. exp-to-pow 79.5%

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

      18. unpow1/2 79.5%

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

      19. exp-sum 79.2%

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

      20. sub-neg 79.2%

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

      21. log-div 80.0%

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

   7. Simplified 85.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.35:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 2: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.4 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 12:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.4e-30)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 12.0)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.4e-30) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 12.0) {
		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) :: tmp
    if (re <= (-9.4d-30)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 12.0d0) 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 tmp;
	if (re <= -9.4e-30) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 12.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.4e-30:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 12.0:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.4e-30)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 12.0)
		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)
	tmp = 0.0;
	if (re <= -9.4e-30)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 12.0)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.4e-30], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 12.0], 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 3 regimes

2. if re < -9.39999999999999938e-30

   1. Initial program 48.5%

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

   2. Taylor expanded in re around -inf 77.6%

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

      1. *-commutative 77.6%

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

   4. Simplified 77.6%

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

   if -9.39999999999999938e-30 < re < 12

   1. Initial program 64.4%

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

   2. Taylor expanded in re around 0 83.8%

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

   if 12 < re

   1. Initial program 10.9%

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

   2. Taylor expanded in re around inf 65.2%

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

      1. unpow2 65.2%

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

   4. Simplified 65.2%

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

   5. Taylor expanded in im around 0 85.2%

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

      1. *-commutative 85.2%

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

      2. rem-exp-log 80.2%

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

      3. unpow1/2 80.2%

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

      4. rem-exp-log 79.5%

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

      5. exp-neg 79.5%

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

      6. exp-prod 79.5%

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

      7. distribute-lft-neg-out 79.5%

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

      8. distribute-rgt-neg-in 79.5%

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

      9. metadata-eval 79.5%

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

      10. *-commutative 79.5%

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

      11. log-pow 79.5%

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

      12. exp-to-pow 79.5%

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

      13. metadata-eval 79.5%

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

      14. distribute-rgt-neg-in 79.5%

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

      15. exp-neg 79.5%

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

      16. log-rec 79.5%

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

      17. exp-to-pow 79.5%

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

      18. unpow1/2 79.5%

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

      19. exp-sum 79.2%

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

      20. sub-neg 79.2%

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

      21. log-div 80.0%

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

   7. Simplified 85.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.4 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 12:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 75.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 27000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.5e-30)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 27000000000000.0)
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.5e-30) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 27000000000000.0) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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) :: tmp
    if (re <= (-7.5d-30)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 27000000000000.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.5e-30) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 27000000000000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.5e-30:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 27000000000000.0:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.5e-30)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 27000000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= -7.5e-30)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 27000000000000.0)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.5e-30], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 27000000000000.0], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.5000000000000006e-30

   1. Initial program 48.5%

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

   2. Taylor expanded in re around -inf 77.6%

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

      1. *-commutative 77.6%

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

   4. Simplified 77.6%

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

   if -7.5000000000000006e-30 < re < 2.7e13

   1. Initial program 63.5%

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

      1. hypot-udef 91.7%

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

      2. add-sqr-sqrt 91.4%

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

      3. pow2 91.4%

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

   3. Applied egg-rr 91.4%

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

      1. unpow2 91.4%

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

      2. add-sqr-sqrt 91.7%

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

      3. add-sqr-sqrt 44.0%

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

      4. add-sqr-sqrt 44.0%

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

      5. sqr-neg 44.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 42.9%

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

      8. distribute-lft-neg-in 42.9%

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

      9. add-sqr-sqrt 82.5%

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

      10. *-un-lft-identity 82.5%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \mathsf{hypot}\left(re, im\right)} - \left(-re\right)\right)} \]

      11. *-commutative 82.5%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right) \cdot 1} - \left(-re\right)\right)} \]

      12. neg-mul-1 82.5%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) \cdot 1 - \color{blue}{-1 \cdot re}\right)} \]

      13. prod-diff 82.5%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(-re, -1, re \cdot -1\right)\right)}} \]

      14. add-sqr-sqrt 42.9%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(-\color{blue}{\sqrt{re} \cdot \sqrt{re}}, -1, re \cdot -1\right)\right)} \]

      15. distribute-rgt-neg-in 42.9%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\color{blue}{\sqrt{re} \cdot \left(-\sqrt{re}\right)}, -1, re \cdot -1\right)\right)} \]

      16. add-sqr-sqrt 0.0%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \color{blue}{\left(\sqrt{-\sqrt{re}} \cdot \sqrt{-\sqrt{re}}\right)}, -1, re \cdot -1\right)\right)} \]

      17. sqrt-unprod 44.0%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \color{blue}{\sqrt{\left(-\sqrt{re}\right) \cdot \left(-\sqrt{re}\right)}}, -1, re \cdot -1\right)\right)} \]

      18. sqr-neg 44.0%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \sqrt{\color{blue}{\sqrt{re} \cdot \sqrt{re}}}, -1, re \cdot -1\right)\right)} \]

      19. add-sqr-sqrt 44.0%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \sqrt{\color{blue}{re}}, -1, re \cdot -1\right)\right)} \]

      20. add-sqr-sqrt 91.7%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\color{blue}{re}, -1, re \cdot -1\right)\right)} \]

   5. Applied egg-rr 91.7%

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

      1. fma-udef 91.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + \left(-re \cdot -1\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      2. distribute-rgt-neg-in 91.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + \color{blue}{re \cdot \left(--1\right)}\right) + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      3. metadata-eval 91.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + re \cdot \color{blue}{1}\right) + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      4. distribute-rgt-in 91.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      5. +-commutative 91.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      6. *-lft-identity 91.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      7. *-lft-identity 91.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      8. +-commutative 91.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(\mathsf{hypot}\left(re, im\right) + re\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      9. distribute-rgt-in 91.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + re \cdot 1\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      10. *-rgt-identity 91.7%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} + re \cdot 1\right) + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      11. *-commutative 91.7%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) + re \cdot 1\right) + \mathsf{fma}\left(re, -1, \color{blue}{-1 \cdot re}\right)\right)} \]

      12. mul-1-neg 91.7%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) + re \cdot 1\right) + \mathsf{fma}\left(re, -1, \color{blue}{-re}\right)\right)} \]

   7. Simplified 91.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\mathsf{hypot}\left(re, im\right) + re \cdot 1\right) + \mathsf{fma}\left(re, -1, -re\right)\right)}} \]

   8. Taylor expanded in re around 0 83.0%

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

   if 2.7e13 < re

   1. Initial program 11.1%

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

   2. Taylor expanded in re around inf 65.6%

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

      1. unpow2 65.6%

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

   4. Simplified 65.6%

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

   5. Taylor expanded in im around 0 86.3%

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

      1. *-commutative 86.3%

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

      2. rem-exp-log 81.1%

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

      3. unpow1/2 81.1%

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

      4. rem-exp-log 80.4%

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

      5. exp-neg 80.4%

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

      6. exp-prod 80.4%

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

      7. distribute-lft-neg-out 80.4%

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

      8. distribute-rgt-neg-in 80.4%

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

      9. metadata-eval 80.4%

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

      10. *-commutative 80.4%

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

      11. log-pow 80.4%

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

      12. exp-to-pow 80.4%

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

      13. metadata-eval 80.4%

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

      14. distribute-rgt-neg-in 80.4%

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

      15. exp-neg 80.4%

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

      16. log-rec 80.4%

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

      17. exp-to-pow 80.4%

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

      18. unpow1/2 80.4%

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

      19. exp-sum 80.2%

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

      20. sub-neg 80.2%

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

      21. log-div 81.0%

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

   7. Simplified 86.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 27000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 30000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 30000000000000.0)
   (* 0.5 (sqrt (* 2.0 im)))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 30000000000000.0) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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) :: tmp
    if (re <= 30000000000000.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 30000000000000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 30000000000000.0:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 30000000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= 30000000000000.0)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 30000000000000.0], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3e13

   1. Initial program 58.4%

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

      1. hypot-udef 94.5%

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

      2. add-sqr-sqrt 94.3%

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

      3. pow2 94.3%

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

   3. Applied egg-rr 94.3%

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

      1. unpow2 94.3%

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

      2. add-sqr-sqrt 94.5%

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

      3. add-sqr-sqrt 29.2%

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

      4. add-sqr-sqrt 29.2%

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

      5. sqr-neg 29.2%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 28.5%

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

      8. distribute-lft-neg-in 28.5%

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

      9. add-sqr-sqrt 62.9%

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

      10. *-un-lft-identity 62.9%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \mathsf{hypot}\left(re, im\right)} - \left(-re\right)\right)} \]

      11. *-commutative 62.9%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right) \cdot 1} - \left(-re\right)\right)} \]

      12. neg-mul-1 62.9%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) \cdot 1 - \color{blue}{-1 \cdot re}\right)} \]

      13. prod-diff 62.9%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(-re, -1, re \cdot -1\right)\right)}} \]

      14. add-sqr-sqrt 28.5%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(-\color{blue}{\sqrt{re} \cdot \sqrt{re}}, -1, re \cdot -1\right)\right)} \]

      15. distribute-rgt-neg-in 28.5%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\color{blue}{\sqrt{re} \cdot \left(-\sqrt{re}\right)}, -1, re \cdot -1\right)\right)} \]

      16. add-sqr-sqrt 0.0%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \color{blue}{\left(\sqrt{-\sqrt{re}} \cdot \sqrt{-\sqrt{re}}\right)}, -1, re \cdot -1\right)\right)} \]

      17. sqrt-unprod 29.1%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \color{blue}{\sqrt{\left(-\sqrt{re}\right) \cdot \left(-\sqrt{re}\right)}}, -1, re \cdot -1\right)\right)} \]

      18. sqr-neg 29.1%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \sqrt{\color{blue}{\sqrt{re} \cdot \sqrt{re}}}, -1, re \cdot -1\right)\right)} \]

      19. add-sqr-sqrt 29.1%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \sqrt{\color{blue}{re}}, -1, re \cdot -1\right)\right)} \]

      20. add-sqr-sqrt 94.5%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\color{blue}{re}, -1, re \cdot -1\right)\right)} \]

   5. Applied egg-rr 94.5%

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

      1. fma-udef 94.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + \left(-re \cdot -1\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      2. distribute-rgt-neg-in 94.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + \color{blue}{re \cdot \left(--1\right)}\right) + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      3. metadata-eval 94.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + re \cdot \color{blue}{1}\right) + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      4. distribute-rgt-in 94.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      5. +-commutative 94.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      6. *-lft-identity 94.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      7. *-lft-identity 94.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      8. +-commutative 94.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(\mathsf{hypot}\left(re, im\right) + re\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      9. distribute-rgt-in 94.5%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + re \cdot 1\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      10. *-rgt-identity 94.5%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} + re \cdot 1\right) + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

      11. *-commutative 94.5%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) + re \cdot 1\right) + \mathsf{fma}\left(re, -1, \color{blue}{-1 \cdot re}\right)\right)} \]

      12. mul-1-neg 94.5%

          \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) + re \cdot 1\right) + \mathsf{fma}\left(re, -1, \color{blue}{-re}\right)\right)} \]

   7. Simplified 94.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\mathsf{hypot}\left(re, im\right) + re \cdot 1\right) + \mathsf{fma}\left(re, -1, -re\right)\right)}} \]

   8. Taylor expanded in re around 0 64.1%

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

   if 3e13 < re

   1. Initial program 11.1%

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

   2. Taylor expanded in re around inf 65.6%

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

      1. unpow2 65.6%

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

   4. Simplified 65.6%

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

   5. Taylor expanded in im around 0 86.3%

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

      1. *-commutative 86.3%

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

      2. rem-exp-log 81.1%

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

      3. unpow1/2 81.1%

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

      4. rem-exp-log 80.4%

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

      5. exp-neg 80.4%

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

      6. exp-prod 80.4%

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

      7. distribute-lft-neg-out 80.4%

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

      8. distribute-rgt-neg-in 80.4%

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

      9. metadata-eval 80.4%

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

      10. *-commutative 80.4%

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

      11. log-pow 80.4%

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

      12. exp-to-pow 80.4%

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

      13. metadata-eval 80.4%

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

      14. distribute-rgt-neg-in 80.4%

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

      15. exp-neg 80.4%

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

      16. log-rec 80.4%

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

      17. exp-to-pow 80.4%

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

      18. unpow1/2 80.4%

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

      19. exp-sum 80.2%

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

      20. sub-neg 80.2%

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

      21. log-div 81.0%

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

   7. Simplified 86.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 30000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 52.1% 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 46.8%

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

   1. hypot-udef 78.9%

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

   2. add-sqr-sqrt 76.3%

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

   3. pow2 76.3%

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

3. Applied egg-rr 76.3%

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

   1. unpow2 76.3%

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

   2. add-sqr-sqrt 78.9%

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

   3. add-sqr-sqrt 27.2%

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

   4. add-sqr-sqrt 27.2%

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

   5. sqr-neg 27.2%

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

   6. sqrt-unprod 0.0%

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

   7. add-sqr-sqrt 25.5%

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

   8. distribute-lft-neg-in 25.5%

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

   9. add-sqr-sqrt 51.4%

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

   10. *-un-lft-identity 51.4%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \mathsf{hypot}\left(re, im\right)} - \left(-re\right)\right)} \]

   11. *-commutative 51.4%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right) \cdot 1} - \left(-re\right)\right)} \]

   12. neg-mul-1 51.4%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) \cdot 1 - \color{blue}{-1 \cdot re}\right)} \]

   13. prod-diff 51.4%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(-re, -1, re \cdot -1\right)\right)}} \]

   14. add-sqr-sqrt 25.5%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(-\color{blue}{\sqrt{re} \cdot \sqrt{re}}, -1, re \cdot -1\right)\right)} \]

   15. distribute-rgt-neg-in 25.5%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\color{blue}{\sqrt{re} \cdot \left(-\sqrt{re}\right)}, -1, re \cdot -1\right)\right)} \]

   16. add-sqr-sqrt 0.0%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \color{blue}{\left(\sqrt{-\sqrt{re}} \cdot \sqrt{-\sqrt{re}}\right)}, -1, re \cdot -1\right)\right)} \]

   17. sqrt-unprod 28.7%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \color{blue}{\sqrt{\left(-\sqrt{re}\right) \cdot \left(-\sqrt{re}\right)}}, -1, re \cdot -1\right)\right)} \]

   18. sqr-neg 28.7%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \sqrt{\color{blue}{\sqrt{re} \cdot \sqrt{re}}}, -1, re \cdot -1\right)\right)} \]

   19. add-sqr-sqrt 28.7%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\sqrt{re} \cdot \sqrt{\color{blue}{re}}, -1, re \cdot -1\right)\right)} \]

   20. add-sqr-sqrt 78.9%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(\mathsf{hypot}\left(re, im\right), 1, -re \cdot -1\right) + \mathsf{fma}\left(\color{blue}{re}, -1, re \cdot -1\right)\right)} \]

5. Applied egg-rr 78.9%

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

   1. fma-udef 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + \left(-re \cdot -1\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

   2. distribute-rgt-neg-in 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + \color{blue}{re \cdot \left(--1\right)}\right) + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

   3. metadata-eval 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + re \cdot \color{blue}{1}\right) + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

   4. distribute-rgt-in 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

   5. +-commutative 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

   6. *-lft-identity 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

   7. *-lft-identity 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

   8. +-commutative 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(\mathsf{hypot}\left(re, im\right) + re\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

   9. distribute-rgt-in 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{hypot}\left(re, im\right) \cdot 1 + re \cdot 1\right)} + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

   10. *-rgt-identity 78.9%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} + re \cdot 1\right) + \mathsf{fma}\left(re, -1, re \cdot -1\right)\right)} \]

   11. *-commutative 78.9%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) + re \cdot 1\right) + \mathsf{fma}\left(re, -1, \color{blue}{-1 \cdot re}\right)\right)} \]

   12. mul-1-neg 78.9%

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) + re \cdot 1\right) + \mathsf{fma}\left(re, -1, \color{blue}{-re}\right)\right)} \]

7. Simplified 78.9%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\mathsf{hypot}\left(re, im\right) + re \cdot 1\right) + \mathsf{fma}\left(re, -1, -re\right)\right)}} \]

8. Taylor expanded in re around 0 52.7%

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

9. Final simplification 52.7%

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

Reproduce

herbie shell --seed 2023189 
(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)))))