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

Percentage Accurate: 41.2% → 87.9%

Time: 5.4s

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: 41.2% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e-21)
   (sqrt (* 0.5 (- (hypot re im) re)))
   (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e-21) {
		tmp = sqrt((0.5 * (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 <= 6.2e-21) {
		tmp = Math.sqrt((0.5 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e-21:
		tmp = math.sqrt((0.5 * (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 <= 6.2e-21)
		tmp = sqrt(Float64(0.5 * 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 <= 6.2e-21)
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e-21], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.1999999999999997e-21

   1. Initial program 54.1%

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

      1. *-commutative 54.1%

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

      2. hypot-udef 95.4%

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

      3. *-commutative 95.4%

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

      4. add-sqr-sqrt 94.6%

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

      5. sqrt-unprod 95.4%

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

      6. *-commutative 95.4%

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

      7. *-commutative 95.4%

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

      8. swap-sqr 95.4%

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

      9. add-sqr-sqrt 95.4%

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

      10. metadata-eval 95.4%

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

   3. Applied egg-rr 95.4%

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

      1. *-commutative 95.4%

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

      2. associate-*r* 95.9%

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

      3. metadata-eval 95.9%

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

   5. Simplified 95.9%

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

   if 6.1999999999999997e-21 < re

   1. Initial program 10.3%

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

   2. Taylor expanded in im around 0 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-*l* 81.2%

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

      3. *-commutative 81.2%

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

   4. Simplified 81.2%

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

      1. expm1-log1p-u 80.2%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      2. expm1-udef 36.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} - 1\right)} \]

   6. Applied egg-rr 36.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \sqrt{0.5 \cdot \frac{2}{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 80.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \sqrt{0.5 \cdot \frac{2}{re}}\right)\right)} \]

      2. expm1-log1p 81.8%

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

      3. associate-*r/ 81.8%

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

      4. metadata-eval 81.8%

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

      5. unpow-1 81.8%

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

      6. exp-to-pow 76.7%

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

      7. *-commutative 76.7%

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

      8. neg-mul-1 76.7%

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

      9. unpow1/2 76.7%

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

      10. exp-prod 76.7%

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

      11. distribute-lft-neg-out 76.7%

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

      12. exp-neg 76.7%

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

      13. exp-to-pow 81.8%

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

      14. unpow1/2 81.8%

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

      15. associate-*r/ 81.9%

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

      16. *-rgt-identity 81.9%

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

   8. Simplified 81.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 2: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -9.2 \cdot 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -8.8 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -1.1 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

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 -9.2e+103)
     t_0
     (if (<= re -8.8e-33)
       t_1
       (if (<= re -1.1e-82)
         t_0
         (if (<= re 7.5e-22) 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 <= -9.2e+103) {
		tmp = t_0;
	} else if (re <= -8.8e-33) {
		tmp = t_1;
	} else if (re <= -1.1e-82) {
		tmp = t_0;
	} else if (re <= 7.5e-22) {
		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 <= (-9.2d+103)) then
        tmp = t_0
    else if (re <= (-8.8d-33)) then
        tmp = t_1
    else if (re <= (-1.1d-82)) then
        tmp = t_0
    else if (re <= 7.5d-22) 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 <= -9.2e+103) {
		tmp = t_0;
	} else if (re <= -8.8e-33) {
		tmp = t_1;
	} else if (re <= -1.1e-82) {
		tmp = t_0;
	} else if (re <= 7.5e-22) {
		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 <= -9.2e+103:
		tmp = t_0
	elif re <= -8.8e-33:
		tmp = t_1
	elif re <= -1.1e-82:
		tmp = t_0
	elif re <= 7.5e-22:
		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 <= -9.2e+103)
		tmp = t_0;
	elseif (re <= -8.8e-33)
		tmp = t_1;
	elseif (re <= -1.1e-82)
		tmp = t_0;
	elseif (re <= 7.5e-22)
		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 <= -9.2e+103)
		tmp = t_0;
	elseif (re <= -8.8e-33)
		tmp = t_1;
	elseif (re <= -1.1e-82)
		tmp = t_0;
	elseif (re <= 7.5e-22)
		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, -9.2e+103], t$95$0, If[LessEqual[re, -8.8e-33], t$95$1, If[LessEqual[re, -1.1e-82], t$95$0, If[LessEqual[re, 7.5e-22], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.20000000000000034e103 or -8.80000000000000022e-33 < re < -1.09999999999999993e-82

   1. Initial program 40.4%

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

      1. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around -inf 87.8%

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

      1. *-commutative 87.8%

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

   6. Simplified 87.8%

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

   if -9.20000000000000034e103 < re < -8.80000000000000022e-33 or -1.09999999999999993e-82 < re < 7.49999999999999978e-22

   1. Initial program 58.5%

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

   2. Taylor expanded in re around 0 82.1%

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

   if 7.49999999999999978e-22 < re

   1. Initial program 10.3%

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

   2. Taylor expanded in im around 0 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-*l* 81.2%

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

      3. *-commutative 81.2%

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

   4. Simplified 81.2%

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

      1. expm1-log1p-u 80.2%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      2. expm1-udef 36.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} - 1\right)} \]

   6. Applied egg-rr 36.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \sqrt{0.5 \cdot \frac{2}{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 80.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \sqrt{0.5 \cdot \frac{2}{re}}\right)\right)} \]

      2. expm1-log1p 81.8%

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

      3. associate-*r/ 81.8%

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

      4. metadata-eval 81.8%

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

      5. unpow-1 81.8%

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

      6. exp-to-pow 76.7%

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

      7. *-commutative 76.7%

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

      8. neg-mul-1 76.7%

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

      9. unpow1/2 76.7%

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

      10. exp-prod 76.7%

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

      11. distribute-lft-neg-out 76.7%

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

      12. exp-neg 76.7%

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

      13. exp-to-pow 81.8%

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

      14. unpow1/2 81.8%

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

      15. associate-*r/ 81.9%

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

      16. *-rgt-identity 81.9%

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

   8. Simplified 81.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -8.8 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -1.1 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 75.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.15e-82)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 5e-16) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.15e-82) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5e-16) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= (-2.15d-82)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5d-16) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 <= -2.15e-82) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5e-16) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.15e-82:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5e-16:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.15e-82)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5e-16)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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 <= -2.15e-82)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5e-16)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.15e-82], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e-16], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.15000000000000009e-82

   1. Initial program 55.9%

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

      1. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around -inf 71.1%

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

      1. *-commutative 71.1%

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

   6. Simplified 71.1%

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

   if -2.15000000000000009e-82 < re < 5.0000000000000004e-16

   1. Initial program 52.8%

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

      1. hypot-def 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in re around 0 85.9%

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

      1. *-commutative 85.9%

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

   6. Simplified 85.9%

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

   if 5.0000000000000004e-16 < re

   1. Initial program 9.1%

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

   2. Taylor expanded in im around 0 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*l* 82.0%

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

      3. *-commutative 82.0%

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

   4. Simplified 82.0%

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

      1. expm1-log1p-u 81.0%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)\right)} \]

      2. expm1-udef 37.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(im \cdot \sqrt{0.5}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} - 1\right)} \]

   6. Applied egg-rr 37.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(im \cdot \sqrt{0.5 \cdot \frac{2}{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 81.5%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(im \cdot \sqrt{0.5 \cdot \frac{2}{re}}\right)\right)} \]

      2. expm1-log1p 82.6%

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

      3. associate-*r/ 82.6%

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

      4. metadata-eval 82.6%

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

      5. unpow-1 82.6%

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

      6. exp-to-pow 77.4%

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

      7. *-commutative 77.4%

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

      8. neg-mul-1 77.4%

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

      9. unpow1/2 77.4%

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

      10. exp-prod 77.5%

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

      11. distribute-lft-neg-out 77.5%

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

      12. exp-neg 77.4%

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

      13. exp-to-pow 82.6%

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

      14. unpow1/2 82.6%

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

      15. associate-*r/ 82.7%

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

      16. *-rgt-identity 82.7%

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

   8. Simplified 82.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 63.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e-82) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-82) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5d-82)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e-82) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-82:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-82)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e-82)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-82], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.9999999999999998e-82

   1. Initial program 55.9%

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

      1. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around -inf 71.1%

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

      1. *-commutative 71.1%

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

   6. Simplified 71.1%

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

   if -4.9999999999999998e-82 < re

   1. Initial program 35.8%

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

      1. hypot-def 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in re around 0 61.4%

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

      1. *-commutative 61.4%

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

   6. Simplified 61.4%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5: 52.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

   1. hypot-def 78.2%

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

3. Simplified 78.2%

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

4. Taylor expanded in re around 0 53.5%

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

   1. *-commutative 53.5%

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

6. Simplified 53.5%

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

7. Final simplification 53.5%

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

Reproduce

herbie shell --seed 2023229 
(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)))))