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

Percentage Accurate: 41.0% → 88.1%

Time: 5.6s

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: 41.0% 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: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.9 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.9e-20)
   (sqrt (* 0.5 (- (hypot re im) re)))
   (/ (* 0.5 im) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.9e-20) {
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.9e-20) {
		tmp = Math.sqrt((0.5 * (Math.hypot(re, im) - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.9e-20:
		tmp = math.sqrt((0.5 * (math.hypot(re, im) - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.9e-20)
		tmp = sqrt(Float64(0.5 * Float64(hypot(re, im) - re)));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.9e-20)
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.9e-20], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.9000000000000002e-20

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

      2. hypot-udef 91.9%

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

      3. *-commutative 91.9%

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

      4. add-sqr-sqrt 91.1%

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

      5. sqrt-unprod 91.9%

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

      6. *-commutative 91.9%

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

      7. *-commutative 91.9%

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

      8. swap-sqr 91.9%

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

      9. add-sqr-sqrt 91.9%

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

      10. metadata-eval 91.9%

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

   3. Applied egg-rr 91.9%

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

      1. *-commutative 91.9%

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

      2. associate-*r* 91.9%

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

      3. metadata-eval 91.9%

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

   5. Simplified 91.9%

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

   if 4.9000000000000002e-20 < re

   1. Initial program 16.7%

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

   2. Taylor expanded in im around 0 89.1%

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

      1. *-commutative 89.1%

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

      2. associate-*l* 89.1%

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

      3. *-commutative 89.1%

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

   4. Simplified 89.1%

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

      1. sqrt-unprod 89.9%

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

      2. metadata-eval 89.9%

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

      3. metadata-eval 89.9%

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

      4. *-un-lft-identity 89.9%

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

      5. associate-*r* 89.9%

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

      6. sqrt-div 89.8%

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

      7. metadata-eval 89.8%

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

      8. un-div-inv 90.0%

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

   6. Applied egg-rr 90.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.9 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

Alternative 2: 76.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\frac{im}{\frac{re}{im}} \cdot -0.5 - re\right) - re\right)}\\ \mathbf{elif}\;re \leq 0.00075:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8e+19)
   (* 0.5 (sqrt (* 2.0 (- (- (* (/ im (/ re im)) -0.5) re) re))))
   (if (<= re 0.00075)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8e+19) {
		tmp = 0.5 * sqrt((2.0 * ((((im / (re / im)) * -0.5) - re) - re)));
	} else if (re <= 0.00075) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-8d+19)) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((((im / (re / im)) * (-0.5d0)) - re) - re)))
    else if (re <= 0.00075d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (0.5d0 * im) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8e+19) {
		tmp = 0.5 * Math.sqrt((2.0 * ((((im / (re / im)) * -0.5) - re) - re)));
	} else if (re <= 0.00075) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8e+19:
		tmp = 0.5 * math.sqrt((2.0 * ((((im / (re / im)) * -0.5) - re) - re)))
	elif re <= 0.00075:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8e+19)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(im / Float64(re / im)) * -0.5) - re) - re))));
	elseif (re <= 0.00075)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8e+19)
		tmp = 0.5 * sqrt((2.0 * ((((im / (re / im)) * -0.5) - re) - re)));
	elseif (re <= 0.00075)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8e+19], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(N[(N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] - re), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.00075], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8e19

   1. Initial program 35.6%

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

   2. Taylor expanded in re around -inf 67.1%

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

      1. neg-mul-1 67.1%

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

      2. +-commutative 67.1%

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

      3. unsub-neg 67.1%

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

      4. *-commutative 67.1%

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

      5. unpow2 67.1%

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

      6. associate-/l* 79.7%

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

   4. Simplified 79.7%

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

   if -8e19 < re < 7.5000000000000002e-4

   1. Initial program 53.6%

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

   2. Taylor expanded in re around 0 77.7%

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

   if 7.5000000000000002e-4 < re

   1. Initial program 16.7%

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

   2. Taylor expanded in im around 0 89.1%

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

      1. *-commutative 89.1%

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

      2. associate-*l* 89.1%

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

      3. *-commutative 89.1%

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

   4. Simplified 89.1%

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

      1. sqrt-unprod 89.9%

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

      2. metadata-eval 89.9%

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

      3. metadata-eval 89.9%

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

      4. *-un-lft-identity 89.9%

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

      5. associate-*r* 89.9%

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

      6. sqrt-div 89.8%

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

      7. metadata-eval 89.8%

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

      8. un-div-inv 90.0%

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

   6. Applied egg-rr 90.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\frac{im}{\frac{re}{im}} \cdot -0.5 - re\right) - re\right)}\\ \mathbf{elif}\;re \leq 0.00075:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7e+19)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 5.2e-13)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7e+19) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 5.2e-13) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-7d+19)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 5.2d-13) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (0.5d0 * im) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7e+19) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 5.2e-13) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7e+19:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 5.2e-13:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7e+19)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 5.2e-13)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7e+19)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 5.2e-13)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7e+19], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.2e-13], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7e19

   1. Initial program 35.6%

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

   2. Taylor expanded in re around -inf 79.3%

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

      1. *-commutative 79.3%

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

   4. Simplified 79.3%

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

   if -7e19 < re < 5.2000000000000001e-13

   1. Initial program 53.6%

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

   2. Taylor expanded in re around 0 77.7%

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

   if 5.2000000000000001e-13 < re

   1. Initial program 16.7%

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

   2. Taylor expanded in im around 0 89.1%

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

      1. *-commutative 89.1%

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

      2. associate-*l* 89.1%

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

      3. *-commutative 89.1%

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

   4. Simplified 89.1%

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

      1. sqrt-unprod 89.9%

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

      2. metadata-eval 89.9%

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

      3. metadata-eval 89.9%

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

      4. *-un-lft-identity 89.9%

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

      5. associate-*r* 89.9%

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

      6. sqrt-div 89.8%

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

      7. metadata-eval 89.8%

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

      8. un-div-inv 90.0%

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

   6. Applied egg-rr 90.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{-13}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 53.5% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e-310)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (/ (* 0.5 im) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5d-310)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else
        tmp = (0.5d0 * im) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e-310)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 53.4%

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

   2. Taylor expanded in re around -inf 47.1%

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

      1. *-commutative 47.1%

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

   4. Simplified 47.1%

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

   if -4.999999999999985e-310 < re

   1. Initial program 27.3%

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

   2. Taylor expanded in im around 0 60.1%

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

      1. *-commutative 60.1%

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

      2. associate-*l* 60.2%

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

      3. *-commutative 60.2%

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

   4. Simplified 60.2%

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

      1. sqrt-unprod 60.7%

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

      2. metadata-eval 60.7%

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

      3. metadata-eval 60.7%

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

      4. *-un-lft-identity 60.7%

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

      5. associate-*r* 60.7%

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

      6. sqrt-div 60.7%

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

      7. metadata-eval 60.7%

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

      8. un-div-inv 60.8%

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

   6. Applied egg-rr 60.8%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 26.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{\sqrt{re}}{im}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ 0.5 (/ (sqrt re) im)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

2. Taylor expanded in im around 0 27.5%

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

   1. *-commutative 27.5%

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

   2. associate-*l* 27.5%

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

   3. *-commutative 27.5%

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

4. Simplified 27.5%

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

   1. sqrt-unprod 27.7%

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

   2. metadata-eval 27.7%

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

   3. metadata-eval 27.7%

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

   4. *-un-lft-identity 27.7%

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

   5. *-un-lft-identity 27.7%

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

   6. metadata-eval 27.7%

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

   7. metadata-eval 27.7%

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

   8. sqrt-unprod 27.5%

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

   9. *-commutative 27.5%

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

   10. associate-*r* 27.6%

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

   11. associate-*l* 27.6%

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

   12. add-log-exp 8.4%

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

   13. *-un-lft-identity 8.4%

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

   14. log-prod 8.4%

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

   15. metadata-eval 8.4%

       \[\leadsto \color{blue}{0} + \log \left(e^{\left(0.5 \cdot im\right) \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)}\right) \]

   16. add-log-exp 27.6%

       \[\leadsto 0 + \color{blue}{\left(0.5 \cdot im\right) \cdot \left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{re}}\right)\right)} \]

6. Applied egg-rr 27.8%

   \[\leadsto \color{blue}{0 + \frac{0.5 \cdot im}{\sqrt{re}}} \]
7. Step-by-step derivation

   1. +-lft-identity 27.8%

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

   2. associate-/l* 27.6%

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

8. Simplified 27.6%

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

9. Final simplification 27.6%

   \[\leadsto \frac{0.5}{\frac{\sqrt{re}}{im}} \]

Alternative 6: 26.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

2. Taylor expanded in im around 0 27.5%

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

   1. *-commutative 27.5%

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

   2. associate-*l* 27.5%

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

   3. *-commutative 27.5%

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

4. Simplified 27.5%

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

   1. sqrt-unprod 27.7%

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

   2. metadata-eval 27.7%

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

   3. metadata-eval 27.7%

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

   4. *-un-lft-identity 27.7%

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

   5. associate-*r* 27.7%

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

   6. sqrt-div 27.7%

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

   7. metadata-eval 27.7%

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

   8. un-div-inv 27.8%

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

6. Applied egg-rr 27.8%

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

7. Final simplification 27.8%

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

Reproduce

herbie shell --seed 2023264 
(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)))))