math.sqrt on complex, real part

Average Error: 39.1 → 25.3

Time: 12.0s

Precision: binary64

Cost: 14952

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ t_1 := im \cdot \sqrt{\frac{-1}{re}}\\ t_2 := 0.5 \cdot \left(-t_1\right)\\ t_3 := 0.5 \cdot t_1\\ \mathbf{if}\;im \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq -2.7 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + re \cdot 4}\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq -2.65 \cdot 10^{-295}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-180}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-69}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

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 (* im (sqrt (/ -1.0 re))))
        (t_2 (* 0.5 (- t_1)))
        (t_3 (* 0.5 t_1)))
   (if (<= im -1.05e-15)
     (* 0.5 (sqrt (* 2.0 (+ (- im) re))))
     (if (<= im -2.7e-94)
       (* 0.5 (sqrt (+ (/ (pow im 2.0) re) (* re 4.0))))
       (if (<= im -6e-100)
         (* 0.5 (sqrt (* im -2.0)))
         (if (<= im -1.7e-187)
           t_2
           (if (<= im -6e-253)
             (* 0.5 (* 2.0 (sqrt re)))
             (if (<= im -2.65e-295)
               t_2
               (if (<= im 1.45e-180)
                 t_3
                 (if (<= im 1.4e-82)
                   t_0
                   (if (<= im 1.7e-69)
                     t_3
                     (if (<= im 3.2e+38)
                       t_0
                       (* 0.5 (sqrt (* im 2.0)))))))))))))))

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 = im * sqrt((-1.0 / re));
	double t_2 = 0.5 * -t_1;
	double t_3 = 0.5 * t_1;
	double tmp;
	if (im <= -1.05e-15) {
		tmp = 0.5 * sqrt((2.0 * (-im + re)));
	} else if (im <= -2.7e-94) {
		tmp = 0.5 * sqrt(((pow(im, 2.0) / re) + (re * 4.0)));
	} else if (im <= -6e-100) {
		tmp = 0.5 * sqrt((im * -2.0));
	} else if (im <= -1.7e-187) {
		tmp = t_2;
	} else if (im <= -6e-253) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else if (im <= -2.65e-295) {
		tmp = t_2;
	} else if (im <= 1.45e-180) {
		tmp = t_3;
	} else if (im <= 1.4e-82) {
		tmp = t_0;
	} else if (im <= 1.7e-69) {
		tmp = t_3;
	} else if (im <= 3.2e+38) {
		tmp = t_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
    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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
    t_1 = im * sqrt(((-1.0d0) / re))
    t_2 = 0.5d0 * -t_1
    t_3 = 0.5d0 * t_1
    if (im <= (-1.05d-15)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (-im + re)))
    else if (im <= (-2.7d-94)) then
        tmp = 0.5d0 * sqrt((((im ** 2.0d0) / re) + (re * 4.0d0)))
    else if (im <= (-6d-100)) then
        tmp = 0.5d0 * sqrt((im * (-2.0d0)))
    else if (im <= (-1.7d-187)) then
        tmp = t_2
    else if (im <= (-6d-253)) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else if (im <= (-2.65d-295)) then
        tmp = t_2
    else if (im <= 1.45d-180) then
        tmp = t_3
    else if (im <= 1.4d-82) then
        tmp = t_0
    else if (im <= 1.7d-69) then
        tmp = t_3
    else if (im <= 3.2d+38) then
        tmp = t_0
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 = im * Math.sqrt((-1.0 / re));
	double t_2 = 0.5 * -t_1;
	double t_3 = 0.5 * t_1;
	double tmp;
	if (im <= -1.05e-15) {
		tmp = 0.5 * Math.sqrt((2.0 * (-im + re)));
	} else if (im <= -2.7e-94) {
		tmp = 0.5 * Math.sqrt(((Math.pow(im, 2.0) / re) + (re * 4.0)));
	} else if (im <= -6e-100) {
		tmp = 0.5 * Math.sqrt((im * -2.0));
	} else if (im <= -1.7e-187) {
		tmp = t_2;
	} else if (im <= -6e-253) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else if (im <= -2.65e-295) {
		tmp = t_2;
	} else if (im <= 1.45e-180) {
		tmp = t_3;
	} else if (im <= 1.4e-82) {
		tmp = t_0;
	} else if (im <= 1.7e-69) {
		tmp = t_3;
	} else if (im <= 3.2e+38) {
		tmp = t_0;
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	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 = im * math.sqrt((-1.0 / re))
	t_2 = 0.5 * -t_1
	t_3 = 0.5 * t_1
	tmp = 0
	if im <= -1.05e-15:
		tmp = 0.5 * math.sqrt((2.0 * (-im + re)))
	elif im <= -2.7e-94:
		tmp = 0.5 * math.sqrt(((math.pow(im, 2.0) / re) + (re * 4.0)))
	elif im <= -6e-100:
		tmp = 0.5 * math.sqrt((im * -2.0))
	elif im <= -1.7e-187:
		tmp = t_2
	elif im <= -6e-253:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	elif im <= -2.65e-295:
		tmp = t_2
	elif im <= 1.45e-180:
		tmp = t_3
	elif im <= 1.4e-82:
		tmp = t_0
	elif im <= 1.7e-69:
		tmp = t_3
	elif im <= 3.2e+38:
		tmp = t_0
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	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(im * sqrt(Float64(-1.0 / re)))
	t_2 = Float64(0.5 * Float64(-t_1))
	t_3 = Float64(0.5 * t_1)
	tmp = 0.0
	if (im <= -1.05e-15)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(-im) + re))));
	elseif (im <= -2.7e-94)
		tmp = Float64(0.5 * sqrt(Float64(Float64((im ^ 2.0) / re) + Float64(re * 4.0))));
	elseif (im <= -6e-100)
		tmp = Float64(0.5 * sqrt(Float64(im * -2.0)));
	elseif (im <= -1.7e-187)
		tmp = t_2;
	elseif (im <= -6e-253)
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	elseif (im <= -2.65e-295)
		tmp = t_2;
	elseif (im <= 1.45e-180)
		tmp = t_3;
	elseif (im <= 1.4e-82)
		tmp = t_0;
	elseif (im <= 1.7e-69)
		tmp = t_3;
	elseif (im <= 3.2e+38)
		tmp = t_0;
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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 = im * sqrt((-1.0 / re));
	t_2 = 0.5 * -t_1;
	t_3 = 0.5 * t_1;
	tmp = 0.0;
	if (im <= -1.05e-15)
		tmp = 0.5 * sqrt((2.0 * (-im + re)));
	elseif (im <= -2.7e-94)
		tmp = 0.5 * sqrt((((im ^ 2.0) / re) + (re * 4.0)));
	elseif (im <= -6e-100)
		tmp = 0.5 * sqrt((im * -2.0));
	elseif (im <= -1.7e-187)
		tmp = t_2;
	elseif (im <= -6e-253)
		tmp = 0.5 * (2.0 * sqrt(re));
	elseif (im <= -2.65e-295)
		tmp = t_2;
	elseif (im <= 1.45e-180)
		tmp = t_3;
	elseif (im <= 1.4e-82)
		tmp = t_0;
	elseif (im <= 1.7e-69)
		tmp = t_3;
	elseif (im <= 3.2e+38)
		tmp = t_0;
	else
		tmp = 0.5 * sqrt((im * 2.0));
	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[(im * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * (-t$95$1)), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * t$95$1), $MachinePrecision]}, If[LessEqual[im, -1.05e-15], N[(0.5 * N[Sqrt[N[(2.0 * N[((-im) + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -2.7e-94], N[(0.5 * N[Sqrt[N[(N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision] + N[(re * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -6e-100], N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -1.7e-187], t$95$2, If[LessEqual[im, -6e-253], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -2.65e-295], t$95$2, If[LessEqual[im, 1.45e-180], t$95$3, If[LessEqual[im, 1.4e-82], t$95$0, If[LessEqual[im, 1.7e-69], t$95$3, If[LessEqual[im, 3.2e+38], t$95$0, N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Target

Original	39.1
Target	34.2
Herbie	25.3

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

Derivation

1. Split input into 8 regimes

2. if im < -1.0499999999999999e-15

   1. Initial program 40.9

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

   2. Taylor expanded in im around -inf 15.5

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

   3. Simplified 15.5

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

      [Start] 15.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot im + re\right)} \]
      rational.json-simplify-2 [=>] 15.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im \cdot -1} + re\right)} \]
      rational.json-simplify-9 [=>] 15.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-im\right)} + re\right)} \]

   if -1.0499999999999999e-15 < im < -2.7000000000000001e-94

   1. Initial program 26.8

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

   2. Taylor expanded in re around inf 45.0

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

   3. Simplified 45.0

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

      [Start] 45.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re} + 2 \cdot re\right)} \]
      rational.json-simplify-2 [=>] 45.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{{im}^{2}}{re} + \color{blue}{re \cdot 2}\right)} \]

   4. Applied egg-rr 45.0

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

   5. Simplified 45.0

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{\frac{{im}^{2}}{re} + re \cdot 4}} \]
      Proof

      [Start] 45.0	\[ 0.5 \cdot \left(\sqrt{\left(re + re\right) + \left(\frac{{im}^{2}}{re} + \left(re + re\right)\right)} + 0\right) \]
      rational.json-simplify-4 [=>] 45.0	\[ 0.5 \cdot \color{blue}{\sqrt{\left(re + re\right) + \left(\frac{{im}^{2}}{re} + \left(re + re\right)\right)}} \]
      rational.json-simplify-41 [=>] 45.0	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re} + \left(\left(re + re\right) + \left(re + re\right)\right)}} \]
      rational.json-simplify-7 [<=] 45.0	\[ 0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + \left(\left(re + re\right) + \color{blue}{\frac{re + re}{1}}\right)} \]
      rational.json-simplify-30 [<=] 45.0	\[ 0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + \color{blue}{\left(1 + 1\right) \cdot \frac{re + re}{1}}} \]
      metadata-eval [=>] 45.0	\[ 0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + \color{blue}{2} \cdot \frac{re + re}{1}} \]
      rational.json-simplify-7 [=>] 45.0	\[ 0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + 2 \cdot \color{blue}{\left(re + re\right)}} \]
      rational.json-simplify-51 [<=] 45.0	\[ 0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + \color{blue}{\left(2 \cdot re + re \cdot 2\right)}} \]
      rational.json-simplify-2 [<=] 45.0	\[ 0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + \left(\color{blue}{re \cdot 2} + re \cdot 2\right)} \]
      rational.json-simplify-2 [=>] 45.0	\[ 0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + \left(re \cdot 2 + \color{blue}{2 \cdot re}\right)} \]
      rational.json-simplify-51 [=>] 45.0	\[ 0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + \color{blue}{re \cdot \left(2 + 2\right)}} \]
      metadata-eval [=>] 45.0	\[ 0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + re \cdot \color{blue}{4}} \]

   if -2.7000000000000001e-94 < im < -6.0000000000000001e-100

   1. Initial program 29.4

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

   2. Taylor expanded in im around -inf 39.7

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

   3. Simplified 39.7

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

      [Start] 39.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot im + re\right)} \]
      rational.json-simplify-2 [=>] 39.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im \cdot -1} + re\right)} \]
      rational.json-simplify-9 [=>] 39.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-im\right)} + re\right)} \]

   4. Taylor expanded in re around 0 64.0

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

   5. Simplified 40.6

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

      [Start] 64.0	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-1}\right) \cdot \sqrt{im}\right) \]
      exponential.json-simplify-20 [=>] 64.0	\[ 0.5 \cdot \left(\color{blue}{\sqrt{-1 \cdot 2}} \cdot \sqrt{im}\right) \]
      exponential.json-simplify-20 [=>] 40.6	\[ 0.5 \cdot \color{blue}{\sqrt{im \cdot \left(-1 \cdot 2\right)}} \]
      metadata-eval [=>] 40.6	\[ 0.5 \cdot \sqrt{im \cdot \color{blue}{-2}} \]

   if -6.0000000000000001e-100 < im < -1.7000000000000001e-187 or -6.0000000000000004e-253 < im < -2.6500000000000002e-295

   1. Initial program 40.1

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

   2. Taylor expanded in re around -inf 53.1

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

   3. Simplified 53.1

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

      [Start] 53.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      rational.json-simplify-2 [=>] 53.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]

   4. Taylor expanded in im around -inf 64.0

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

   5. Simplified 36.0

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

      [Start] 64.0	\[ 0.5 \cdot \left(-1 \cdot \left(\left(\sqrt{-1} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right)\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(\left(\left(\sqrt{-1} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right) \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(-\left(\sqrt{-1} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \left(-\color{blue}{\sqrt{\frac{1}{re}} \cdot \left(\sqrt{-1} \cdot im\right)}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \left(-\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(im \cdot \sqrt{-1}\right)}\right) \]
      rational.json-simplify-43 [=>] 64.0	\[ 0.5 \cdot \left(-\color{blue}{im \cdot \left(\sqrt{-1} \cdot \sqrt{\frac{1}{re}}\right)}\right) \]
      exponential.json-simplify-20 [=>] 36.0	\[ 0.5 \cdot \left(-im \cdot \color{blue}{\sqrt{\frac{1}{re} \cdot -1}}\right) \]
      rational.json-simplify-8 [<=] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\color{blue}{-\frac{1}{re}}}\right) \]
      rational.json-simplify-10 [=>] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\color{blue}{\frac{\frac{1}{re}}{-1}}}\right) \]
      rational.json-simplify-35 [=>] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\frac{\color{blue}{\frac{1 + 1}{re + re}}}{-1}}\right) \]
      metadata-eval [=>] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\frac{\frac{\color{blue}{2}}{re + re}}{-1}}\right) \]
      rational.json-simplify-46 [<=] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\color{blue}{\frac{2}{\left(re + re\right) \cdot -1}}}\right) \]
      rational.json-simplify-8 [<=] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\frac{2}{\color{blue}{-\left(re + re\right)}}}\right) \]
      rational.json-simplify-10 [=>] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\frac{2}{\color{blue}{\frac{re + re}{-1}}}}\right) \]
      rational.json-simplify-61 [=>] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\color{blue}{\frac{-1}{\frac{re + re}{2}}}}\right) \]
      metadata-eval [<=] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\frac{-1}{\frac{re + re}{\color{blue}{1 + 1}}}}\right) \]
      rational.json-simplify-35 [<=] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\frac{-1}{\color{blue}{\frac{re}{1}}}}\right) \]
      rational.json-simplify-7 [=>] 36.0	\[ 0.5 \cdot \left(-im \cdot \sqrt{\frac{-1}{\color{blue}{re}}}\right) \]

   if -1.7000000000000001e-187 < im < -6.0000000000000004e-253

   1. Initial program 47.5

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

   2. Taylor expanded in im around 0 37.4

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

   3. Simplified 36.9

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

      [Start] 37.4	\[ 0.5 \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right) \]
      exponential.json-simplify-24 [=>] 36.9	\[ 0.5 \cdot \left(\color{blue}{\sqrt{{2}^{2}}} \cdot \sqrt{re}\right) \]
      metadata-eval [=>] 36.9	\[ 0.5 \cdot \left(\sqrt{\color{blue}{4}} \cdot \sqrt{re}\right) \]
      metadata-eval [=>] 36.9	\[ 0.5 \cdot \left(\color{blue}{2} \cdot \sqrt{re}\right) \]

   if -2.6500000000000002e-295 < im < 1.4499999999999999e-180 or 1.40000000000000012e-82 < im < 1.70000000000000004e-69

   1. Initial program 42.0

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

   2. Taylor expanded in re around -inf 51.8

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

   3. Simplified 51.8

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

      [Start] 51.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      rational.json-simplify-2 [=>] 51.8	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]

   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 37.8

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

      [Start] 64.0	\[ 0.5 \cdot \left(\left(\sqrt{-1} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{-1} \cdot im\right)\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(im \cdot \sqrt{-1}\right)}\right) \]
      rational.json-simplify-43 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(im \cdot \left(\sqrt{-1} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]
      exponential.json-simplify-20 [=>] 37.8	\[ 0.5 \cdot \left(im \cdot \color{blue}{\sqrt{\frac{1}{re} \cdot -1}}\right) \]
      rational.json-simplify-8 [<=] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{-\frac{1}{re}}}\right) \]
      rational.json-simplify-10 [=>] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{\frac{\frac{1}{re}}{-1}}}\right) \]
      rational.json-simplify-35 [=>] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\frac{\color{blue}{\frac{1 + 1}{re + re}}}{-1}}\right) \]
      metadata-eval [=>] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\frac{\frac{\color{blue}{2}}{re + re}}{-1}}\right) \]
      rational.json-simplify-46 [<=] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{\frac{2}{\left(re + re\right) \cdot -1}}}\right) \]
      rational.json-simplify-8 [<=] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\frac{2}{\color{blue}{-\left(re + re\right)}}}\right) \]
      rational.json-simplify-10 [=>] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\frac{2}{\color{blue}{\frac{re + re}{-1}}}}\right) \]
      rational.json-simplify-61 [=>] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{\frac{-1}{\frac{re + re}{2}}}}\right) \]
      metadata-eval [<=] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{\frac{re + re}{\color{blue}{1 + 1}}}}\right) \]
      rational.json-simplify-35 [<=] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{\color{blue}{\frac{re}{1}}}}\right) \]
      rational.json-simplify-7 [=>] 37.8	\[ 0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{\color{blue}{re}}}\right) \]

   if 1.4499999999999999e-180 < im < 1.40000000000000012e-82 or 1.70000000000000004e-69 < im < 3.19999999999999985e38

   1. Initial program 28.6

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

   if 3.19999999999999985e38 < im

   1. Initial program 45.0

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

   2. Taylor expanded in re around 0 14.1

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

   3. Simplified 13.8

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

      [Start] 14.1	\[ 0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right) \]
      exponential.json-simplify-20 [=>] 13.8	\[ 0.5 \cdot \color{blue}{\sqrt{im \cdot 2}} \]

3. Recombined 8 regimes into one program.

4. Final simplification 25.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq -2.7 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + re \cdot 4}\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \left(-im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq -2.65 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \left(-im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-180}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.1
Cost	13704

\[\begin{array}{l} t_0 := im \cdot \sqrt{\frac{-1}{re}}\\ t_1 := 0.5 \cdot \left(-t_0\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq -2.8 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{re} + re \cdot 4}\\ \mathbf{elif}\;im \leq -8.6 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -2.05 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-251}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq -2.65 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-157}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Error	26.9
Cost	8300

\[\begin{array}{l} t_0 := im \cdot \sqrt{\frac{-1}{re}}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ t_2 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_3 := 0.5 \cdot \left(-t_0\right)\\ t_4 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_5 := 0.5 \cdot t_0\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+51}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;im \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -9 \cdot 10^{-94}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;im \leq -7.1 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -1.1 \cdot 10^{-188}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq -1.25 \cdot 10^{-252}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;im \leq -2.65 \cdot 10^{-295}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{-157}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 6 \cdot 10^{-69}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	27.6
Cost	7772

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ t_1 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_2 := 0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ t_3 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.32 \cdot 10^{+25}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq -1.4 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1 \cdot 10^{-270}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{-157}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	25.4
Cost	7772

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ t_1 := im \cdot \sqrt{\frac{-1}{re}}\\ t_2 := 0.5 \cdot \left(-t_1\right)\\ t_3 := 0.5 \cdot t_1\\ \mathbf{if}\;im \leq -5.9 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq -8.6 \cdot 10^{-188}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -2.5 \cdot 10^{-252}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq -2.65 \cdot 10^{-295}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{-156}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-69}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	27.3
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -3.2 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.12 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6
Error	27.6
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -9 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7
Error	30.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 8
Error	47.3
Cost	6720

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

Reproduce

herbie shell --seed 2023073 
(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)))))