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

Percentage Accurate: 41.1% → 87.7%

Time: 7.3s

Alternatives: 6

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: 41.1% 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: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.55 \cdot 10^{+66}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.55e+66)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.55e+66) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.55e+66) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.55e+66:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.55e+66)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.55e+66)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.55e+66], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.55000000000000009e66

   1. Initial program 53.6%

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

      1. sub-neg 53.6%

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

      2. sqr-neg 53.6%

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

      3. sub-neg 53.6%

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

      4. sqr-neg 53.6%

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

      5. hypot-def 93.7%

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

   3. Simplified 93.7%

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

   if 1.55000000000000009e66 < re

   1. Initial program 10.8%

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

   3. Taylor expanded in re around inf 44.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. div-inv 44.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{im}^{2} \cdot \frac{1}{re}}} \]

      2. sqrt-prod 67.6%

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

      3. unpow2 67.6%

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

      4. sqrt-prod 86.7%

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

      5. add-sqr-sqrt 87.2%

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

      6. *-commutative 87.2%

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

      7. inv-pow 87.2%

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

      8. sqrt-pow1 87.2%

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

      9. metadata-eval 87.2%

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

   5. Applied egg-rr 87.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.55 \cdot 10^{+66}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -1.45 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -3.4 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2.7 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

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 -1.45e+63)
     t_0
     (if (<= re -3.4e-60)
       t_1
       (if (<= re -2.7e-117)
         t_0
         (if (<= re 8.8e+21) t_1 (* 0.5 (* im (pow re -0.5)))))))))

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 <= -1.45e+63) {
		tmp = t_0;
	} else if (re <= -3.4e-60) {
		tmp = t_1;
	} else if (re <= -2.7e-117) {
		tmp = t_0;
	} else if (re <= 8.8e+21) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	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 <= (-1.45d+63)) then
        tmp = t_0
    else if (re <= (-3.4d-60)) then
        tmp = t_1
    else if (re <= (-2.7d-117)) then
        tmp = t_0
    else if (re <= 8.8d+21) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    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 <= -1.45e+63) {
		tmp = t_0;
	} else if (re <= -3.4e-60) {
		tmp = t_1;
	} else if (re <= -2.7e-117) {
		tmp = t_0;
	} else if (re <= 8.8e+21) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	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 <= -1.45e+63:
		tmp = t_0
	elif re <= -3.4e-60:
		tmp = t_1
	elif re <= -2.7e-117:
		tmp = t_0
	elif re <= 8.8e+21:
		tmp = t_1
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	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 <= -1.45e+63)
		tmp = t_0;
	elseif (re <= -3.4e-60)
		tmp = t_1;
	elseif (re <= -2.7e-117)
		tmp = t_0;
	elseif (re <= 8.8e+21)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	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 <= -1.45e+63)
		tmp = t_0;
	elseif (re <= -3.4e-60)
		tmp = t_1;
	elseif (re <= -2.7e-117)
		tmp = t_0;
	elseif (re <= 8.8e+21)
		tmp = t_1;
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	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, -1.45e+63], t$95$0, If[LessEqual[re, -3.4e-60], t$95$1, If[LessEqual[re, -2.7e-117], t$95$0, If[LessEqual[re, 8.8e+21], t$95$1, N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.45e63 or -3.40000000000000007e-60 < re < -2.70000000000000003e-117

   1. Initial program 42.9%

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

   3. Taylor expanded in re around -inf 83.1%

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

      1. *-commutative 83.1%

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

   5. Simplified 83.1%

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

   if -1.45e63 < re < -3.40000000000000007e-60 or -2.70000000000000003e-117 < re < 8.8e21

   1. Initial program 61.2%

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

   3. Taylor expanded in re around 0 83.0%

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

   if 8.8e21 < re

   1. Initial program 11.8%

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

   3. Taylor expanded in re around inf 44.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. div-inv 44.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{im}^{2} \cdot \frac{1}{re}}} \]

      2. sqrt-prod 64.2%

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

      3. unpow2 64.2%

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

      4. sqrt-prod 82.0%

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

      5. add-sqr-sqrt 82.4%

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

      6. *-commutative 82.4%

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

      7. inv-pow 82.4%

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

      8. sqrt-pow1 82.4%

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

      9. metadata-eval 82.4%

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

   5. Applied egg-rr 82.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.5} \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{+63}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -3.4 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq -2.7 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -3.5 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -2.2 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2.7 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (* 0.5 (sqrt (* 2.0 im)))))
   (if (<= re -3.5e+55)
     t_0
     (if (<= re -2.2e-61)
       t_1
       (if (<= re -2.7e-117)
         t_0
         (if (<= re 7.5e+54) t_1 (* 0.5 (* im (pow re -0.5)))))))))

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));
	double tmp;
	if (re <= -3.5e+55) {
		tmp = t_0;
	} else if (re <= -2.2e-61) {
		tmp = t_1;
	} else if (re <= -2.7e-117) {
		tmp = t_0;
	} else if (re <= 7.5e+54) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	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))
    if (re <= (-3.5d+55)) then
        tmp = t_0
    else if (re <= (-2.2d-61)) then
        tmp = t_1
    else if (re <= (-2.7d-117)) then
        tmp = t_0
    else if (re <= 7.5d+54) then
        tmp = t_1
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    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));
	double tmp;
	if (re <= -3.5e+55) {
		tmp = t_0;
	} else if (re <= -2.2e-61) {
		tmp = t_1;
	} else if (re <= -2.7e-117) {
		tmp = t_0;
	} else if (re <= 7.5e+54) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	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))
	tmp = 0
	if re <= -3.5e+55:
		tmp = t_0
	elif re <= -2.2e-61:
		tmp = t_1
	elif re <= -2.7e-117:
		tmp = t_0
	elif re <= 7.5e+54:
		tmp = t_1
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	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 * im)))
	tmp = 0.0
	if (re <= -3.5e+55)
		tmp = t_0;
	elseif (re <= -2.2e-61)
		tmp = t_1;
	elseif (re <= -2.7e-117)
		tmp = t_0;
	elseif (re <= 7.5e+54)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	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));
	tmp = 0.0;
	if (re <= -3.5e+55)
		tmp = t_0;
	elseif (re <= -2.2e-61)
		tmp = t_1;
	elseif (re <= -2.7e-117)
		tmp = t_0;
	elseif (re <= 7.5e+54)
		tmp = t_1;
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	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 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -3.5e+55], t$95$0, If[LessEqual[re, -2.2e-61], t$95$1, If[LessEqual[re, -2.7e-117], t$95$0, If[LessEqual[re, 7.5e+54], t$95$1, N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.5000000000000001e55 or -2.20000000000000009e-61 < re < -2.70000000000000003e-117

   1. Initial program 44.7%

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

   3. Taylor expanded in re around -inf 81.5%

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

      1. *-commutative 81.5%

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

   5. Simplified 81.5%

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

   if -3.5000000000000001e55 < re < -2.20000000000000009e-61 or -2.70000000000000003e-117 < re < 7.50000000000000042e54

   1. Initial program 58.9%

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

   3. Taylor expanded in re around 0 81.0%

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

      1. *-commutative 81.0%

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

   5. Simplified 81.0%

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

   if 7.50000000000000042e54 < re

   1. Initial program 10.9%

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

   3. Taylor expanded in re around inf 43.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. div-inv 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{im}^{2} \cdot \frac{1}{re}}} \]

      2. sqrt-prod 65.5%

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

      3. unpow2 65.5%

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

      4. sqrt-prod 85.4%

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

      5. add-sqr-sqrt 85.8%

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

      6. *-commutative 85.8%

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

      7. inv-pow 85.8%

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

      8. sqrt-pow1 85.8%

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

      9. metadata-eval 85.8%

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

   5. Applied egg-rr 85.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{-0.5} \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -2.2 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq -2.7 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+54}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{re \cdot -4}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{if}\;re \leq -1.32 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;re \leq -2.7 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* re -4.0)))) (t_1 (* 0.5 (sqrt (* 2.0 im)))))
   (if (<= re -1.32e+54)
     t_0
     (if (<= re -1.9e-61)
       t_1
       (if (<= re -2.7e-117)
         t_0
         (if (<= re 1.2e+55) 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));
	double tmp;
	if (re <= -1.32e+54) {
		tmp = t_0;
	} else if (re <= -1.9e-61) {
		tmp = t_1;
	} else if (re <= -2.7e-117) {
		tmp = t_0;
	} else if (re <= 1.2e+55) {
		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))
    if (re <= (-1.32d+54)) then
        tmp = t_0
    else if (re <= (-1.9d-61)) then
        tmp = t_1
    else if (re <= (-2.7d-117)) then
        tmp = t_0
    else if (re <= 1.2d+55) 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));
	double tmp;
	if (re <= -1.32e+54) {
		tmp = t_0;
	} else if (re <= -1.9e-61) {
		tmp = t_1;
	} else if (re <= -2.7e-117) {
		tmp = t_0;
	} else if (re <= 1.2e+55) {
		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))
	tmp = 0
	if re <= -1.32e+54:
		tmp = t_0
	elif re <= -1.9e-61:
		tmp = t_1
	elif re <= -2.7e-117:
		tmp = t_0
	elif re <= 1.2e+55:
		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 * im)))
	tmp = 0.0
	if (re <= -1.32e+54)
		tmp = t_0;
	elseif (re <= -1.9e-61)
		tmp = t_1;
	elseif (re <= -2.7e-117)
		tmp = t_0;
	elseif (re <= 1.2e+55)
		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));
	tmp = 0.0;
	if (re <= -1.32e+54)
		tmp = t_0;
	elseif (re <= -1.9e-61)
		tmp = t_1;
	elseif (re <= -2.7e-117)
		tmp = t_0;
	elseif (re <= 1.2e+55)
		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 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.32e+54], t$95$0, If[LessEqual[re, -1.9e-61], t$95$1, If[LessEqual[re, -2.7e-117], t$95$0, If[LessEqual[re, 1.2e+55], t$95$1, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.3200000000000001e54 or -1.8999999999999999e-61 < re < -2.70000000000000003e-117

   1. Initial program 44.7%

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

   3. Taylor expanded in re around -inf 81.5%

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

      1. *-commutative 81.5%

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

   5. Simplified 81.5%

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

   if -1.3200000000000001e54 < re < -1.8999999999999999e-61 or -2.70000000000000003e-117 < re < 1.2e55

   1. Initial program 58.9%

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

   3. Taylor expanded in re around 0 81.0%

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

      1. *-commutative 81.0%

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

   5. Simplified 81.0%

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

   if 1.2e55 < re

   1. Initial program 10.9%

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

   3. Taylor expanded in re around inf 43.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. div-inv 43.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{im}^{2} \cdot \frac{1}{re}}} \]

      2. sqrt-prod 65.5%

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

      3. unpow2 65.5%

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

      4. sqrt-prod 85.4%

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

      5. add-sqr-sqrt 85.8%

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

      6. expm1-log1p-u 84.7%

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

      7. expm1-udef 34.1%

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

      8. sqrt-div 34.1%

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

      9. metadata-eval 34.1%

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

      10. div-inv 34.1%

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

   5. Applied egg-rr 34.1%

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

      1. expm1-def 84.6%

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

      2. expm1-log1p 85.7%

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

   7. Simplified 85.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{+54}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq -1.9 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{elif}\;re \leq -2.7 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+46} \lor \neg \left(re \leq -7.5 \cdot 10^{-59}\right) \land re \leq -2.7 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -3.1e+46) (and (not (<= re -7.5e-59)) (<= re -2.7e-117)))
   (* 0.5 (sqrt (* re -4.0)))
   (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -3.1e+46) || (!(re <= -7.5e-59) && (re <= -2.7e-117))) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-3.1d+46)) .or. (.not. (re <= (-7.5d-59))) .and. (re <= (-2.7d-117))) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -3.1e+46) || (!(re <= -7.5e-59) && (re <= -2.7e-117))) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -3.1e+46) or (not (re <= -7.5e-59) and (re <= -2.7e-117)):
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -3.1e+46) || (!(re <= -7.5e-59) && (re <= -2.7e-117)))
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -3.1e+46) || (~((re <= -7.5e-59)) && (re <= -2.7e-117)))
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -3.1e+46], And[N[Not[LessEqual[re, -7.5e-59]], $MachinePrecision], LessEqual[re, -2.7e-117]]], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.09999999999999975e46 or -7.50000000000000019e-59 < re < -2.70000000000000003e-117

   1. Initial program 44.7%

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

   3. Taylor expanded in re around -inf 81.5%

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

      1. *-commutative 81.5%

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

   5. Simplified 81.5%

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

   if -3.09999999999999975e46 < re < -7.50000000000000019e-59 or -2.70000000000000003e-117 < re

   1. Initial program 45.7%

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

   3. Taylor expanded in re around 0 63.8%

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

      1. *-commutative 63.8%

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

   5. Simplified 63.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+46} \lor \neg \left(re \leq -7.5 \cdot 10^{-59}\right) \land re \leq -2.7 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.4%

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

3. Taylor expanded in re around 0 52.4%

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

   1. *-commutative 52.4%

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

5. Simplified 52.4%

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

6. Final simplification 52.4%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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)))))