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

Percentage Accurate: 41.1% → 90.1%

Time: 6.6s

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: 41.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = (im * 0.5) / sqrt(re);
	} else {
		tmp = sqrt((0.5 * (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 = (im * 0.5) / Math.sqrt(re);
	} else {
		tmp = Math.sqrt((0.5 * (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 = (im * 0.5) / math.sqrt(re)
	else:
		tmp = math.sqrt((0.5 * (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(Float64(im * 0.5) / sqrt(re));
	else
		tmp = sqrt(Float64(0.5 * 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 = (im * 0.5) / sqrt(re);
	else
		tmp = sqrt((0.5 * (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[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $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 14.4%

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

   2. Taylor expanded in im around 0 96.5%

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

      1. associate-*l* 96.8%

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

      2. *-commutative 96.8%

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

   4. Simplified 96.8%

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

      1. add-log-exp 18.1%

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

      2. *-un-lft-identity 18.1%

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

      3. log-prod 18.1%

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

      4. metadata-eval 18.1%

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

      5. add-log-exp 96.8%

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

      6. associate-*l* 96.6%

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

      7. sqrt-div 96.7%

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

      8. metadata-eval 96.7%

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

      9. un-div-inv 96.8%

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

   6. Applied egg-rr 96.8%

      \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(0 + im \cdot \frac{\sqrt{0.5}}{\sqrt{re}}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lft-identity 96.8%

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

      2. associate-*r/ 96.8%

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

   8. Simplified 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*r/ 96.6%

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

      3. associate-*l/ 96.6%

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

      4. *-commutative 96.6%

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

      5. associate-*r* 96.4%

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

      6. pow1/2 96.4%

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

      7. pow1/2 96.4%

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

      8. pow-prod-down 97.6%

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

      9. metadata-eval 97.6%

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

      10. pow-to-exp 97.6%

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

      11. metadata-eval 97.6%

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

      12. metadata-eval 97.6%

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

      13. 1-exp 97.6%

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

      14. *-un-lft-identity 97.6%

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

   10. Applied egg-rr 97.6%

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

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

   1. Initial program 49.0%

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

      1. hypot-def 90.6%

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

   3. Simplified 90.6%

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

      1. add-sqr-sqrt 89.8%

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

      2. sqrt-unprod 90.6%

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

      3. *-commutative 90.6%

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

      4. *-commutative 90.6%

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

      5. swap-sqr 90.6%

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

      6. add-sqr-sqrt 90.6%

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

      7. metadata-eval 90.6%

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

   5. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. metadata-eval 90.6%

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

   7. Simplified 90.6%

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

4. Final simplification 91.7%

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

Alternative 2: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+178}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 160:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+178)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 160.0)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+178) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 160.0) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / 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 <= (-1.8d+178)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 160.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+178) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 160.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+178:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 160.0:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+178)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 160.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e+178)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 160.0)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+178], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 160.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.7999999999999999e178

   1. Initial program 4.3%

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

   2. Taylor expanded in re around -inf 87.2%

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

      1. *-commutative 87.2%

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

   4. Simplified 87.2%

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

   if -1.7999999999999999e178 < re < 160

   1. Initial program 62.5%

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

   2. Taylor expanded in re around 0 76.5%

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

   if 160 < re

   1. Initial program 14.3%

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

   2. Taylor expanded in im around 0 75.9%

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

      1. associate-*l* 76.0%

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

      2. *-commutative 76.0%

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

   4. Simplified 76.0%

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

      1. add-log-exp 17.0%

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

      2. *-un-lft-identity 17.0%

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

      3. log-prod 17.0%

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

      4. metadata-eval 17.0%

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

      5. add-log-exp 76.0%

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

      6. associate-*l* 75.9%

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

      7. sqrt-div 75.9%

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

      8. metadata-eval 75.9%

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

      9. un-div-inv 75.9%

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

   6. Applied egg-rr 75.9%

      \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(0 + im \cdot \frac{\sqrt{0.5}}{\sqrt{re}}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lft-identity 75.9%

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

      2. associate-*r/ 76.0%

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

   8. Simplified 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-*r/ 76.0%

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

      3. associate-*l/ 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-*r* 75.7%

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

      6. pow1/2 75.7%

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

      7. pow1/2 75.7%

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

      8. pow-prod-down 76.5%

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

      9. metadata-eval 76.5%

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

      10. pow-to-exp 76.5%

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

      11. metadata-eval 76.5%

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

      12. metadata-eval 76.5%

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

      13. 1-exp 76.5%

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

      14. *-un-lft-identity 76.5%

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

   10. Applied egg-rr 76.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+178}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 160:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 3: 73.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 25.5:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.7e+146)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 25.5) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+146) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 25.5) {
		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 <= (-2.7d+146)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 25.5d0) 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 <= -2.7e+146) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 25.5) {
		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 <= -2.7e+146:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 25.5:
		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 <= -2.7e+146)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 25.5)
		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 <= -2.7e+146)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 25.5)
		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, -2.7e+146], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 25.5], 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 < -2.69999999999999989e146

   1. Initial program 4.3%

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

   2. Taylor expanded in re around -inf 83.0%

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

      1. *-commutative 83.0%

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

   4. Simplified 83.0%

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

   if -2.69999999999999989e146 < re < 25.5

   1. Initial program 64.0%

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

   2. Taylor expanded in re around 0 74.6%

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

      1. *-commutative 74.6%

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

   4. Simplified 74.6%

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

   if 25.5 < re

   1. Initial program 14.3%

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

   2. Taylor expanded in re around inf 42.7%

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

      1. unpow2 42.7%

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

   4. Simplified 42.7%

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

      1. add-log-exp 17.3%

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

      2. *-un-lft-identity 17.3%

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

      3. log-prod 17.3%

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

      4. metadata-eval 17.3%

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

      5. add-log-exp 42.7%

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

      6. sqrt-div 56.8%

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

      7. sqrt-prod 76.2%

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

      8. add-sqr-sqrt 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. +-lft-identity 76.5%

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

   8. Simplified 76.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 25.5:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 4: 73.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 82:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.7e+146)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 82.0) (* 0.5 (sqrt (* im 2.0))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+146) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 82.0) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = (im * 0.5) / 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 <= (-2.7d+146)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 82.0d0) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+146) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 82.0) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.7e+146:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 82.0:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.7e+146)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 82.0)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.7e+146)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 82.0)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.7e+146], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 82.0], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.69999999999999989e146

   1. Initial program 4.3%

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

   2. Taylor expanded in re around -inf 83.0%

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

      1. *-commutative 83.0%

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

   4. Simplified 83.0%

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

   if -2.69999999999999989e146 < re < 82

   1. Initial program 64.0%

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

   2. Taylor expanded in re around 0 74.6%

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

      1. *-commutative 74.6%

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

   4. Simplified 74.6%

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

   if 82 < re

   1. Initial program 14.3%

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

   2. Taylor expanded in im around 0 75.9%

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

      1. associate-*l* 76.0%

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

      2. *-commutative 76.0%

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

   4. Simplified 76.0%

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

      1. add-log-exp 17.0%

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

      2. *-un-lft-identity 17.0%

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

      3. log-prod 17.0%

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

      4. metadata-eval 17.0%

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

      5. add-log-exp 76.0%

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

      6. associate-*l* 75.9%

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

      7. sqrt-div 75.9%

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

      8. metadata-eval 75.9%

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

      9. un-div-inv 75.9%

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

   6. Applied egg-rr 75.9%

      \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(0 + im \cdot \frac{\sqrt{0.5}}{\sqrt{re}}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lft-identity 75.9%

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

      2. associate-*r/ 76.0%

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

   8. Simplified 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-*r/ 76.0%

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

      3. associate-*l/ 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-*r* 75.7%

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

      6. pow1/2 75.7%

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

      7. pow1/2 75.7%

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

      8. pow-prod-down 76.5%

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

      9. metadata-eval 76.5%

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

      10. pow-to-exp 76.5%

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

      11. metadata-eval 76.5%

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

      12. metadata-eval 76.5%

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

      13. 1-exp 76.5%

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

      14. *-un-lft-identity 76.5%

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

   10. Applied egg-rr 76.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 82:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]

Alternative 5: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.7e+146) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.7e+146) {
		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 <= (-2.7d+146)) 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 <= -2.7e+146) {
		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 <= -2.7e+146:
		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 <= -2.7e+146)
		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 <= -2.7e+146)
		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, -2.7e+146], 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 < -2.69999999999999989e146

   1. Initial program 4.3%

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

   2. Taylor expanded in re around -inf 83.0%

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

      1. *-commutative 83.0%

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

   4. Simplified 83.0%

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

   if -2.69999999999999989e146 < re

   1. Initial program 48.9%

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

   2. Taylor expanded in re around 0 60.8%

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

      1. *-commutative 60.8%

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

   4. Simplified 60.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6: 52.1% 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 43.4%

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

2. Taylor expanded in re around 0 56.1%

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

   1. *-commutative 56.1%

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

4. Simplified 56.1%

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

5. Final simplification 56.1%

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

Reproduce

herbie shell --seed 2023207 
(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)))))