math.sqrt on complex, real part

Average Error: 38.6 → 26.2

Time: 43.0s

Precision: binary64

Cost: 14424

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ t_1 := 0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{if}\;im \leq -1.55 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{-0.25}{re}} \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq -2.75 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))
        (t_1 (* 0.5 (* im (sqrt (/ -1.0 re))))))
   (if (<= im -1.55e+112)
     (* 0.5 (sqrt (* 2.0 (- re im))))
     (if (<= im -5.5e+22)
       (* (sqrt (/ -0.25 re)) (- im))
       (if (<= im -2.75e-176)
         t_0
         (if (<= im 1.75e-296)
           (* 0.5 (* 2.0 (sqrt re)))
           (if (<= im 5.6e-149)
             t_1
             (if (<= im 5.3e-114)
               t_0
               (if (<= im 2.5e-45)
                 t_1
                 (* 0.5 (sqrt (* 2.0 (+ re im)))))))))))))

C

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

↓

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	double t_1 = 0.5 * (im * sqrt((-1.0 / re)));
	double tmp;
	if (im <= -1.55e+112) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= -5.5e+22) {
		tmp = sqrt((-0.25 / re)) * -im;
	} else if (im <= -2.75e-176) {
		tmp = t_0;
	} else if (im <= 1.75e-296) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else if (im <= 5.6e-149) {
		tmp = t_1;
	} else if (im <= 5.3e-114) {
		tmp = t_0;
	} else if (im <= 2.5e-45) {
		tmp = t_1;
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Fortran

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

↓

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((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
    t_1 = 0.5d0 * (im * sqrt(((-1.0d0) / re)))
    if (im <= (-1.55d+112)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= (-5.5d+22)) then
        tmp = sqrt(((-0.25d0) / re)) * -im
    else if (im <= (-2.75d-176)) then
        tmp = t_0
    else if (im <= 1.75d-296) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else if (im <= 5.6d-149) then
        tmp = t_1
    else if (im <= 5.3d-114) then
        tmp = t_0
    else if (im <= 2.5d-45) then
        tmp = t_1
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
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)));
}

↓

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
	double t_1 = 0.5 * (im * Math.sqrt((-1.0 / re)));
	double tmp;
	if (im <= -1.55e+112) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= -5.5e+22) {
		tmp = Math.sqrt((-0.25 / re)) * -im;
	} else if (im <= -2.75e-176) {
		tmp = t_0;
	} else if (im <= 1.75e-296) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else if (im <= 5.6e-149) {
		tmp = t_1;
	} else if (im <= 5.3e-114) {
		tmp = t_0;
	} else if (im <= 2.5e-45) {
		tmp = t_1;
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

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

↓

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))
	t_1 = 0.5 * (im * math.sqrt((-1.0 / re)))
	tmp = 0
	if im <= -1.55e+112:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= -5.5e+22:
		tmp = math.sqrt((-0.25 / re)) * -im
	elif im <= -2.75e-176:
		tmp = t_0
	elif im <= 1.75e-296:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	elif im <= 5.6e-149:
		tmp = t_1
	elif im <= 5.3e-114:
		tmp = t_0
	elif im <= 2.5e-45:
		tmp = t_1
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

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

↓

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
	t_1 = Float64(0.5 * Float64(im * sqrt(Float64(-1.0 / re))))
	tmp = 0.0
	if (im <= -1.55e+112)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= -5.5e+22)
		tmp = Float64(sqrt(Float64(-0.25 / re)) * Float64(-im));
	elseif (im <= -2.75e-176)
		tmp = t_0;
	elseif (im <= 1.75e-296)
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	elseif (im <= 5.6e-149)
		tmp = t_1;
	elseif (im <= 5.3e-114)
		tmp = t_0;
	elseif (im <= 2.5e-45)
		tmp = t_1;
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	t_1 = 0.5 * (im * sqrt((-1.0 / re)));
	tmp = 0.0;
	if (im <= -1.55e+112)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= -5.5e+22)
		tmp = sqrt((-0.25 / re)) * -im;
	elseif (im <= -2.75e-176)
		tmp = t_0;
	elseif (im <= 1.75e-296)
		tmp = 0.5 * (2.0 * sqrt(re));
	elseif (im <= 5.6e-149)
		tmp = t_1;
	elseif (im <= 5.3e-114)
		tmp = t_0;
	elseif (im <= 2.5e-45)
		tmp = t_1;
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
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]

↓

code[re_, im_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(0.5 * N[(im * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.55e+112], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -5.5e+22], N[(N[Sqrt[N[(-0.25 / re), $MachinePrecision]], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, -2.75e-176], t$95$0, If[LessEqual[im, 1.75e-296], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e-149], t$95$1, If[LessEqual[im, 5.3e-114], t$95$0, If[LessEqual[im, 2.5e-45], t$95$1, N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Target

Original	38.6
Target	33.6
Herbie	26.2

\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array} \]

Derivation

1. Split input into 6 regimes

2. if im < -1.54999999999999991e112

   1. Initial program 52.5

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

   2. Taylor expanded in im around -inf 9.0

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

   3. Simplified 9.0

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

      [Start] 9.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + -1 \cdot im\right)} \]
      rational_best-simplify-59 [=>] 9.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-1 \cdot im - \left(-re\right)\right)}} \]
      rational_best-simplify-1 [=>] 9.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im \cdot -1} - \left(-re\right)\right)} \]
      rational_best-simplify-11 [<=] 9.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-im\right)} - \left(-re\right)\right)} \]
      rational_best-simplify-14 [=>] 9.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - im\right)} - \left(-re\right)\right)} \]
      rational_best-simplify-48 [=>] 9.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(0 - \left(-re\right)\right) - im\right)}} \]
      rational_best-simplify-14 [=>] 9.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(0 - \color{blue}{\left(0 - re\right)}\right) - im\right)} \]
      rational_best-simplify-51 [=>] 9.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(re - \left(0 - 0\right)\right)} - im\right)} \]
      metadata-eval [=>] 9.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\left(re - \color{blue}{0}\right) - im\right)} \]
      rational_best-simplify-9 [=>] 9.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} - im\right)} \]

   if -1.54999999999999991e112 < im < -5.50000000000000021e22

   1. Initial program 20.4

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

   2. Taylor expanded in re around -inf 50.6

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

   3. Applied egg-rr 50.3

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{-0.25}{re}} \cdot \sqrt{{im}^{2} \cdot 4}\right)} \]

   4. Taylor expanded in im around -inf 64.0

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot \left(\left(\sqrt{-0.25} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

   5. Simplified 50.3

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot \left(im \cdot \sqrt{\frac{-0.25}{re}}\right)\right)} \]
      Proof

      [Start] 64.0	\[ 0.5 \cdot \left(-2 \cdot \left(\left(\sqrt{-0.25} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right)\right) \]
      rational_best-simplify-1 [=>] 64.0	\[ 0.5 \cdot \left(-2 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{-0.25} \cdot im\right)\right)}\right) \]
      rational_best-simplify-50 [=>] 64.0	\[ 0.5 \cdot \left(-2 \cdot \color{blue}{\left(im \cdot \left(\sqrt{-0.25} \cdot \sqrt{\frac{1}{re}}\right)\right)}\right) \]
      exponential-simplify-21 [=>] 50.3	\[ 0.5 \cdot \left(-2 \cdot \left(im \cdot \color{blue}{\sqrt{-0.25 \cdot \frac{1}{re}}}\right)\right) \]
      rational_best-simplify-55 [=>] 50.3	\[ 0.5 \cdot \left(-2 \cdot \left(im \cdot \sqrt{\color{blue}{1 \cdot \frac{-0.25}{re}}}\right)\right) \]
      exponential-simplify-20 [=>] 50.3	\[ 0.5 \cdot \left(-2 \cdot \left(im \cdot \color{blue}{\left(\sqrt{1} \cdot \sqrt{\frac{-0.25}{re}}\right)}\right)\right) \]
      metadata-eval [=>] 50.3	\[ 0.5 \cdot \left(-2 \cdot \left(im \cdot \left(\color{blue}{1} \cdot \sqrt{\frac{-0.25}{re}}\right)\right)\right) \]
      rational_best-simplify-1 [<=] 50.3	\[ 0.5 \cdot \left(-2 \cdot \left(im \cdot \color{blue}{\left(\sqrt{\frac{-0.25}{re}} \cdot 1\right)}\right)\right) \]
      rational_best-simplify-7 [=>] 50.3	\[ 0.5 \cdot \left(-2 \cdot \left(im \cdot \color{blue}{\sqrt{\frac{-0.25}{re}}}\right)\right) \]

   6. Applied egg-rr 50.3

      \[\leadsto \color{blue}{\sqrt{\frac{-0.25}{re}} \cdot \left(-im\right) + 0} \]

   7. Simplified 50.3

      \[\leadsto \color{blue}{\sqrt{\frac{-0.25}{re}} \cdot \left(-im\right)} \]
      Proof

      [Start] 50.3	\[ \sqrt{\frac{-0.25}{re}} \cdot \left(-im\right) + 0 \]
      rational_best-simplify-3 [=>] 50.3	\[ \color{blue}{0 + \sqrt{\frac{-0.25}{re}} \cdot \left(-im\right)} \]
      rational_best-simplify-6 [=>] 50.3	\[ \color{blue}{\sqrt{\frac{-0.25}{re}} \cdot \left(-im\right)} \]

   if -5.50000000000000021e22 < im < -2.75e-176 or 5.5999999999999997e-149 < im < 5.29999999999999973e-114

   1. Initial program 29.6

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

   if -2.75e-176 < im < 1.7499999999999999e-296

   1. Initial program 42.6

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

   2. Taylor expanded in im around 0 33.9

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

   3. Simplified 33.3

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

      [Start] 33.9	\[ 0.5 \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right) \]
      exponential-simplify-25 [=>] 33.3	\[ 0.5 \cdot \left(\color{blue}{\sqrt{{2}^{2}}} \cdot \sqrt{re}\right) \]
      metadata-eval [=>] 33.3	\[ 0.5 \cdot \left(\sqrt{\color{blue}{4}} \cdot \sqrt{re}\right) \]
      metadata-eval [=>] 33.3	\[ 0.5 \cdot \left(\color{blue}{2} \cdot \sqrt{re}\right) \]

   if 1.7499999999999999e-296 < im < 5.5999999999999997e-149 or 5.29999999999999973e-114 < im < 2.49999999999999988e-45

   1. Initial program 38.7

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

   2. Taylor expanded in re around -inf 52.4

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

   3. Applied egg-rr 38.6

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

   4. Taylor expanded in im around 0 64.0

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

   5. Simplified 38.5

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

      [Start] 64.0	\[ 0.5 \cdot \left(\left(\sqrt{-1} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      rational_best-simplify-1 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{-1} \cdot im\right)\right)} \]
      rational_best-simplify-50 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(im \cdot \left(\sqrt{-1} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
      exponential-simplify-21 [=>] 38.5	\[ 0.5 \cdot \left(im \cdot \color{blue}{\sqrt{-1 \cdot \frac{1}{re}}}\right) \]
      rational_best-simplify-55 [=>] 38.5	\[ 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{1 \cdot \frac{-1}{re}}}\right) \]
      rational_best-simplify-1 [=>] 38.5	\[ 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{\frac{-1}{re} \cdot 1}}\right) \]
      rational_best-simplify-7 [=>] 38.5	\[ 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{\frac{-1}{re}}}\right) \]

   if 2.49999999999999988e-45 < im

   1. Initial program 39.8

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

   2. Taylor expanded in re around 0 16.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + im\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 26.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.55 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{-0.25}{re}} \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq -2.75 \cdot 10^{-176}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{-114}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.6
Cost	7508

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{if}\;im \leq -1.55 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{-0.25}{re}} \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{-184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	28.0
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{if}\;im \leq -3.5 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{\frac{-0.25}{re}} \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{-184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	27.8
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{if}\;im \leq -1.55 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{-0.25}{re}} \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{-184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	28.2
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{if}\;im \leq -3.5 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{-0.25}{re}} \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{-184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Error	26.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{-184}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 4.6 \cdot 10^{-64}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Error	30.5
Cost	6852

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

Alternative 7
Error	47.7
Cost	6720

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

Reproduce

herbie shell --seed 2023099 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))