math.sqrt on complex, real part

Percentage Accurate: 41.3% → 82.6%

Time: 8.4s

Alternatives: 9

Speedup: 1.7×

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: 41.3% 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: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.8 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\mathsf{hypot}\left(im, re\right) + re} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.8e+100)
   (* (sqrt (* (/ (- im) re) im)) 0.5)
   (* (sqrt 2.0) (* (sqrt (+ (hypot im re) re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.8e+100) {
		tmp = sqrt(((-im / re) * im)) * 0.5;
	} else {
		tmp = sqrt(2.0) * (sqrt((hypot(im, re) + re)) * 0.5);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.8e+100) {
		tmp = Math.sqrt(((-im / re) * im)) * 0.5;
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt((Math.hypot(im, re) + re)) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.8e+100:
		tmp = math.sqrt(((-im / re) * im)) * 0.5
	else:
		tmp = math.sqrt(2.0) * (math.sqrt((math.hypot(im, re) + re)) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.8e+100)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im) / re) * im)) * 0.5);
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(hypot(im, re) + re)) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.8e+100)
		tmp = sqrt(((-im / re) * im)) * 0.5;
	else
		tmp = sqrt(2.0) * (sqrt((hypot(im, re) + re)) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.8e+100], N[(N[Sqrt[N[(N[((-im) / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] + re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.79999999999999963e100

   1. Initial program 5.5%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -3.79999999999999963e100 < re

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 90.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.8 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\mathsf{hypot}\left(im, re\right) + re} \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.8 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.8e+100)
   (* (sqrt (* (/ (- im) re) im)) 0.5)
   (* (sqrt (* (+ (hypot im re) re) 2.0)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.8e+100) {
		tmp = sqrt(((-im / re) * im)) * 0.5;
	} else {
		tmp = sqrt(((hypot(im, re) + re) * 2.0)) * 0.5;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.8e+100) {
		tmp = Math.sqrt(((-im / re) * im)) * 0.5;
	} else {
		tmp = Math.sqrt(((Math.hypot(im, re) + re) * 2.0)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.8e+100:
		tmp = math.sqrt(((-im / re) * im)) * 0.5
	else:
		tmp = math.sqrt(((math.hypot(im, re) + re) * 2.0)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.8e+100)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im) / re) * im)) * 0.5);
	else
		tmp = Float64(sqrt(Float64(Float64(hypot(im, re) + re) * 2.0)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.8e+100)
		tmp = sqrt(((-im / re) * im)) * 0.5;
	else
		tmp = sqrt(((hypot(im, re) + re) * 2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.8e+100], N[(N[Sqrt[N[(N[((-im) / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.79999999999999963e100

   1. Initial program 5.5%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -3.79999999999999963e100 < re

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 48.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-hypot.f64 90.1

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

   4. Applied rewrites 90.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.8 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{im + re} \cdot 0.5\right) \cdot \sqrt{2}\\ \mathbf{elif}\;re \leq 10^{+125}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.45e+95)
   (* (sqrt (* (/ (- im) re) im)) 0.5)
   (if (<= re 1.55e-131)
     (* (* (sqrt (+ im re)) 0.5) (sqrt 2.0))
     (if (<= re 1e+125)
       (* (sqrt (* (+ (sqrt (fma re re (* im im))) re) 2.0)) 0.5)
       (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+95) {
		tmp = sqrt(((-im / re) * im)) * 0.5;
	} else if (re <= 1.55e-131) {
		tmp = (sqrt((im + re)) * 0.5) * sqrt(2.0);
	} else if (re <= 1e+125) {
		tmp = sqrt(((sqrt(fma(re, re, (im * im))) + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.45e+95)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im) / re) * im)) * 0.5);
	elseif (re <= 1.55e-131)
		tmp = Float64(Float64(sqrt(Float64(im + re)) * 0.5) * sqrt(2.0));
	elseif (re <= 1e+125)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(re, re, Float64(im * im))) + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.45e+95], N[(N[Sqrt[N[(N[((-im) / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.55e-131], N[(N[(N[Sqrt[N[(im + re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+125], N[(N[Sqrt[N[(N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.4499999999999999e95

   1. Initial program 5.6%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -2.4499999999999999e95 < re < 1.5500000000000001e-131

   1. Initial program 47.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 83.5%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 42.6

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

   7. Applied rewrites 42.6%

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

   if 1.5500000000000001e-131 < re < 9.9999999999999992e124

   1. Initial program 84.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.0

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

   4. Applied rewrites 84.0%

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

   if 9.9999999999999992e124 < re

   1. Initial program 13.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 82.0

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 82.0%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{-131}:\\ \;\;\;\;\left(\sqrt{im + re} \cdot 0.5\right) \cdot \sqrt{2}\\ \mathbf{elif}\;re \leq 10^{+125}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 130000000:\\ \;\;\;\;\left(\sqrt{im + re} \cdot 0.5\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.45e+95)
   (* (sqrt (* (/ (- im) re) im)) 0.5)
   (if (<= re 130000000.0) (* (* (sqrt (+ im re)) 0.5) (sqrt 2.0)) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+95) {
		tmp = sqrt(((-im / re) * im)) * 0.5;
	} else if (re <= 130000000.0) {
		tmp = (sqrt((im + re)) * 0.5) * sqrt(2.0);
	} else {
		tmp = 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+95)) then
        tmp = sqrt(((-im / re) * im)) * 0.5d0
    else if (re <= 130000000.0d0) then
        tmp = (sqrt((im + re)) * 0.5d0) * sqrt(2.0d0)
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+95) {
		tmp = Math.sqrt(((-im / re) * im)) * 0.5;
	} else if (re <= 130000000.0) {
		tmp = (Math.sqrt((im + re)) * 0.5) * Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.45e+95:
		tmp = math.sqrt(((-im / re) * im)) * 0.5
	elif re <= 130000000.0:
		tmp = (math.sqrt((im + re)) * 0.5) * math.sqrt(2.0)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.45e+95)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im) / re) * im)) * 0.5);
	elseif (re <= 130000000.0)
		tmp = Float64(Float64(sqrt(Float64(im + re)) * 0.5) * sqrt(2.0));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.45e+95)
		tmp = sqrt(((-im / re) * im)) * 0.5;
	elseif (re <= 130000000.0)
		tmp = (sqrt((im + re)) * 0.5) * sqrt(2.0);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.45e+95], N[(N[Sqrt[N[(N[((-im) / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 130000000.0], N[(N[(N[Sqrt[N[(im + re), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.4499999999999999e95

   1. Initial program 5.6%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -2.4499999999999999e95 < re < 1.3e8

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 86.5%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 40.1

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

   7. Applied rewrites 40.1%

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

   if 1.3e8 < re

   1. Initial program 39.0%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 77.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 77.8%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 130000000:\\ \;\;\;\;\left(\sqrt{im + re} \cdot 0.5\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.04 \cdot 10^{-120}:\\ \;\;\;\;\left(\sqrt{im} \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.45e+95)
   (* (sqrt (* (/ (- im) re) im)) 0.5)
   (if (<= re 1.04e-120) (* (* (sqrt im) (sqrt 2.0)) 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+95) {
		tmp = sqrt(((-im / re) * im)) * 0.5;
	} else if (re <= 1.04e-120) {
		tmp = (sqrt(im) * sqrt(2.0)) * 0.5;
	} else {
		tmp = 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+95)) then
        tmp = sqrt(((-im / re) * im)) * 0.5d0
    else if (re <= 1.04d-120) then
        tmp = (sqrt(im) * sqrt(2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+95) {
		tmp = Math.sqrt(((-im / re) * im)) * 0.5;
	} else if (re <= 1.04e-120) {
		tmp = (Math.sqrt(im) * Math.sqrt(2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.45e+95:
		tmp = math.sqrt(((-im / re) * im)) * 0.5
	elif re <= 1.04e-120:
		tmp = (math.sqrt(im) * math.sqrt(2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.45e+95)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im) / re) * im)) * 0.5);
	elseif (re <= 1.04e-120)
		tmp = Float64(Float64(sqrt(im) * sqrt(2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.45e+95)
		tmp = sqrt(((-im / re) * im)) * 0.5;
	elseif (re <= 1.04e-120)
		tmp = (sqrt(im) * sqrt(2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.45e+95], N[(N[Sqrt[N[(N[((-im) / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.04e-120], N[(N[(N[Sqrt[im], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.4499999999999999e95

   1. Initial program 5.6%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -2.4499999999999999e95 < re < 1.03999999999999994e-120

   1. Initial program 47.7%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 41.8

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

   5. Applied rewrites 41.8%

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

   if 1.03999999999999994e-120 < re

   1. Initial program 49.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 71.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 71.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.04 \cdot 10^{-120}:\\ \;\;\;\;\left(\sqrt{im} \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.04 \cdot 10^{-120}:\\ \;\;\;\;\left(\sqrt{im} \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.45e+188)
   (* (sqrt (* (+ (- re) re) 2.0)) 0.5)
   (if (<= re 1.04e-120) (* (* (sqrt im) (sqrt 2.0)) 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+188) {
		tmp = sqrt(((-re + re) * 2.0)) * 0.5;
	} else if (re <= 1.04e-120) {
		tmp = (sqrt(im) * sqrt(2.0)) * 0.5;
	} else {
		tmp = 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+188)) then
        tmp = sqrt(((-re + re) * 2.0d0)) * 0.5d0
    else if (re <= 1.04d-120) then
        tmp = (sqrt(im) * sqrt(2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.45e+188) {
		tmp = Math.sqrt(((-re + re) * 2.0)) * 0.5;
	} else if (re <= 1.04e-120) {
		tmp = (Math.sqrt(im) * Math.sqrt(2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.45e+188:
		tmp = math.sqrt(((-re + re) * 2.0)) * 0.5
	elif re <= 1.04e-120:
		tmp = (math.sqrt(im) * math.sqrt(2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.45e+188)
		tmp = Float64(sqrt(Float64(Float64(Float64(-re) + re) * 2.0)) * 0.5);
	elseif (re <= 1.04e-120)
		tmp = Float64(Float64(sqrt(im) * sqrt(2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.45e+188)
		tmp = sqrt(((-re + re) * 2.0)) * 0.5;
	elseif (re <= 1.04e-120)
		tmp = (sqrt(im) * sqrt(2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.45e+188], N[(N[Sqrt[N[(N[((-re) + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.04e-120], N[(N[(N[Sqrt[im], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.45e188

   1. Initial program 2.3%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(re\right)\right)} + re\right)} \]

      2. lower-neg.f64 39.6

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

   5. Applied rewrites 39.6%

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

   if -2.45e188 < re < 1.03999999999999994e-120

   1. Initial program 43.4%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 38.4

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

   5. Applied rewrites 38.4%

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

   if 1.03999999999999994e-120 < re

   1. Initial program 49.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 71.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 71.8%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.45 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.04 \cdot 10^{-120}:\\ \;\;\;\;\left(\sqrt{im} \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 130000000:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.3e+161)
   (* (sqrt (* (+ (- re) re) 2.0)) 0.5)
   (if (<= re 130000000.0) (* (sqrt (* (+ im re) 2.0)) 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.3e+161) {
		tmp = sqrt(((-re + re) * 2.0)) * 0.5;
	} else if (re <= 130000000.0) {
		tmp = sqrt(((im + re) * 2.0)) * 0.5;
	} else {
		tmp = 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 <= (-1.3d+161)) then
        tmp = sqrt(((-re + re) * 2.0d0)) * 0.5d0
    else if (re <= 130000000.0d0) then
        tmp = sqrt(((im + re) * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.3e+161) {
		tmp = Math.sqrt(((-re + re) * 2.0)) * 0.5;
	} else if (re <= 130000000.0) {
		tmp = Math.sqrt(((im + re) * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.3e+161:
		tmp = math.sqrt(((-re + re) * 2.0)) * 0.5
	elif re <= 130000000.0:
		tmp = math.sqrt(((im + re) * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.3e+161)
		tmp = Float64(sqrt(Float64(Float64(Float64(-re) + re) * 2.0)) * 0.5);
	elseif (re <= 130000000.0)
		tmp = Float64(sqrt(Float64(Float64(im + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.3e+161)
		tmp = sqrt(((-re + re) * 2.0)) * 0.5;
	elseif (re <= 130000000.0)
		tmp = sqrt(((im + re) * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.3e+161], N[(N[Sqrt[N[(N[((-re) + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 130000000.0], N[(N[Sqrt[N[(N[(im + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2999999999999999e161

   1. Initial program 2.3%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(re\right)\right)} + re\right)} \]

      2. lower-neg.f64 35.1

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

   5. Applied rewrites 35.1%

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

   if -1.2999999999999999e161 < re < 1.3e8

   1. Initial program 51.2%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 38.7

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

   5. Applied rewrites 38.7%

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

   if 1.3e8 < re

   1. Initial program 39.0%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 77.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 77.8%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 130000000:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.04 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.04e-120) (* (sqrt (* im 2.0)) 0.5) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.04e-120) {
		tmp = sqrt((im * 2.0)) * 0.5;
	} else {
		tmp = 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 <= 1.04d-120) then
        tmp = sqrt((im * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.04e-120) {
		tmp = Math.sqrt((im * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.04e-120:
		tmp = math.sqrt((im * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.04e-120)
		tmp = Float64(sqrt(Float64(im * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.04e-120)
		tmp = sqrt((im * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.04e-120], N[(N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.03999999999999994e-120

   1. Initial program 36.1%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 32.4

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

   5. Applied rewrites 32.4%

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

   if 1.03999999999999994e-120 < re

   1. Initial program 49.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 71.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 71.8%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.04 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt re))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(re)
end function

Java

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

Python

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

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 41.6%

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

3. Taylor expanded in re around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

      \[\leadsto \color{blue}{\sqrt{re}} \]

   7. lower-sqrt.f64 31.4

      \[\leadsto \color{blue}{\sqrt{re}} \]

5. Applied rewrites 31.4%

   \[\leadsto \color{blue}{\sqrt{re}} \]
6. Add Preprocessing

Developer Target 1: 48.6% accurate, 0.6× 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 2024276 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< re 0) (* 1/2 (* (sqrt 2) (sqrt (/ (* im im) (- (modulus re im) re))))) (* 1/2 (sqrt (* 2 (+ (modulus re im) re))))))

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