math.sqrt on complex, real part

Percentage Accurate: 41.7% → 89.1%

Time: 6.3s

Alternatives: 9

Speedup: 1.9×

Specification

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: 41.7% 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: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{\frac{im}{re}} \cdot \sqrt[3]{-im}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im im)))) 0.0)
   (* 0.5 (pow (* (cbrt (/ im re)) (cbrt (- im))) 1.5))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	double tmp;
	if ((re + sqrt(((re * re) + (im * im)))) <= 0.0) {
		tmp = 0.5 * pow((cbrt((im / re)) * cbrt(-im)), 1.5);
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((re + Math.sqrt(((re * re) + (im * im)))) <= 0.0) {
		tmp = 0.5 * Math.pow((Math.cbrt((im / re)) * Math.cbrt(-im)), 1.5);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im)))) <= 0.0)
		tmp = Float64(0.5 * (Float64(cbrt(Float64(im / re)) * cbrt(Float64(-im))) ^ 1.5));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[N[(N[Power[N[(im / re), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[(-im), 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $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 6.0%

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

      1. +-commutative 6.0%

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

      2. hypot-def 13.6%

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

   3. Simplified 13.6%

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

   4. Taylor expanded in re around -inf 46.4%

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

      1. *-commutative 46.4%

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

      2. unpow2 46.4%

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

   6. Simplified 46.4%

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

      1. pow1/2 46.4%

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

      2. add-cube-cbrt 45.7%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\left(\sqrt[3]{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \cdot \sqrt[3]{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right) \cdot \sqrt[3]{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right)}}^{0.5} \]

      3. pow3 45.8%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(\sqrt[3]{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right)}^{3}\right)}}^{0.5} \]

      4. metadata-eval 45.8%

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

      5. pow-pow 45.8%

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

      6. *-commutative 45.8%

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

      7. associate-*l* 45.8%

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

      8. associate-/l* 57.8%

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

      9. associate-/r/ 57.7%

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

      10. metadata-eval 57.7%

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

      11. metadata-eval 57.7%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\frac{im}{re} \cdot im\right) \cdot -1}\right)}^{\left(\color{blue}{3} \cdot 0.5\right)} \]

      12. metadata-eval 57.7%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\frac{im}{re} \cdot im\right) \cdot -1}\right)}^{\color{blue}{1.5}} \]

   8. Applied egg-rr 57.7%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{\left(\frac{im}{re} \cdot im\right) \cdot -1}\right)}^{1.5}} \]
   9. Step-by-step derivation

      1. associate-*l* 57.7%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\color{blue}{\frac{im}{re} \cdot \left(im \cdot -1\right)}}\right)}^{1.5} \]

      2. cbrt-prod 86.5%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\frac{im}{re}} \cdot \sqrt[3]{im \cdot -1}\right)}}^{1.5} \]

   10. Applied egg-rr 86.5%

       \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\frac{im}{re}} \cdot \sqrt[3]{im \cdot -1}\right)}}^{1.5} \]
   11. Step-by-step derivation

       1. *-commutative 86.5%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{im}{re}} \cdot \sqrt[3]{\color{blue}{-1 \cdot im}}\right)}^{1.5} \]

       2. neg-mul-1 86.5%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{im}{re}} \cdot \sqrt[3]{\color{blue}{-im}}\right)}^{1.5} \]

   12. Simplified 86.5%

       \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\frac{im}{re}} \cdot \sqrt[3]{-im}\right)}}^{1.5} \]

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

   1. Initial program 48.1%

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

      1. +-commutative 48.1%

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

      2. hypot-def 90.2%

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

   3. Simplified 90.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{\frac{im}{re}} \cdot \sqrt[3]{-im}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{-re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9e+52)
   (* 0.5 (sqrt (/ im (/ (- re) im))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9e+52) {
		tmp = 0.5 * sqrt((im / (-re / im)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9e+52) {
		tmp = 0.5 * Math.sqrt((im / (-re / im)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9e+52:
		tmp = 0.5 * math.sqrt((im / (-re / im)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9e+52)
		tmp = Float64(0.5 * sqrt(Float64(im / Float64(Float64(-re) / im))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9e+52)
		tmp = 0.5 * sqrt((im / (-re / im)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9e+52], N[(0.5 * N[Sqrt[N[(im / N[((-re) / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -8.9999999999999999e52

   1. Initial program 8.5%

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

      1. +-commutative 8.5%

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

      2. hypot-def 25.1%

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

   3. Simplified 25.1%

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

   4. Taylor expanded in re around -inf 59.3%

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

      1. *-commutative 59.3%

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

      2. unpow2 59.3%

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

   6. Simplified 59.3%

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

      1. add-log-exp 21.8%

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

      2. *-un-lft-identity 21.8%

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

      3. log-prod 21.8%

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

      4. metadata-eval 21.8%

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

      5. add-log-exp 59.3%

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

      6. *-commutative 59.3%

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

      7. associate-*l* 59.3%

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

      8. associate-/l* 68.4%

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

      9. associate-/r/ 68.2%

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

      10. metadata-eval 68.2%

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

   8. Applied egg-rr 68.2%

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

      1. +-lft-identity 68.2%

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

      2. *-commutative 68.2%

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

      3. metadata-eval 68.2%

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

      4. associate-*l/ 59.3%

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

      5. times-frac 59.3%

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

      6. neg-mul-1 59.3%

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

      7. *-lft-identity 59.3%

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

      8. associate-/l* 68.4%

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

      9. distribute-frac-neg 68.4%

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

   10. Simplified 68.4%

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

   if -8.9999999999999999e52 < re

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      2. hypot-def 92.0%

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

   3. Simplified 92.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{-re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3: 43.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-57} \lor \neg \left(re \leq 3.6 \cdot 10^{-35}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.22e-116)
   (* 0.5 (sqrt (* im 2.0)))
   (if (or (<= re 2.1e-57) (not (<= re 3.6e-35)))
     (* 0.5 (* 2.0 (sqrt re)))
     (* 0.5 (sqrt (* 2.0 (+ re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.22e-116) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else if ((re <= 2.1e-57) || !(re <= 3.6e-35)) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.22d-116) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else if ((re <= 2.1d-57) .or. (.not. (re <= 3.6d-35))) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.22e-116) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else if ((re <= 2.1e-57) || !(re <= 3.6e-35)) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.22e-116:
		tmp = 0.5 * math.sqrt((im * 2.0))
	elif (re <= 2.1e-57) or not (re <= 3.6e-35):
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.22e-116)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	elseif ((re <= 2.1e-57) || !(re <= 3.6e-35))
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.22e-116)
		tmp = 0.5 * sqrt((im * 2.0));
	elseif ((re <= 2.1e-57) || ~((re <= 3.6e-35)))
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.22e-116], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 2.1e-57], N[Not[LessEqual[re, 3.6e-35]], $MachinePrecision]], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.22e-116

   1. Initial program 38.1%

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

      1. +-commutative 38.1%

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

      2. hypot-def 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in re around 0 24.1%

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

   if 1.22e-116 < re < 2.0999999999999999e-57 or 3.60000000000000019e-35 < re

   1. Initial program 47.5%

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

      1. +-commutative 47.5%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 79.4%

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

      1. unpow2 79.4%

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

      2. rem-square-sqrt 81.0%

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

   6. Simplified 81.0%

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

   if 2.0999999999999999e-57 < re < 3.60000000000000019e-35

   1. Initial program 58.9%

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

      1. +-commutative 58.9%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 45.8%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.22 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-57} \lor \neg \left(re \leq 3.6 \cdot 10^{-35}\right):\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4: 43.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.22 \cdot 10^{-116} \lor \neg \left(re \leq 4.2 \cdot 10^{-57}\right) \land re \leq 3.7 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 1.22e-116) (and (not (<= re 4.2e-57)) (<= re 3.7e-36)))
   (* 0.5 (sqrt (* im 2.0)))
   (* 0.5 (* 2.0 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 1.22e-116) || (!(re <= 4.2e-57) && (re <= 3.7e-36))) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (2.0 * 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 <= 1.22d-116) .or. (.not. (re <= 4.2d-57)) .and. (re <= 3.7d-36)) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 1.22e-116) || (!(re <= 4.2e-57) && (re <= 3.7e-36))) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 1.22e-116) or (not (re <= 4.2e-57) and (re <= 3.7e-36)):
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 1.22e-116) || (!(re <= 4.2e-57) && (re <= 3.7e-36)))
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 1.22e-116) || (~((re <= 4.2e-57)) && (re <= 3.7e-36)))
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 1.22e-116], And[N[Not[LessEqual[re, 4.2e-57]], $MachinePrecision], LessEqual[re, 3.7e-36]]], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.22e-116 or 4.1999999999999999e-57 < re < 3.70000000000000002e-36

   1. Initial program 38.9%

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

      1. +-commutative 38.9%

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

      2. hypot-def 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in re around 0 24.9%

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

   if 1.22e-116 < re < 4.1999999999999999e-57 or 3.70000000000000002e-36 < re

   1. Initial program 47.5%

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

      1. +-commutative 47.5%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 79.4%

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

      1. unpow2 79.4%

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

      2. rem-square-sqrt 81.0%

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

   6. Simplified 81.0%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.22 \cdot 10^{-116} \lor \neg \left(re \leq 4.2 \cdot 10^{-57}\right) \land re \leq 3.7 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5: 44.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+125}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e+125)
   (* 0.5 (sqrt (/ (* im im) re)))
   (if (<= re 6.6e-36)
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e+125) {
		tmp = 0.5 * sqrt(((im * im) / re));
	} else if (re <= 6.6e-36) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5d+125)) then
        tmp = 0.5d0 * sqrt(((im * im) / re))
    else if (re <= 6.6d-36) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e+125) {
		tmp = 0.5 * Math.sqrt(((im * im) / re));
	} else if (re <= 6.6e-36) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e+125:
		tmp = 0.5 * math.sqrt(((im * im) / re))
	elif re <= 6.6e-36:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e+125)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * im) / re)));
	elseif (re <= 6.6e-36)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e+125)
		tmp = 0.5 * sqrt(((im * im) / re));
	elseif (re <= 6.6e-36)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e+125], N[(0.5 * N[Sqrt[N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.6e-36], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.99999999999999962e125

   1. Initial program 5.1%

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

      1. +-commutative 5.1%

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

      2. hypot-def 26.5%

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

   3. Simplified 26.5%

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

   4. Taylor expanded in re around -inf 61.6%

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

      1. *-commutative 61.6%

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

      2. unpow2 61.6%

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

   6. Simplified 61.6%

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

      1. pow1/2 61.6%

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

      2. add-cube-cbrt 61.2%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\left(\sqrt[3]{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \cdot \sqrt[3]{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right) \cdot \sqrt[3]{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right)}}^{0.5} \]

      3. pow3 61.3%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(\sqrt[3]{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right)}^{3}\right)}}^{0.5} \]

      4. metadata-eval 61.3%

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

      5. pow-pow 61.2%

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

      6. *-commutative 61.2%

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

      7. associate-*l* 61.2%

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

      8. associate-/l* 75.9%

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

      9. associate-/r/ 76.0%

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

      10. metadata-eval 76.0%

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

      11. metadata-eval 76.0%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\frac{im}{re} \cdot im\right) \cdot -1}\right)}^{\left(\color{blue}{3} \cdot 0.5\right)} \]

      12. metadata-eval 76.0%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\frac{im}{re} \cdot im\right) \cdot -1}\right)}^{\color{blue}{1.5}} \]

   8. Applied egg-rr 76.0%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{\left(\frac{im}{re} \cdot im\right) \cdot -1}\right)}^{1.5}} \]
   9. Step-by-step derivation

      1. pow1/3 72.2%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(\left(\frac{im}{re} \cdot im\right) \cdot -1\right)}^{0.3333333333333333}\right)}}^{1.5} \]

      2. pow-pow 76.4%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\frac{im}{re} \cdot im\right) \cdot -1\right)}^{\left(0.3333333333333333 \cdot 1.5\right)}} \]

      3. metadata-eval 76.4%

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

      4. pow1/2 76.4%

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

      5. *-commutative 76.4%

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

      6. metadata-eval 76.4%

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

      7. *-commutative 76.4%

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

      8. clear-num 76.4%

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

      9. un-div-inv 76.5%

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

      10. times-frac 76.5%

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

      11. *-un-lft-identity 76.5%

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

      12. neg-mul-1 76.5%

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

      13. div-inv 76.4%

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

      14. sqrt-prod 33.0%

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

      15. add-sqr-sqrt 33.0%

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

      16. sqrt-unprod 27.2%

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

      17. sqr-neg 27.2%

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

      18. sqrt-unprod 0.0%

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

      19. add-sqr-sqrt 14.9%

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

      20. clear-num 14.9%

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

   10. Applied egg-rr 21.7%

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

       1. associate-*r/ 21.7%

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

   12. Simplified 21.7%

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

   if -4.99999999999999962e125 < re < 6.59999999999999981e-36

   1. Initial program 48.8%

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

      1. +-commutative 48.8%

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

      2. hypot-def 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in re around 0 31.2%

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

   if 6.59999999999999981e-36 < re

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 79.7%

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

      1. unpow2 79.7%

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

      2. rem-square-sqrt 81.2%

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

   6. Simplified 81.2%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+125}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 6: 44.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 7.4 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.45e+122)
   (* 0.5 (sqrt (* 2.0 (- re re))))
   (if (<= re 7.4e-37)
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+122) {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	} else if (re <= 7.4e-37) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * 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 <= (-2.45d+122)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    else if (re <= 7.4d-37) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+122) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	} else if (re <= 7.4e-37) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.45e+122:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	elif re <= 7.4e-37:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.45e+122)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	elseif (re <= 7.4e-37)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.45e+122)
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	elseif (re <= 7.4e-37)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.45e+122], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.4e-37], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.4499999999999999e122

   1. Initial program 5.1%

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

   2. Taylor expanded in re around -inf 23.0%

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

      1. mul-1-neg 23.0%

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

   4. Simplified 23.0%

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

   if -2.4499999999999999e122 < re < 7.4e-37

   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. +-commutative 49.0%

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

      2. hypot-def 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in re around 0 31.4%

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

   if 7.4e-37 < re

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 79.7%

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

      1. unpow2 79.7%

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

      2. rem-square-sqrt 81.2%

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

   6. Simplified 81.2%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 7.4 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 7: 52.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.95 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.95e+48)
   (* 0.5 (sqrt (* im (/ (- im) re))))
   (if (<= re 1.65e-35)
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.95e+48) {
		tmp = 0.5 * sqrt((im * (-im / re)));
	} else if (re <= 1.65e-35) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * 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 <= (-1.95d+48)) then
        tmp = 0.5d0 * sqrt((im * (-im / re)))
    else if (re <= 1.65d-35) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.95e+48) {
		tmp = 0.5 * Math.sqrt((im * (-im / re)));
	} else if (re <= 1.65e-35) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.95e+48:
		tmp = 0.5 * math.sqrt((im * (-im / re)))
	elif re <= 1.65e-35:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.95e+48)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(Float64(-im) / re))));
	elseif (re <= 1.65e-35)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.95e+48)
		tmp = 0.5 * sqrt((im * (-im / re)));
	elseif (re <= 1.65e-35)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.95e+48], N[(0.5 * N[Sqrt[N[(im * N[((-im) / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.65e-35], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.95e48

   1. Initial program 8.5%

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

      1. +-commutative 8.5%

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

      2. hypot-def 25.1%

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

   3. Simplified 25.1%

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

   4. Taylor expanded in re around -inf 59.3%

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

      1. *-commutative 59.3%

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

      2. unpow2 59.3%

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

   6. Simplified 59.3%

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

      1. add-log-exp 21.8%

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

      2. *-un-lft-identity 21.8%

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

      3. log-prod 21.8%

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

      4. metadata-eval 21.8%

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

      5. add-log-exp 59.3%

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

      6. *-commutative 59.3%

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

      7. associate-*l* 59.3%

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

      8. associate-/l* 68.4%

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

      9. associate-/r/ 68.2%

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

      10. metadata-eval 68.2%

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

   8. Applied egg-rr 68.2%

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

      1. +-lft-identity 68.2%

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

      2. associate-*l* 68.2%

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

      3. *-commutative 68.2%

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

      4. neg-mul-1 68.2%

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

   10. Simplified 68.2%

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

   if -1.95e48 < re < 1.65e-35

   1. Initial program 54.1%

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

      1. +-commutative 54.1%

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

      2. hypot-def 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in re around 0 35.2%

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

   if 1.65e-35 < re

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 79.7%

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

      1. unpow2 79.7%

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

      2. rem-square-sqrt 81.2%

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

   6. Simplified 81.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.95 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 8: 51.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.75 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{-re}{im}}}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.75e+50)
   (* 0.5 (sqrt (/ im (/ (- re) im))))
   (if (<= re 7e-36)
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.75e+50) {
		tmp = 0.5 * sqrt((im / (-re / im)));
	} else if (re <= 7e-36) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * 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 <= (-3.75d+50)) then
        tmp = 0.5d0 * sqrt((im / (-re / im)))
    else if (re <= 7d-36) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.75e+50) {
		tmp = 0.5 * Math.sqrt((im / (-re / im)));
	} else if (re <= 7e-36) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.75e+50:
		tmp = 0.5 * math.sqrt((im / (-re / im)))
	elif re <= 7e-36:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.75e+50)
		tmp = Float64(0.5 * sqrt(Float64(im / Float64(Float64(-re) / im))));
	elseif (re <= 7e-36)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.75e+50)
		tmp = 0.5 * sqrt((im / (-re / im)));
	elseif (re <= 7e-36)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.75e+50], N[(0.5 * N[Sqrt[N[(im / N[((-re) / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7e-36], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.75e50

   1. Initial program 8.5%

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

      1. +-commutative 8.5%

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

      2. hypot-def 25.1%

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

   3. Simplified 25.1%

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

   4. Taylor expanded in re around -inf 59.3%

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

      1. *-commutative 59.3%

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

      2. unpow2 59.3%

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

   6. Simplified 59.3%

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

      1. add-log-exp 21.8%

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

      2. *-un-lft-identity 21.8%

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

      3. log-prod 21.8%

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

      4. metadata-eval 21.8%

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

      5. add-log-exp 59.3%

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

      6. *-commutative 59.3%

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

      7. associate-*l* 59.3%

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

      8. associate-/l* 68.4%

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

      9. associate-/r/ 68.2%

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

      10. metadata-eval 68.2%

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

   8. Applied egg-rr 68.2%

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

      1. +-lft-identity 68.2%

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

      2. *-commutative 68.2%

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

      3. metadata-eval 68.2%

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

      4. associate-*l/ 59.3%

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

      5. times-frac 59.3%

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

      6. neg-mul-1 59.3%

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

      7. *-lft-identity 59.3%

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

      8. associate-/l* 68.4%

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

      9. distribute-frac-neg 68.4%

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

   10. Simplified 68.4%

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

   if -3.75e50 < re < 6.9999999999999999e-36

   1. Initial program 54.1%

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

      1. +-commutative 54.1%

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

      2. hypot-def 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in re around 0 35.2%

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

   if 6.9999999999999999e-36 < re

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 79.7%

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

      1. unpow2 79.7%

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

      2. rem-square-sqrt 81.2%

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

   6. Simplified 81.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.75 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{-re}{im}}}\\ \mathbf{elif}\;re \leq 7 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 9: 25.7% 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 41.3%

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

   1. +-commutative 41.3%

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

   2. hypot-def 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in re around 0 21.4%

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

5. Final simplification 21.4%

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

Developer target: 48.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + 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 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023222 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))