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

Percentage Accurate: 41.0% → 88.0%

Time: 9.2s

Alternatives: 4

Speedup: 1.9×

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: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.2e+48)
   (* 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 <= 1.2e+48) {
		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 <= 1.2e+48) {
		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 <= 1.2e+48:
		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 <= 1.2e+48)
		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 <= 1.2e+48)
		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, 1.2e+48], 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 < 1.2000000000000001e48

   1. Initial program 50.0%

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

      1. sub-neg 50.0%

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

      2. sqr-neg 50.0%

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

      3. sub-neg 50.0%

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

      4. sqr-neg 50.0%

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

      5. hypot-define 90.9%

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

   3. Simplified 90.9%

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

   if 1.2000000000000001e48 < re

   1. Initial program 9.8%

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

   3. Taylor expanded in re around inf 52.0%

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

      1. metadata-eval 52.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\sqrt{4}} \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)} \]

      2. metadata-eval 52.0%

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

      3. sqrt-undiv 52.0%

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

      4. associate-*r* 52.0%

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

      5. sqrt-undiv 52.0%

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

      6. metadata-eval 52.0%

         \[\leadsto 0.5 \cdot \sqrt{\left(\sqrt{\color{blue}{4}} \cdot 0.5\right) \cdot \frac{{im}^{2}}{re}} \]

      7. metadata-eval 52.0%

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

      8. metadata-eval 52.0%

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

      9. *-un-lft-identity 52.0%

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

      10. div-inv 52.0%

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

      11. sqrt-prod 64.0%

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

      12. unpow2 64.0%

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

      13. sqrt-prod 83.8%

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

      14. add-sqr-sqrt 84.1%

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

      15. *-commutative 84.1%

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

      16. inv-pow 84.1%

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

      17. sqrt-pow1 84.2%

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

      18. metadata-eval 84.2%

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

   5. Applied egg-rr 84.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 8.7 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7e+85)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 8.7e+46)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7e+85) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 8.7e+46) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} 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) :: tmp
    if (re <= (-7d+85)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 8.7d+46) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7e+85) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 8.7e+46) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7e+85:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 8.7e+46:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7e+85)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 8.7e+46)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(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 <= -7e+85)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 8.7e+46)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7e+85], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.7e+46], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.0000000000000001e85

   1. Initial program 34.6%

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

   3. Taylor expanded in re around -inf 91.0%

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

      1. *-commutative 91.0%

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

   5. Simplified 91.0%

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

   if -7.0000000000000001e85 < re < 8.69999999999999961e46

   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 0 79.6%

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

   if 8.69999999999999961e46 < re

   1. Initial program 9.8%

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

   3. Taylor expanded in re around inf 52.0%

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

      1. metadata-eval 52.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\sqrt{4}} \cdot \left(0.5 \cdot \frac{{im}^{2}}{re}\right)} \]

      2. metadata-eval 52.0%

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

      3. sqrt-undiv 52.0%

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

      4. associate-*r* 52.0%

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

      5. sqrt-undiv 52.0%

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

      6. metadata-eval 52.0%

         \[\leadsto 0.5 \cdot \sqrt{\left(\sqrt{\color{blue}{4}} \cdot 0.5\right) \cdot \frac{{im}^{2}}{re}} \]

      7. metadata-eval 52.0%

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

      8. metadata-eval 52.0%

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

      9. *-un-lft-identity 52.0%

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

      10. div-inv 52.0%

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

      11. sqrt-prod 64.0%

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

      12. unpow2 64.0%

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

      13. sqrt-prod 83.8%

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

      14. add-sqr-sqrt 84.1%

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

      15. *-commutative 84.1%

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

      16. inv-pow 84.1%

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

      17. sqrt-pow1 84.2%

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

      18. metadata-eval 84.2%

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

   5. Applied egg-rr 84.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 8.7 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-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 <= -2e-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 <= (-2d-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 <= -2e-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 <= -2e-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 <= -2e-310)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -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 <= -2e-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, -2e-310], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.999999999999994e-310

   1. Initial program 54.9%

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

   3. Taylor expanded in re around -inf 46.9%

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

      1. *-commutative 46.9%

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

   5. Simplified 46.9%

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

   if -1.999999999999994e-310 < re

   1. Initial program 28.2%

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

   3. Taylor expanded in re around inf 29.9%

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

   4. Taylor expanded in im around 0 53.2%

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

      1. unpow1/2 53.2%

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

      2. rem-exp-log 50.5%

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

      3. exp-neg 50.5%

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

      4. exp-prod 50.4%

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

      5. distribute-lft-neg-out 50.4%

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

      6. exp-neg 50.4%

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

      7. exp-to-pow 53.2%

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

      8. unpow1/2 53.2%

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

      9. associate-/l* 53.2%

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

      10. *-rgt-identity 53.2%

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

   6. Simplified 53.2%

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

4. Final simplification 50.2%

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

Alternative 4: 26.3% accurate, 2.0× speedup

Math

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

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(0.5 * Float64(im / sqrt(re)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

3. Taylor expanded in re around inf 16.0%

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

4. Taylor expanded in im around 0 27.9%

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

   1. unpow1/2 27.9%

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

   2. rem-exp-log 26.4%

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

   3. exp-neg 26.4%

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

   4. exp-prod 26.4%

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

   5. distribute-lft-neg-out 26.4%

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

   6. exp-neg 26.4%

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

   7. exp-to-pow 27.8%

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

   8. unpow1/2 27.8%

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

   9. associate-/l* 27.9%

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

   10. *-rgt-identity 27.9%

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

6. Simplified 27.9%

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

7. Final simplification 27.9%

   \[\leadsto 0.5 \cdot \frac{im}{\sqrt{re}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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)))))