math.sqrt on complex, real part

Percentage Accurate: 41.6% → 82.9%

Time: 10.2s

Alternatives: 7

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: 41.6% 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: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8600000000000:\\ \;\;\;\;0.5 \cdot {\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(im \cdot im\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8600000000000.0)
   (* 0.5 (pow (exp (* 0.25 (+ (log (/ -1.0 re)) (log (* im im))))) 2.0))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8600000000000.0) {
		tmp = 0.5 * pow(exp((0.25 * (log((-1.0 / re)) + log((im * im))))), 2.0);
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8600000000000.0) {
		tmp = 0.5 * Math.pow(Math.exp((0.25 * (Math.log((-1.0 / re)) + Math.log((im * im))))), 2.0);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8600000000000.0:
		tmp = 0.5 * math.pow(math.exp((0.25 * (math.log((-1.0 / re)) + math.log((im * im))))), 2.0)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8600000000000.0)
		tmp = Float64(0.5 * (exp(Float64(0.25 * Float64(log(Float64(-1.0 / re)) + log(Float64(im * im))))) ^ 2.0));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8600000000000.0)
		tmp = 0.5 * (exp((0.25 * (log((-1.0 / re)) + log((im * im))))) ^ 2.0);
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8600000000000.0], N[(0.5 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] + N[Log[N[(im * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -8.6e12

   1. Initial program 8.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 7.6

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

   5. Applied rewrites 7.6%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {\left({\mathsf{Rewrite=>}\left(lower-*.f64, \left(2 \cdot im\right)\right)}^{\color{blue}{\frac{1}{4}}}\right)}^{2} \]

   7. Applied rewrites 7.6%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot im\right)}^{0.25}\right)}^{2}} \]

   8. Taylor expanded in re around -inf

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

      1. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot {\color{blue}{\left(e^{\frac{1}{4} \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}}^{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot {\left(e^{\color{blue}{\frac{1}{4} \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}}\right)}^{2} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot {\left(e^{\frac{1}{4} \cdot \color{blue}{\left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}}\right)}^{2} \]

      4. lower-log.f64 N/A

         \[\leadsto \frac{1}{2} \cdot {\left(e^{\frac{1}{4} \cdot \left(\color{blue}{\log \left(\frac{-1}{re}\right)} + \log \left({im}^{2}\right)\right)}\right)}^{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot {\left(e^{\frac{1}{4} \cdot \left(\log \color{blue}{\left(\frac{-1}{re}\right)} + \log \left({im}^{2}\right)\right)}\right)}^{2} \]

      6. lower-log.f64 N/A

         \[\leadsto \frac{1}{2} \cdot {\left(e^{\frac{1}{4} \cdot \left(\log \left(\frac{-1}{re}\right) + \color{blue}{\log \left({im}^{2}\right)}\right)}\right)}^{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot {\left(e^{\frac{1}{4} \cdot \left(\log \left(\frac{-1}{re}\right) + \log \color{blue}{\left(im \cdot im\right)}\right)}\right)}^{2} \]

      8. lower-*.f64 60.2

         \[\leadsto 0.5 \cdot {\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \color{blue}{\left(im \cdot im\right)}\right)}\right)}^{2} \]

   10. Applied rewrites 60.2%

       \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(im \cdot im\right)\right)}\right)}}^{2} \]

   if -8.6e12 < re

   1. Initial program 54.2%

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

      1. lift-sqrt.f64 N/A

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

      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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 94.6

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

   4. Applied rewrites 94.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8600000000000:\\ \;\;\;\;0.5 \cdot {\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(im \cdot im\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7000000000000.0)
   (* 0.5 (sqrt (* im (/ (- im) re))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7000000000000.0) {
		tmp = 0.5 * sqrt((im * (-im / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7000000000000.0) {
		tmp = 0.5 * Math.sqrt((im * (-im / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7000000000000.0:
		tmp = 0.5 * math.sqrt((im * (-im / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7000000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(Float64(-im) / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7000000000000.0)
		tmp = 0.5 * sqrt((im * (-im / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7000000000000.0], N[(0.5 * N[Sqrt[N[(im * N[((-im) / re), $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 < -7e12

   1. Initial program 8.0%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.7

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 57.7%

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

   if -7e12 < re

   1. Initial program 54.2%

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

      1. lift-sqrt.f64 N/A

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

      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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 94.6

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

   4. Applied rewrites 94.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-139}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7000000000000.0)
   (* 0.5 (sqrt (* im (/ (- im) re))))
   (if (<= re 1.4e-139)
     (* 0.5 (sqrt (fma 2.0 (+ re im) (/ (* re re) im))))
     (if (<= re 8.5e+93)
       (* 0.5 (sqrt (* 2.0 (+ re (sqrt (fma im im (* re re)))))))
       (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7000000000000.0) {
		tmp = 0.5 * sqrt((im * (-im / re)));
	} else if (re <= 1.4e-139) {
		tmp = 0.5 * sqrt(fma(2.0, (re + im), ((re * re) / im)));
	} else if (re <= 8.5e+93) {
		tmp = 0.5 * sqrt((2.0 * (re + sqrt(fma(im, im, (re * re))))));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7000000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(Float64(-im) / re))));
	elseif (re <= 1.4e-139)
		tmp = Float64(0.5 * sqrt(fma(2.0, Float64(re + im), Float64(Float64(re * re) / im))));
	elseif (re <= 8.5e+93)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + sqrt(fma(im, im, Float64(re * re)))))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7000000000000.0], N[(0.5 * N[Sqrt[N[(im * N[((-im) / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.4e-139], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e+93], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7e12

   1. Initial program 8.0%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.7

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 57.7%

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

   if -7e12 < re < 1.3999999999999999e-139

   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. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 45.6

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

   5. Applied rewrites 45.6%

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

   if 1.3999999999999999e-139 < re < 8.5000000000000005e93

   1. Initial program 88.0%

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

      1. lift-*.f64 N/A

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

      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 88.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 88.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 88.0

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

   4. Applied rewrites 88.0%

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

   if 8.5000000000000005e93 < 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 84.6

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

   5. Applied rewrites 84.6%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-139}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7000000000000.0)
   (* 0.5 (sqrt (* im (/ (- im) re))))
   (if (<= re 2.3e-44) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7000000000000.0) {
		tmp = 0.5 * sqrt((im * (-im / re)));
	} else if (re <= 2.3e-44) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} 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 <= (-7000000000000.0d0)) then
        tmp = 0.5d0 * sqrt((im * (-im / re)))
    else if (re <= 2.3d-44) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7000000000000.0) {
		tmp = 0.5 * Math.sqrt((im * (-im / re)));
	} else if (re <= 2.3e-44) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7000000000000.0:
		tmp = 0.5 * math.sqrt((im * (-im / re)))
	elif re <= 2.3e-44:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7000000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(Float64(-im) / re))));
	elseif (re <= 2.3e-44)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7000000000000.0)
		tmp = 0.5 * sqrt((im * (-im / re)));
	elseif (re <= 2.3e-44)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7000000000000.0], N[(0.5 * N[Sqrt[N[(im * N[((-im) / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.3e-44], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7e12

   1. Initial program 8.0%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.7

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 57.7%

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

   if -7e12 < re < 2.29999999999999998e-44

   1. Initial program 59.7%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 45.3

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

   5. Applied rewrites 45.3%

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

   if 2.29999999999999998e-44 < re

   1. Initial program 43.2%

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

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im \cdot im}{re}}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7000000000000.0)
   (* 0.5 (sqrt (- (/ (* im im) re))))
   (if (<= re 2.3e-44) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7000000000000.0) {
		tmp = 0.5 * sqrt(-((im * im) / re));
	} else if (re <= 2.3e-44) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} 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 <= (-7000000000000.0d0)) then
        tmp = 0.5d0 * sqrt(-((im * im) / re))
    else if (re <= 2.3d-44) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7000000000000.0) {
		tmp = 0.5 * Math.sqrt(-((im * im) / re));
	} else if (re <= 2.3e-44) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7000000000000.0:
		tmp = 0.5 * math.sqrt(-((im * im) / re))
	elif re <= 2.3e-44:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7000000000000.0)
		tmp = Float64(0.5 * sqrt(Float64(-Float64(Float64(im * im) / re))));
	elseif (re <= 2.3e-44)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7000000000000.0)
		tmp = 0.5 * sqrt(-((im * im) / re));
	elseif (re <= 2.3e-44)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7000000000000.0], N[(0.5 * N[Sqrt[(-N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.3e-44], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7e12

   1. Initial program 8.0%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.7

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

   5. Applied rewrites 47.7%

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

   if -7e12 < re < 2.29999999999999998e-44

   1. Initial program 59.7%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 45.3

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

   5. Applied rewrites 45.3%

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

   if 2.29999999999999998e-44 < re

   1. Initial program 43.2%

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

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7000000000000:\\ \;\;\;\;0.5 \cdot \sqrt{-\frac{im \cdot im}{re}}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-52) (* 0.5 (sqrt (* 2.0 im))) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e-52) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= 1d-52) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1e-52) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1e-52:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e-52)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1e-52)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e-52], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1e-52

   1. Initial program 41.9%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 32.0

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

   5. Applied rewrites 32.0%

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

   if 1e-52 < re

   1. Initial program 44.9%

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

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.1% 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 42.6%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
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 26.6

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

5. Applied rewrites 26.6%

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

Developer Target 1: 48.5% 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 2024234 
(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)))))