math.sqrt on complex, real part

Percentage Accurate: 41.1% → 99.7%

Time: 10.6s

Alternatives: 7

Speedup: 2.0×

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: 41.1% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{-301}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot 2\right) \cdot \sqrt{\frac{0.5}{\mathsf{hypot}\left(re, im\right) - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -1.75e-301)
   (* 0.5 (* (* im 2.0) (sqrt (/ 0.5 (- (hypot re im) re)))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -1.75e-301) {
		tmp = 0.5 * ((im * 2.0) * sqrt((0.5 / (hypot(re, im) - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -1.75e-301) {
		tmp = 0.5 * ((im * 2.0) * Math.sqrt((0.5 / (Math.hypot(re, im) - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -1.75e-301:
		tmp = 0.5 * ((im * 2.0) * math.sqrt((0.5 / (math.hypot(re, im) - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -1.75e-301)
		tmp = Float64(0.5 * Float64(Float64(im * 2.0) * sqrt(Float64(0.5 / Float64(hypot(re, im) - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.75e-301)
		tmp = 0.5 * ((im * 2.0) * sqrt((0.5 / (hypot(re, im) - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -1.75e-301], N[(0.5 * N[(N[(im * 2.0), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $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 < -1.74999999999999996e-301

   1. Initial program 27.5%

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

   3. Simplified 56.6%

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

      1. +-commutative 56.6%

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

      2. distribute-rgt-in 56.6%

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

      3. flip-+ 27.1%

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

   5. Applied egg-rr 27.1%

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

   6. Taylor expanded in re around 0 47.4%

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

      1. *-commutative 47.4%

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

   8. Simplified 47.4%

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

      1. pow1/2 47.4%

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

      2. div-inv 47.4%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\left({im}^{2} \cdot 4\right) \cdot \frac{1}{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}\right)}}^{0.5} \]

      3. unpow-prod-down 55.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left({\left({im}^{2} \cdot 4\right)}^{0.5} \cdot {\left(\frac{1}{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}\right)}^{0.5}\right)} \]

      4. pow1/2 55.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{{im}^{2} \cdot 4}} \cdot {\left(\frac{1}{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}\right)}^{0.5}\right) \]

      5. sqrt-prod 55.6%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{{im}^{2}} \cdot \sqrt{4}\right)} \cdot {\left(\frac{1}{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}\right)}^{0.5}\right) \]

      6. unpow2 55.6%

         \[\leadsto 0.5 \cdot \left(\left(\sqrt{\color{blue}{im \cdot im}} \cdot \sqrt{4}\right) \cdot {\left(\frac{1}{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}\right)}^{0.5}\right) \]

      7. sqrt-prod 50.7%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \sqrt{4}\right) \cdot {\left(\frac{1}{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}\right)}^{0.5}\right) \]

      8. add-sqr-sqrt 52.2%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{im} \cdot \sqrt{4}\right) \cdot {\left(\frac{1}{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}\right)}^{0.5}\right) \]

      9. metadata-eval 52.2%

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

      10. distribute-rgt-out-- 52.2%

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

   10. Applied egg-rr 52.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot 2\right) \cdot {\left(\frac{1}{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\right)}^{0.5}\right)} \]
   11. Step-by-step derivation

       1. unpow1/2 52.2%

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

       2. associate-/r* 52.2%

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

       3. metadata-eval 52.2%

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

   12. Simplified 52.2%

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

   if -1.74999999999999996e-301 < re

   1. Initial program 50.3%

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

   3. Simplified 100.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{-301}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot 2\right) \cdot \sqrt{\frac{0.5}{\mathsf{hypot}\left(re, im\right) - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{-302}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -7e-302)
   (* 0.5 (/ im (sqrt (* 0.5 (- (hypot re im) re)))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -7e-302) {
		tmp = 0.5 * (im / sqrt((0.5 * (hypot(re, im) - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -7e-302) {
		tmp = 0.5 * (im / Math.sqrt((0.5 * (Math.hypot(re, im) - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -7e-302:
		tmp = 0.5 * (im / math.sqrt((0.5 * (math.hypot(re, im) - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -7e-302)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(0.5 * Float64(hypot(re, im) - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7e-302)
		tmp = 0.5 * (im / sqrt((0.5 * (hypot(re, im) - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -7e-302], N[(0.5 * N[(im / N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $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 < -7.0000000000000003e-302

   1. Initial program 27.5%

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

   3. Simplified 56.6%

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

      1. +-commutative 56.6%

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

      2. distribute-rgt-in 56.6%

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

      3. flip-+ 27.1%

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

   5. Applied egg-rr 27.1%

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

   6. Taylor expanded in re around 0 47.4%

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

      1. *-commutative 47.4%

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

   8. Simplified 47.4%

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

      1. expm1-log1p-u 46.9%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{im}^{2} \cdot 4}{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}}\right)\right)} \]

      2. expm1-udef 20.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{{im}^{2} \cdot 4}{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}}\right)} - 1\right)} \]

      3. associate-/l* 20.8%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{{im}^{2}}{\frac{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}{4}}}}\right)} - 1\right) \]

      4. sqrt-div 20.8%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{{im}^{2}}}{\sqrt{\frac{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}{4}}}}\right)} - 1\right) \]

      5. unpow2 20.8%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{im \cdot im}}}{\sqrt{\frac{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}{4}}}\right)} - 1\right) \]

      6. sqrt-prod 26.1%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}{\sqrt{\frac{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}{4}}}\right)} - 1\right) \]

      7. add-sqr-sqrt 27.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{im}}{\sqrt{\frac{\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2}{4}}}\right)} - 1\right) \]

      8. div-inv 27.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{\color{blue}{\left(\mathsf{hypot}\left(re, im\right) \cdot 2 - re \cdot 2\right) \cdot \frac{1}{4}}}}\right)} - 1\right) \]

      9. distribute-rgt-out-- 27.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{\color{blue}{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)} \cdot \frac{1}{4}}}\right)} - 1\right) \]

      10. metadata-eval 27.6%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot \color{blue}{0.25}}}\right)} - 1\right) \]

   10. Applied egg-rr 27.6%

       \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 50.5%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}}\right)\right)} \]

       2. expm1-log1p 52.1%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}}} \]

       3. *-commutative 52.1%

          \[\leadsto 0.5 \cdot \frac{im}{\sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}}} \]

       4. associate-*r* 52.1%

          \[\leadsto 0.5 \cdot \frac{im}{\sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}} \]

       5. metadata-eval 52.1%

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

   12. Simplified 52.1%

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

   if -7.0000000000000003e-302 < re

   1. Initial program 50.3%

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

   3. Simplified 100.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{-302}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -4.6e+75)
   (* 0.5 (sqrt (/ (- (pow im 2.0)) re)))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -4.6e+75) {
		tmp = 0.5 * sqrt((-pow(im, 2.0) / re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -4.6e+75) {
		tmp = 0.5 * Math.sqrt((-Math.pow(im, 2.0) / re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -4.6e+75:
		tmp = 0.5 * math.sqrt((-math.pow(im, 2.0) / re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -4.6e+75)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-(im ^ 2.0)) / re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.6e+75)
		tmp = 0.5 * sqrt((-(im ^ 2.0) / re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -4.6e+75], N[(0.5 * N[Sqrt[N[((-N[Power[im, 2.0], $MachinePrecision]) / re), $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 < -4.5999999999999997e75

   1. Initial program 12.1%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in re around -inf 45.3%

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

      1. mul-1-neg 45.3%

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

      2. distribute-neg-frac 45.3%

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

   6. Simplified 45.3%

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

   if -4.5999999999999997e75 < re

   1. Initial program 44.6%

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

   3. Simplified 87.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 4: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-{im}^{2}}{re}}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+148}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re -3e+21)
   (* 0.5 (sqrt (/ (- (pow im 2.0)) re)))
   (if (<= re 9e+148)
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= -3e+21) {
		tmp = 0.5 * sqrt((-pow(im, 2.0) / re));
	} else if (re <= 9e+148) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3d+21)) then
        tmp = 0.5d0 * sqrt((-(im ** 2.0d0) / re))
    else if (re <= 9d+148) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= -3e+21) {
		tmp = 0.5 * Math.sqrt((-Math.pow(im, 2.0) / re));
	} else if (re <= 9e+148) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= -3e+21:
		tmp = 0.5 * math.sqrt((-math.pow(im, 2.0) / re))
	elif re <= 9e+148:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= -3e+21)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-(im ^ 2.0)) / re)));
	elseif (re <= 9e+148)
		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

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3e+21)
		tmp = 0.5 * sqrt((-(im ^ 2.0) / re));
	elseif (re <= 9e+148)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, -3e+21], N[(0.5 * N[Sqrt[N[((-N[Power[im, 2.0], $MachinePrecision]) / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9e+148], 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 < -3e21

   1. Initial program 12.7%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in re around -inf 41.5%

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

      1. mul-1-neg 41.5%

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

      2. distribute-neg-frac 41.5%

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

   6. Simplified 41.5%

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

   if -3e21 < re < 8.99999999999999987e148

   1. Initial program 57.2%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in re around 0 36.8%

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

      1. distribute-lft-out 36.8%

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

      2. *-commutative 36.8%

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

   6. Simplified 36.8%

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

   if 8.99999999999999987e148 < re

   1. Initial program 4.2%

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

   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 75.1%

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

      1. *-commutative 75.1%

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

      2. unpow2 75.1%

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

      3. rem-square-sqrt 76.3%

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

   6. Simplified 76.3%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-{im}^{2}}{re}}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+148}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 8.5e-103)
   (* 0.5 (* 2.0 (sqrt re)))
   (* 0.5 (sqrt (* 2.0 (+ re im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 8.5e-103) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 8.5d-103) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 8.5e-103) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 8.5e-103:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 8.5e-103)
		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

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8.5e-103)
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 8.5e-103], 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 2 regimes

2. if im < 8.50000000000000032e-103

   1. Initial program 38.4%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in im around 0 27.7%

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

      1. *-commutative 27.7%

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

      2. unpow2 27.7%

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

      3. rem-square-sqrt 28.1%

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

   6. Simplified 28.1%

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

   if 8.50000000000000032e-103 < im

   1. Initial program 40.6%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in re around 0 72.9%

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

      1. distribute-lft-out 72.9%

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

      2. *-commutative 72.9%

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

   6. Simplified 72.9%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6: 59.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 4.1e-105) (* 0.5 (* 2.0 (sqrt re))) (* 0.5 (sqrt (* im 2.0)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 4.1e-105) {
		tmp = 0.5 * (2.0 * sqrt(re));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.1d-105) then
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 4.1e-105) {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 4.1e-105:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 4.1e-105)
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.1e-105)
		tmp = 0.5 * (2.0 * sqrt(re));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 4.1e-105], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.1000000000000003e-105

   1. Initial program 38.4%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in im around 0 27.7%

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

      1. *-commutative 27.7%

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

      2. unpow2 27.7%

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

      3. rem-square-sqrt 28.1%

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

   6. Simplified 28.1%

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

   if 4.1000000000000003e-105 < im

   1. Initial program 40.6%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in re around 0 71.4%

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

      1. *-commutative 71.4%

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

   6. Simplified 71.4%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7: 51.6% accurate, 2.0× speedup

Math

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (* 0.5 (sqrt (* im 2.0))))

C

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

Fortran

NOTE: im should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.2%

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

3. Simplified 78.8%

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

4. Taylor expanded in re around 0 27.1%

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

   1. *-commutative 27.1%

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

6. Simplified 27.1%

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

7. Final simplification 27.1%

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

Developer target: 48.0% 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 2023298 
(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)))))