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

Percentage Accurate: 40.8% → 90.4%

Time: 11.9s

Alternatives: 7

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: 40.8% 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: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* 0.5 (* im (sqrt (/ 1.0 re))))
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))

C

double code(double re, double im) {
	double tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sqrt(((re * re) + (im * im))) - re) <= 0.0:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0)
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 4.2%

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

   3. Taylor expanded in re around inf 89.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   4. 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* 89.5%

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

   5. Simplified 89.5%

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

      1. associate-*r* 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. sqrt-unprod 90.3%

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

      3. metadata-eval 90.3%

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

      4. metadata-eval 90.3%

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

      5. add-log-exp 5.1%

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

      6. *-un-lft-identity 5.1%

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

      7. *-un-lft-identity 5.1%

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

      8. log-prod 5.1%

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

      9. metadata-eval 5.1%

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

      10. add-log-exp 90.3%

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

   7. Applied egg-rr 90.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(0 + im\right)} \cdot \sqrt{\frac{1}{re}}\right) \]
   8. Step-by-step derivation

      1. +-lft-identity 90.3%

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

   9. Simplified 90.3%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. sqr-neg 51.9%

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

      3. sub-neg 51.9%

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

      4. sqr-neg 51.9%

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

      5. hypot-define 87.4%

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

   3. Simplified 87.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{-123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2 + re \cdot \left(\frac{re}{im} - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1e-97)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.8e-123)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 2.45e-5)
       (* 0.5 (* im (sqrt (/ 1.0 re))))
       (if (<= re 2.6e+29)
         (* 0.5 (sqrt (+ (* im 2.0) (* re (- (/ re im) 2.0)))))
         (* 0.5 (/ im (sqrt re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e-97) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.8e-123) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 2.45e-5) {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	} else if (re <= 2.6e+29) {
		tmp = 0.5 * sqrt(((im * 2.0) + (re * ((re / im) - 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 <= (-1d-97)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.8d-123) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 2.45d-5) then
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    else if (re <= 2.6d+29) then
        tmp = 0.5d0 * sqrt(((im * 2.0d0) + (re * ((re / im) - 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 <= -1e-97) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.8e-123) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 2.45e-5) {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	} else if (re <= 2.6e+29) {
		tmp = 0.5 * Math.sqrt(((im * 2.0) + (re * ((re / im) - 2.0))));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1e-97:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.8e-123:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 2.45e-5:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	elif re <= 2.6e+29:
		tmp = 0.5 * math.sqrt(((im * 2.0) + (re * ((re / im) - 2.0))))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e-97)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.8e-123)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 2.45e-5)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	elseif (re <= 2.6e+29)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * 2.0) + Float64(re * Float64(Float64(re / im) - 2.0)))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1e-97)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.8e-123)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 2.45e-5)
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	elseif (re <= 2.6e+29)
		tmp = 0.5 * sqrt(((im * 2.0) + (re * ((re / im) - 2.0))));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e-97], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.8e-123], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.45e-5], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e+29], N[(0.5 * N[Sqrt[N[(N[(im * 2.0), $MachinePrecision] + N[(re * N[(N[(re / im), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -1.00000000000000004e-97

   1. Initial program 54.0%

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

   3. Taylor expanded in re around -inf 74.2%

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

      1. *-commutative 74.2%

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

   5. Simplified 74.2%

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

   if -1.00000000000000004e-97 < re < 4.8e-123

   1. Initial program 62.2%

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

   3. Taylor expanded in re around 0 82.7%

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

   if 4.8e-123 < re < 2.45e-5

   1. Initial program 31.2%

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

   3. Taylor expanded in re around inf 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l* 65.2%

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

   5. Simplified 65.2%

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

      1. associate-*r* 64.9%

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

      2. sqrt-unprod 65.9%

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

      3. metadata-eval 65.9%

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

      4. metadata-eval 65.9%

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

      5. add-log-exp 4.6%

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

      6. *-un-lft-identity 4.6%

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

      7. *-un-lft-identity 4.6%

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

      8. log-prod 4.6%

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

      9. metadata-eval 4.6%

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

      10. add-log-exp 65.9%

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

   7. Applied egg-rr 65.9%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(0 + im\right)} \cdot \sqrt{\frac{1}{re}}\right) \]
   8. Step-by-step derivation

      1. +-lft-identity 65.9%

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

   9. Simplified 65.9%

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

   if 2.45e-5 < re < 2.6e29

   1. Initial program 57.7%

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

   3. Taylor expanded in re around 0 89.3%

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

   if 2.6e29 < re

   1. Initial program 9.4%

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

   3. Taylor expanded in re around inf 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-*l* 79.1%

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

   5. Simplified 79.1%

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

      1. add-log-exp 11.0%

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

      2. *-un-lft-identity 11.0%

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

      3. log-prod 11.0%

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

      4. metadata-eval 11.0%

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

      5. add-log-exp 79.1%

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

      6. sqrt-div 79.0%

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

      7. metadata-eval 79.0%

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

      8. un-div-inv 78.9%

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

      9. associate-*r* 78.6%

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

      10. sqrt-unprod 79.7%

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

      11. metadata-eval 79.7%

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

      12. metadata-eval 79.7%

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

      13. *-un-lft-identity 79.7%

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

   7. Applied egg-rr 79.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   8. Step-by-step derivation

      1. +-lft-identity 79.7%

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

   9. Simplified 79.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{-123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2 + re \cdot \left(\frac{re}{im} - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.6 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.2 \cdot 10^{-129} \lor \neg \left(re \leq 2.45 \cdot 10^{-5}\right) \land re \leq 9.5 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.6e-98)
   (* 0.5 (sqrt (* re -4.0)))
   (if (or (<= re 9.2e-129) (and (not (<= re 2.45e-5)) (<= re 9.5e+28)))
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.6e-98) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if ((re <= 9.2e-129) || (!(re <= 2.45e-5) && (re <= 9.5e+28))) {
		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 <= (-8.6d-98)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if ((re <= 9.2d-129) .or. (.not. (re <= 2.45d-5)) .and. (re <= 9.5d+28)) 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 <= -8.6e-98) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if ((re <= 9.2e-129) || (!(re <= 2.45e-5) && (re <= 9.5e+28))) {
		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 <= -8.6e-98:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif (re <= 9.2e-129) or (not (re <= 2.45e-5) and (re <= 9.5e+28)):
		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 <= -8.6e-98)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif ((re <= 9.2e-129) || (!(re <= 2.45e-5) && (re <= 9.5e+28)))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.6e-98)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif ((re <= 9.2e-129) || (~((re <= 2.45e-5)) && (re <= 9.5e+28)))
		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, -8.6e-98], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 9.2e-129], And[N[Not[LessEqual[re, 2.45e-5]], $MachinePrecision], LessEqual[re, 9.5e+28]]], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.59999999999999977e-98

   1. Initial program 54.0%

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

   3. Taylor expanded in re around -inf 74.2%

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

      1. *-commutative 74.2%

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

   5. Simplified 74.2%

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

   if -8.59999999999999977e-98 < re < 9.1999999999999998e-129 or 2.45e-5 < re < 9.49999999999999927e28

   1. Initial program 61.8%

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

   3. Taylor expanded in re around 0 83.3%

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

   if 9.1999999999999998e-129 < re < 2.45e-5 or 9.49999999999999927e28 < re

   1. Initial program 15.1%

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

   3. Taylor expanded in re around inf 75.1%

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

      1. *-commutative 75.1%

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

      2. associate-*l* 75.5%

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

   5. Simplified 75.5%

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

      1. add-log-exp 9.4%

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

      2. *-un-lft-identity 9.4%

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

      3. log-prod 9.4%

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

      4. metadata-eval 9.4%

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

      5. add-log-exp 75.5%

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

      6. sqrt-div 75.4%

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

      7. metadata-eval 75.4%

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

      8. un-div-inv 75.3%

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

      9. associate-*r* 75.0%

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

      10. sqrt-unprod 76.1%

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

      11. metadata-eval 76.1%

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

      12. metadata-eval 76.1%

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

      13. *-un-lft-identity 76.1%

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

   7. Applied egg-rr 76.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   8. Step-by-step derivation

      1. +-lft-identity 76.1%

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

   9. Simplified 76.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.6 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 9.2 \cdot 10^{-129} \lor \neg \left(re \leq 2.45 \cdot 10^{-5}\right) \land re \leq 9.5 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{if}\;re \leq -1 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{-124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 2.75 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 7.4 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (- im re))))))
   (if (<= re -1e-97)
     (* 0.5 (sqrt (* re -4.0)))
     (if (<= re 7.2e-124)
       t_0
       (if (<= re 2.75e-5)
         (* 0.5 (* im (sqrt (/ 1.0 re))))
         (if (<= re 7.4e+28) t_0 (* 0.5 (/ im (sqrt re)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((2.0 * (im - re)));
	double tmp;
	if (re <= -1e-97) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 7.2e-124) {
		tmp = t_0;
	} else if (re <= 2.75e-5) {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	} else if (re <= 7.4e+28) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((2.0d0 * (im - re)))
    if (re <= (-1d-97)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 7.2d-124) then
        tmp = t_0
    else if (re <= 2.75d-5) then
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    else if (re <= 7.4d+28) then
        tmp = t_0
    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((2.0 * (im - re)));
	double tmp;
	if (re <= -1e-97) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 7.2e-124) {
		tmp = t_0;
	} else if (re <= 2.75e-5) {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	} else if (re <= 7.4e+28) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (im - re)))
	tmp = 0
	if re <= -1e-97:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 7.2e-124:
		tmp = t_0
	elif re <= 2.75e-5:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	elif re <= 7.4e+28:
		tmp = t_0
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))))
	tmp = 0.0
	if (re <= -1e-97)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 7.2e-124)
		tmp = t_0;
	elseif (re <= 2.75e-5)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	elseif (re <= 7.4e+28)
		tmp = t_0;
	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((2.0 * (im - re)));
	tmp = 0.0;
	if (re <= -1e-97)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 7.2e-124)
		tmp = t_0;
	elseif (re <= 2.75e-5)
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	elseif (re <= 7.4e+28)
		tmp = t_0;
	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[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1e-97], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.2e-124], t$95$0, If[LessEqual[re, 2.75e-5], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.4e+28], t$95$0, N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.00000000000000004e-97

   1. Initial program 54.0%

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

   3. Taylor expanded in re around -inf 74.2%

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

      1. *-commutative 74.2%

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

   5. Simplified 74.2%

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

   if -1.00000000000000004e-97 < re < 7.20000000000000019e-124 or 2.7500000000000001e-5 < re < 7.3999999999999998e28

   1. Initial program 61.8%

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

   3. Taylor expanded in re around 0 83.3%

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

   if 7.20000000000000019e-124 < re < 2.7500000000000001e-5

   1. Initial program 31.2%

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

   3. Taylor expanded in re around inf 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l* 65.2%

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

   5. Simplified 65.2%

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

      1. associate-*r* 64.9%

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

      2. sqrt-unprod 65.9%

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

      3. metadata-eval 65.9%

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

      4. metadata-eval 65.9%

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

      5. add-log-exp 4.6%

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

      6. *-un-lft-identity 4.6%

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

      7. *-un-lft-identity 4.6%

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

      8. log-prod 4.6%

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

      9. metadata-eval 4.6%

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

      10. add-log-exp 65.9%

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

   7. Applied egg-rr 65.9%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(0 + im\right)} \cdot \sqrt{\frac{1}{re}}\right) \]
   8. Step-by-step derivation

      1. +-lft-identity 65.9%

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

   9. Simplified 65.9%

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

   if 7.3999999999999998e28 < re

   1. Initial program 9.4%

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

   3. Taylor expanded in re around inf 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-*l* 79.1%

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

   5. Simplified 79.1%

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

      1. add-log-exp 11.0%

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

      2. *-un-lft-identity 11.0%

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

      3. log-prod 11.0%

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

      4. metadata-eval 11.0%

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

      5. add-log-exp 79.1%

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

      6. sqrt-div 79.0%

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

      7. metadata-eval 79.0%

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

      8. un-div-inv 78.9%

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

      9. associate-*r* 78.6%

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

      10. sqrt-unprod 79.7%

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

      11. metadata-eval 79.7%

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

      12. metadata-eval 79.7%

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

      13. *-un-lft-identity 79.7%

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

   7. Applied egg-rr 79.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   8. Step-by-step derivation

      1. +-lft-identity 79.7%

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

   9. Simplified 79.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{-124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.75 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;re \leq 7.4 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{-109} \lor \neg \left(re \leq 2.45 \cdot 10^{-5}\right) \land re \leq 1.22 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.5e-98)
   (* 0.5 (sqrt (* re -4.0)))
   (if (or (<= re 4.1e-109) (and (not (<= re 2.45e-5)) (<= re 1.22e+29)))
     (* 0.5 (sqrt (* im 2.0)))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.5e-98) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if ((re <= 4.1e-109) || (!(re <= 2.45e-5) && (re <= 1.22e+29))) {
		tmp = 0.5 * sqrt((im * 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 <= (-5.5d-98)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if ((re <= 4.1d-109) .or. (.not. (re <= 2.45d-5)) .and. (re <= 1.22d+29)) then
        tmp = 0.5d0 * sqrt((im * 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 <= -5.5e-98) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if ((re <= 4.1e-109) || (!(re <= 2.45e-5) && (re <= 1.22e+29))) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.5e-98:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif (re <= 4.1e-109) or (not (re <= 2.45e-5) and (re <= 1.22e+29)):
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.5e-98)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif ((re <= 4.1e-109) || (!(re <= 2.45e-5) && (re <= 1.22e+29)))
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.5e-98)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif ((re <= 4.1e-109) || (~((re <= 2.45e-5)) && (re <= 1.22e+29)))
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.5e-98], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 4.1e-109], And[N[Not[LessEqual[re, 2.45e-5]], $MachinePrecision], LessEqual[re, 1.22e+29]]], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.4999999999999997e-98

   1. Initial program 54.0%

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

   3. Taylor expanded in re around -inf 74.2%

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

      1. *-commutative 74.2%

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

   5. Simplified 74.2%

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

   if -5.4999999999999997e-98 < re < 4.1000000000000002e-109 or 2.45e-5 < re < 1.22e29

   1. Initial program 60.5%

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

   3. Taylor expanded in re around 0 80.9%

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

      1. *-commutative 80.9%

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

   5. Simplified 80.9%

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

   if 4.1000000000000002e-109 < re < 2.45e-5 or 1.22e29 < re

   1. Initial program 14.4%

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

   3. Taylor expanded in re around inf 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*l* 76.7%

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

   5. Simplified 76.7%

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

      1. add-log-exp 9.6%

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

      2. *-un-lft-identity 9.6%

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

      3. log-prod 9.6%

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

      4. metadata-eval 9.6%

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

      5. add-log-exp 76.7%

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

      6. sqrt-div 76.6%

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

      7. metadata-eval 76.6%

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

      8. un-div-inv 76.5%

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

      9. associate-*r* 76.3%

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

      10. sqrt-unprod 77.3%

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

      11. metadata-eval 77.3%

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

      12. metadata-eval 77.3%

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

      13. *-un-lft-identity 77.3%

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

   7. Applied egg-rr 77.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \frac{im}{\sqrt{re}}\right)} \]
   8. Step-by-step derivation

      1. +-lft-identity 77.3%

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

   9. Simplified 77.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{-109} \lor \neg \left(re \leq 2.45 \cdot 10^{-5}\right) \land re \leq 1.22 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.6 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.6e-98) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.6e-98) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-8.6d-98)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8.6e-98) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.6e-98:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.6e-98)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.6e-98)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.6e-98], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -8.59999999999999977e-98

   1. Initial program 54.0%

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

   3. Taylor expanded in re around -inf 74.2%

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

      1. *-commutative 74.2%

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

   5. Simplified 74.2%

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

   if -8.59999999999999977e-98 < re

   1. Initial program 40.6%

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

   3. Taylor expanded in re around 0 57.6%

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

      1. *-commutative 57.6%

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

   5. Simplified 57.6%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.6 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 0 49.2%

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

   1. *-commutative 49.2%

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

5. Simplified 49.2%

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

6. Final simplification 49.2%

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

Reproduce

herbie shell --seed 2024074 
(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)))))