math.sqrt on complex, real part

Average Error: 39.1 → 23.3

Time: 12.0s

Precision: binary64

Cost: 14820

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ t_2 := \sqrt{-\frac{1}{re}}\\ t_3 := im \cdot t_2\\ t_4 := 0.5 \cdot t_3\\ \mathbf{if}\;im \leq -1.3 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.1 \cdot 10^{-268}:\\ \;\;\;\;0.5 \cdot \left(-t_3\right)\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{-283}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \left(-im \cdot \left(\frac{1}{t_2} \cdot \sqrt{\frac{1}{re} \cdot \frac{1}{re}}\right)\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-293}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{-242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{-172}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \end{array} \]

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 (* 2.0 (sqrt re))))
        (t_1 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))
        (t_2 (sqrt (- (/ 1.0 re))))
        (t_3 (* im t_2))
        (t_4 (* 0.5 t_3)))
   (if (<= im -1.3e+48)
     (* 0.5 (sqrt (* im -2.0)))
     (if (<= im -1.7e-161)
       t_1
       (if (<= im -1.1e-268)
         (* 0.5 (- t_3))
         (if (<= im -6.4e-283)
           t_0
           (if (<= im -6e-303)
             (*
              0.5
              (- (* im (* (/ 1.0 t_2) (sqrt (* (/ 1.0 re) (/ 1.0 re)))))))
             (if (<= im 5.5e-293)
               t_4
               (if (<= im 7.6e-242)
                 t_0
                 (if (<= im 5.2e-172)
                   t_4
                   (if (<= im 4e+70)
                     t_1
                     (* 0.5 (* (sqrt 2.0) (sqrt 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 * (2.0 * sqrt(re));
	double t_1 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	double t_2 = sqrt(-(1.0 / re));
	double t_3 = im * t_2;
	double t_4 = 0.5 * t_3;
	double tmp;
	if (im <= -1.3e+48) {
		tmp = 0.5 * sqrt((im * -2.0));
	} else if (im <= -1.7e-161) {
		tmp = t_1;
	} else if (im <= -1.1e-268) {
		tmp = 0.5 * -t_3;
	} else if (im <= -6.4e-283) {
		tmp = t_0;
	} else if (im <= -6e-303) {
		tmp = 0.5 * -(im * ((1.0 / t_2) * sqrt(((1.0 / re) * (1.0 / re)))));
	} else if (im <= 5.5e-293) {
		tmp = t_4;
	} else if (im <= 7.6e-242) {
		tmp = t_0;
	} else if (im <= 5.2e-172) {
		tmp = t_4;
	} else if (im <= 4e+70) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (sqrt(2.0) * sqrt(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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 0.5d0 * (2.0d0 * sqrt(re))
    t_1 = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
    t_2 = sqrt(-(1.0d0 / re))
    t_3 = im * t_2
    t_4 = 0.5d0 * t_3
    if (im <= (-1.3d+48)) then
        tmp = 0.5d0 * sqrt((im * (-2.0d0)))
    else if (im <= (-1.7d-161)) then
        tmp = t_1
    else if (im <= (-1.1d-268)) then
        tmp = 0.5d0 * -t_3
    else if (im <= (-6.4d-283)) then
        tmp = t_0
    else if (im <= (-6d-303)) then
        tmp = 0.5d0 * -(im * ((1.0d0 / t_2) * sqrt(((1.0d0 / re) * (1.0d0 / re)))))
    else if (im <= 5.5d-293) then
        tmp = t_4
    else if (im <= 7.6d-242) then
        tmp = t_0
    else if (im <= 5.2d-172) then
        tmp = t_4
    else if (im <= 4d+70) then
        tmp = t_1
    else
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(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 * (2.0 * Math.sqrt(re));
	double t_1 = 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
	double t_2 = Math.sqrt(-(1.0 / re));
	double t_3 = im * t_2;
	double t_4 = 0.5 * t_3;
	double tmp;
	if (im <= -1.3e+48) {
		tmp = 0.5 * Math.sqrt((im * -2.0));
	} else if (im <= -1.7e-161) {
		tmp = t_1;
	} else if (im <= -1.1e-268) {
		tmp = 0.5 * -t_3;
	} else if (im <= -6.4e-283) {
		tmp = t_0;
	} else if (im <= -6e-303) {
		tmp = 0.5 * -(im * ((1.0 / t_2) * Math.sqrt(((1.0 / re) * (1.0 / re)))));
	} else if (im <= 5.5e-293) {
		tmp = t_4;
	} else if (im <= 7.6e-242) {
		tmp = t_0;
	} else if (im <= 5.2e-172) {
		tmp = t_4;
	} else if (im <= 4e+70) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(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 * (2.0 * math.sqrt(re))
	t_1 = 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))
	t_2 = math.sqrt(-(1.0 / re))
	t_3 = im * t_2
	t_4 = 0.5 * t_3
	tmp = 0
	if im <= -1.3e+48:
		tmp = 0.5 * math.sqrt((im * -2.0))
	elif im <= -1.7e-161:
		tmp = t_1
	elif im <= -1.1e-268:
		tmp = 0.5 * -t_3
	elif im <= -6.4e-283:
		tmp = t_0
	elif im <= -6e-303:
		tmp = 0.5 * -(im * ((1.0 / t_2) * math.sqrt(((1.0 / re) * (1.0 / re)))))
	elif im <= 5.5e-293:
		tmp = t_4
	elif im <= 7.6e-242:
		tmp = t_0
	elif im <= 5.2e-172:
		tmp = t_4
	elif im <= 4e+70:
		tmp = t_1
	else:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(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 * Float64(2.0 * sqrt(re)))
	t_1 = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
	t_2 = sqrt(Float64(-Float64(1.0 / re)))
	t_3 = Float64(im * t_2)
	t_4 = Float64(0.5 * t_3)
	tmp = 0.0
	if (im <= -1.3e+48)
		tmp = Float64(0.5 * sqrt(Float64(im * -2.0)));
	elseif (im <= -1.7e-161)
		tmp = t_1;
	elseif (im <= -1.1e-268)
		tmp = Float64(0.5 * Float64(-t_3));
	elseif (im <= -6.4e-283)
		tmp = t_0;
	elseif (im <= -6e-303)
		tmp = Float64(0.5 * Float64(-Float64(im * Float64(Float64(1.0 / t_2) * sqrt(Float64(Float64(1.0 / re) * Float64(1.0 / re)))))));
	elseif (im <= 5.5e-293)
		tmp = t_4;
	elseif (im <= 7.6e-242)
		tmp = t_0;
	elseif (im <= 5.2e-172)
		tmp = t_4;
	elseif (im <= 4e+70)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(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 * (2.0 * sqrt(re));
	t_1 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
	t_2 = sqrt(-(1.0 / re));
	t_3 = im * t_2;
	t_4 = 0.5 * t_3;
	tmp = 0.0;
	if (im <= -1.3e+48)
		tmp = 0.5 * sqrt((im * -2.0));
	elseif (im <= -1.7e-161)
		tmp = t_1;
	elseif (im <= -1.1e-268)
		tmp = 0.5 * -t_3;
	elseif (im <= -6.4e-283)
		tmp = t_0;
	elseif (im <= -6e-303)
		tmp = 0.5 * -(im * ((1.0 / t_2) * sqrt(((1.0 / re) * (1.0 / re)))));
	elseif (im <= 5.5e-293)
		tmp = t_4;
	elseif (im <= 7.6e-242)
		tmp = t_0;
	elseif (im <= 5.2e-172)
		tmp = t_4;
	elseif (im <= 4e+70)
		tmp = t_1;
	else
		tmp = 0.5 * (sqrt(2.0) * sqrt(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[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$2 = N[Sqrt[(-N[(1.0 / re), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$3 = N[(im * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * t$95$3), $MachinePrecision]}, If[LessEqual[im, -1.3e+48], N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -1.7e-161], t$95$1, If[LessEqual[im, -1.1e-268], N[(0.5 * (-t$95$3)), $MachinePrecision], If[LessEqual[im, -6.4e-283], t$95$0, If[LessEqual[im, -6e-303], N[(0.5 * (-N[(im * N[(N[(1.0 / t$95$2), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / re), $MachinePrecision] * N[(1.0 / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 5.5e-293], t$95$4, If[LessEqual[im, 7.6e-242], t$95$0, If[LessEqual[im, 5.2e-172], t$95$4, If[LessEqual[im, 4e+70], t$95$1, N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Target

Original	39.1
Target	34.1
Herbie	23.3

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

Derivation

1. Split input into 7 regimes

2. if im < -1.29999999999999998e48

   1. Initial program 46.3

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

   2. Taylor expanded in im around -inf 12.8

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

   3. Simplified 12.8

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

      [Start] 12.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot im + re\right)} \]
      rational_best.json-simplify-2 [=>] 12.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im \cdot -1} + re\right)} \]
      rational_best.json-simplify-12 [=>] 12.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-im\right)} + re\right)} \]

   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 13.4

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

      [Start] 64.0	\[ 0.5 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-1}\right) \cdot \sqrt{im}\right) \]
      exponential.json-simplify-19 [=>] 64.0	\[ 0.5 \cdot \left(\color{blue}{\sqrt{-1 \cdot 2}} \cdot \sqrt{im}\right) \]
      metadata-eval [=>] 64.0	\[ 0.5 \cdot \left(\sqrt{\color{blue}{-2}} \cdot \sqrt{im}\right) \]
      exponential.json-simplify-19 [=>] 13.4	\[ 0.5 \cdot \color{blue}{\sqrt{im \cdot -2}} \]

   if -1.29999999999999998e48 < im < -1.69999999999999991e-161 or 5.1999999999999996e-172 < im < 4.00000000000000029e70

   1. Initial program 26.7

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

   if -1.69999999999999991e-161 < im < -1.10000000000000002e-268

   1. Initial program 44.9

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

   2. Taylor expanded in re around -inf 55.2

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

   3. Simplified 55.2

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

      [Start] 55.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      rational_best.json-simplify-2 [=>] 55.2	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]

   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 36.4

      \[\leadsto 0.5 \cdot \color{blue}{\left(-im \cdot \sqrt{-\frac{1}{re}}\right)} \]
      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_best.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_best.json-simplify-12 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(-\left(\sqrt{-1} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
      rational_best.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \left(-\color{blue}{\sqrt{\frac{1}{re}} \cdot \left(\sqrt{-1} \cdot im\right)}\right) \]
      rational_best.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \left(-\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(im \cdot \sqrt{-1}\right)}\right) \]
      rational_best.json-simplify-44 [=>] 64.0	\[ 0.5 \cdot \left(-\color{blue}{im \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{-1}\right)}\right) \]
      exponential.json-simplify-19 [=>] 36.4	\[ 0.5 \cdot \left(-im \cdot \color{blue}{\sqrt{-1 \cdot \frac{1}{re}}}\right) \]
      rational_best.json-simplify-2 [=>] 36.4	\[ 0.5 \cdot \left(-im \cdot \sqrt{\color{blue}{\frac{1}{re} \cdot -1}}\right) \]
      rational_best.json-simplify-12 [=>] 36.4	\[ 0.5 \cdot \left(-im \cdot \sqrt{\color{blue}{-\frac{1}{re}}}\right) \]

   if -1.10000000000000002e-268 < im < -6.40000000000000023e-283 or 5.50000000000000028e-293 < im < 7.6000000000000004e-242

   1. Initial program 43.2

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

   2. Taylor expanded in im around 0 36.0

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

   3. Simplified 35.5

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

      [Start] 36.0	\[ 0.5 \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right) \]
      exponential.json-simplify-24 [=>] 35.5	\[ 0.5 \cdot \left(\color{blue}{\sqrt{{2}^{2}}} \cdot \sqrt{re}\right) \]
      metadata-eval [=>] 35.5	\[ 0.5 \cdot \left(\sqrt{\color{blue}{4}} \cdot \sqrt{re}\right) \]
      metadata-eval [=>] 35.5	\[ 0.5 \cdot \left(\color{blue}{2} \cdot \sqrt{re}\right) \]

   if -6.40000000000000023e-283 < im < -6.00000000000000055e-303

   1. Initial program 39.0

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

   2. Taylor expanded in re around -inf 47.4

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

   3. Simplified 47.4

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

      [Start] 47.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      rational_best.json-simplify-2 [=>] 47.4	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]

   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 32.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(-im \cdot \sqrt{-\frac{1}{re}}\right)} \]
      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_best.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_best.json-simplify-12 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(-\left(\sqrt{-1} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
      rational_best.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \left(-\color{blue}{\sqrt{\frac{1}{re}} \cdot \left(\sqrt{-1} \cdot im\right)}\right) \]
      rational_best.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \left(-\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(im \cdot \sqrt{-1}\right)}\right) \]
      rational_best.json-simplify-44 [=>] 64.0	\[ 0.5 \cdot \left(-\color{blue}{im \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{-1}\right)}\right) \]
      exponential.json-simplify-19 [=>] 32.7	\[ 0.5 \cdot \left(-im \cdot \color{blue}{\sqrt{-1 \cdot \frac{1}{re}}}\right) \]
      rational_best.json-simplify-2 [=>] 32.7	\[ 0.5 \cdot \left(-im \cdot \sqrt{\color{blue}{\frac{1}{re} \cdot -1}}\right) \]
      rational_best.json-simplify-12 [=>] 32.7	\[ 0.5 \cdot \left(-im \cdot \sqrt{\color{blue}{-\frac{1}{re}}}\right) \]

   6. Applied egg-rr 39.0

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

   if -6.00000000000000055e-303 < im < 5.50000000000000028e-293 or 7.6000000000000004e-242 < im < 5.1999999999999996e-172

   1. Initial program 44.0

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

   2. Taylor expanded in re around -inf 53.6

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

   3. Simplified 53.6

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

      [Start] 53.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      rational_best.json-simplify-2 [=>] 53.6	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]

   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 37.5

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

      [Start] 64.0	\[ 0.5 \cdot \left(\left(\sqrt{-1} \cdot im\right) \cdot \sqrt{\frac{1}{re}}\right) \]
      rational_best.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{-1} \cdot im\right)\right)} \]
      rational_best.json-simplify-2 [=>] 64.0	\[ 0.5 \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(im \cdot \sqrt{-1}\right)}\right) \]
      rational_best.json-simplify-44 [=>] 64.0	\[ 0.5 \cdot \color{blue}{\left(im \cdot \left(\sqrt{\frac{1}{re}} \cdot \sqrt{-1}\right)\right)} \]
      exponential.json-simplify-19 [=>] 37.5	\[ 0.5 \cdot \left(im \cdot \color{blue}{\sqrt{-1 \cdot \frac{1}{re}}}\right) \]
      rational_best.json-simplify-2 [=>] 37.5	\[ 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{\frac{1}{re} \cdot -1}}\right) \]
      rational_best.json-simplify-12 [=>] 37.5	\[ 0.5 \cdot \left(im \cdot \sqrt{\color{blue}{-\frac{1}{re}}}\right) \]

   if 4.00000000000000029e70 < im

   1. Initial program 48.6

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

   2. Taylor expanded in re around 0 12.4

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{im}\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 23.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{elif}\;im \leq -1.1 \cdot 10^{-268}:\\ \;\;\;\;0.5 \cdot \left(-im \cdot \sqrt{-\frac{1}{re}}\right)\\ \mathbf{elif}\;im \leq -6.4 \cdot 10^{-283}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-303}:\\ \;\;\;\;0.5 \cdot \left(-im \cdot \left(\frac{1}{\sqrt{-\frac{1}{re}}} \cdot \sqrt{\frac{1}{re} \cdot \frac{1}{re}}\right)\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-293}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{-\frac{1}{re}}\right)\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{-242}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{-172}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{-\frac{1}{re}}\right)\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	23.4
Cost	14424

\[\begin{array}{l} t_0 := im \cdot \sqrt{-\frac{1}{re}}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{if}\;im \leq -2.8 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -2.1 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{-292}:\\ \;\;\;\;0.5 \cdot \left(-t_0\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{-240}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \end{array} \]

Alternative 2
Error	25.9
Cost	7640

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \left(im \cdot \sqrt{-\frac{1}{re}}\right)\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq -3.9 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.46 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Error	26.0
Cost	7508

\[\begin{array}{l} t_0 := im \cdot \sqrt{-\frac{1}{re}}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.62 \cdot 10^{-160}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{-292}:\\ \;\;\;\;0.5 \cdot \left(-t_0\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-193}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Error	26.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Error	25.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -2.4 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-im\right) + re\right)}\\ \mathbf{elif}\;im \leq 1.46 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6
Error	26.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -4.7 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.46 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7
Error	30.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 8
Error	47.5
Cost	6720

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

Reproduce

herbie shell --seed 2023090 
(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)))))