math.sqrt on complex, real part

Percentage Accurate: 40.9% → 73.7%

Time: 10.7s

Alternatives: 7

Speedup: 1.4×

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: 40.9% 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: 73.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+50}:\\ \;\;\;\;{\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(im \cdot im\right)\right)}\right)}^{2} \cdot 0.5\\ \mathbf{elif}\;re \leq 10^{-171}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im\_m, im\_m, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.5e+50)
   (* (pow (exp (* 0.25 (+ (log (/ -1.0 re)) (log (* im im))))) 2.0) 0.5)
   (if (<= re 1e-171)
     (* 0.5 (sqrt (fma 2.0 (+ re im) (/ (* re re) im))))
     (if (<= re 2e+81)
       (* 0.5 (sqrt (* 2.0 (+ re (sqrt (fma im_m im_m (* re re)))))))
       (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.5e+50) {
		tmp = pow(exp((0.25 * (log((-1.0 / re)) + log((im * im))))), 2.0) * 0.5;
	} else if (re <= 1e-171) {
		tmp = 0.5 * sqrt(fma(2.0, (re + im), ((re * re) / im)));
	} else if (re <= 2e+81) {
		tmp = 0.5 * sqrt((2.0 * (re + sqrt(fma(im_m, im_m, (re * re))))));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.5e+50)
		tmp = Float64((exp(Float64(0.25 * Float64(log(Float64(-1.0 / re)) + log(Float64(im * im))))) ^ 2.0) * 0.5);
	elseif (re <= 1e-171)
		tmp = Float64(0.5 * sqrt(fma(2.0, Float64(re + im), Float64(Float64(re * re) / im))));
	elseif (re <= 2e+81)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + sqrt(fma(im_m, im_m, Float64(re * re)))))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.5e+50], N[(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] * 0.5), $MachinePrecision], If[LessEqual[re, 1e-171], 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, 2e+81], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(im$95$m * im$95$m + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.5e50

   1. Initial program 5.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 5.4

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

      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 5.4

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

      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 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 5.4

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

   7. Applied rewrites 5.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

   9. Applied rewrites 5.4%

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

   10. Taylor expanded in re around -inf

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

       1. lower-exp.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-log.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-log.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 71.9

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

   12. Applied rewrites 71.9%

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

   if -2.5e50 < re < 9.9999999999999998e-172

   1. Initial program 43.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + re \cdot \left(2 + \frac{re}{im}\right)}} \]
   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 47.7

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

   5. Applied rewrites 47.7%

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

   if 9.9999999999999998e-172 < re < 1.99999999999999984e81

   1. Initial program 74.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 74.4

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

      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 74.4

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

      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 74.4

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

   4. Applied rewrites 74.4%

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

   if 1.99999999999999984e81 < re

   1. Initial program 27.7%

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

   3. Taylor expanded in re around 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 89.8

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

   5. Applied rewrites 89.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+50}:\\ \;\;\;\;{\left(e^{0.25 \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left(im \cdot im\right)\right)}\right)}^{2} \cdot 0.5\\ \mathbf{elif}\;re \leq 10^{-171}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 10^{-171}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im\_m, im\_m, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.5e+50)
   (* 0.5 (sqrt (* (- im) (/ im re))))
   (if (<= re 1e-171)
     (* 0.5 (sqrt (fma 2.0 (+ re im) (/ (* re re) im))))
     (if (<= re 2e+81)
       (* 0.5 (sqrt (* 2.0 (+ re (sqrt (fma im_m im_m (* re re)))))))
       (sqrt re)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.5e+50) {
		tmp = 0.5 * sqrt((-im * (im / re)));
	} else if (re <= 1e-171) {
		tmp = 0.5 * sqrt(fma(2.0, (re + im), ((re * re) / im)));
	} else if (re <= 2e+81) {
		tmp = 0.5 * sqrt((2.0 * (re + sqrt(fma(im_m, im_m, (re * re))))));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.5e+50)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) * Float64(im / re))));
	elseif (re <= 1e-171)
		tmp = Float64(0.5 * sqrt(fma(2.0, Float64(re + im), Float64(Float64(re * re) / im))));
	elseif (re <= 2e+81)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + sqrt(fma(im_m, im_m, Float64(re * re)))))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.5e+50], N[(0.5 * N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e-171], 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, 2e+81], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(im$95$m * im$95$m + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.5e50

   1. Initial program 5.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 5.4

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

      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 5.4

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

      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 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in re around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 65.2

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

   7. Applied rewrites 65.2%

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

   9. Applied rewrites 67.2%

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

   if -2.5e50 < re < 9.9999999999999998e-172

   1. Initial program 43.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + re \cdot \left(2 + \frac{re}{im}\right)}} \]
   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 47.7

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

   5. Applied rewrites 47.7%

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

   if 9.9999999999999998e-172 < re < 1.99999999999999984e81

   1. Initial program 74.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 74.4

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

      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 74.4

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

      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 74.4

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

   4. Applied rewrites 74.4%

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

   if 1.99999999999999984e81 < re

   1. Initial program 27.7%

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

   3. Taylor expanded in re around 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 89.8

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

   5. Applied rewrites 89.8%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 10^{-171}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{+81}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.5e+50)
   (* 0.5 (sqrt (* (- im) (/ im re))))
   (if (<= re 5e+82) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.5e+50) {
		tmp = 0.5 * sqrt((-im * (im / re)));
	} else if (re <= 5e+82) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-2.5d+50)) then
        tmp = 0.5d0 * sqrt((-im * (im / re)))
    else if (re <= 5d+82) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -2.5e+50) {
		tmp = 0.5 * Math.sqrt((-im * (im / re)));
	} else if (re <= 5e+82) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -2.5e+50:
		tmp = 0.5 * math.sqrt((-im * (im / re)))
	elif re <= 5e+82:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.5e+50)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) * Float64(im / re))));
	elseif (re <= 5e+82)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -2.5e+50)
		tmp = 0.5 * sqrt((-im * (im / re)));
	elseif (re <= 5e+82)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.5e+50], N[(0.5 * N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+82], 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 < -2.5e50

   1. Initial program 5.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 5.4

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

      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 5.4

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

      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 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in re around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 65.2

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

   7. Applied rewrites 65.2%

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

   9. Applied rewrites 67.2%

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

   if -2.5e50 < re < 5.00000000000000015e82

   1. Initial program 52.1%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 42.5

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

   5. Applied rewrites 42.5%

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

   if 5.00000000000000015e82 < re

   1. Initial program 26.5%

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

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   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 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{-re}}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.5e+50)
   (* 0.5 (sqrt (/ (* im im) (- re))))
   (if (<= re 5e+82) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.5e+50) {
		tmp = 0.5 * sqrt(((im * im) / -re));
	} else if (re <= 5e+82) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-2.5d+50)) then
        tmp = 0.5d0 * sqrt(((im * im) / -re))
    else if (re <= 5d+82) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -2.5e+50) {
		tmp = 0.5 * Math.sqrt(((im * im) / -re));
	} else if (re <= 5e+82) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -2.5e+50:
		tmp = 0.5 * math.sqrt(((im * im) / -re))
	elif re <= 5e+82:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.5e+50)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * im) / Float64(-re))));
	elseif (re <= 5e+82)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -2.5e+50)
		tmp = 0.5 * sqrt(((im * im) / -re));
	elseif (re <= 5e+82)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.5e+50], N[(0.5 * N[Sqrt[N[(N[(im * im), $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5e+82], 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 < -2.5e50

   1. Initial program 5.4%

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

   3. Taylor expanded in re around -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 65.2

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

   5. Applied rewrites 65.2%

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

   if -2.5e50 < re < 5.00000000000000015e82

   1. Initial program 52.1%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 42.5

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

   5. Applied rewrites 42.5%

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

   if 5.00000000000000015e82 < re

   1. Initial program 26.5%

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

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   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 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+50}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{-re}}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{+82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{+223}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{-131}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2e+223)
   (* 0.5 (sqrt 0.0))
   (if (<= re 2e-131) (* 0.5 (sqrt (* 2.0 im))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2e+223) {
		tmp = 0.5 * sqrt(0.0);
	} else if (re <= 2e-131) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-2d+223)) then
        tmp = 0.5d0 * sqrt(0.0d0)
    else if (re <= 2d-131) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -2e+223) {
		tmp = 0.5 * Math.sqrt(0.0);
	} else if (re <= 2e-131) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -2e+223:
		tmp = 0.5 * math.sqrt(0.0)
	elif re <= 2e-131:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2e+223)
		tmp = Float64(0.5 * sqrt(0.0));
	elseif (re <= 2e-131)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -2e+223)
		tmp = 0.5 * sqrt(0.0);
	elseif (re <= 2e-131)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2e+223], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2e-131], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.00000000000000009e223

   1. Initial program 2.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \left(\frac{\color{blue}{0} \cdot im}{im} \cdot \frac{{re}^{2}}{im}\right)} \]

      6. mul0-lft N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \left(\frac{\color{blue}{0}}{im} \cdot \frac{{re}^{2}}{im}\right)} \]

      7. times-frac N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \color{blue}{\frac{0 \cdot {re}^{2}}{im \cdot im}}} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \frac{\color{blue}{0}}{im \cdot im}} \]

      9. mul0-lft N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \frac{\color{blue}{0 \cdot re}}{im \cdot im}} \]

      10. mul0-lft N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \frac{\color{blue}{\left(0 \cdot {re}^{2}\right)} \cdot re}{im \cdot im}} \]

      11. metadata-eval N/A

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

      12. distribute-lft1-in N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

   7. Applied rewrites 48.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

   if -2.00000000000000009e223 < re < 2e-131

   1. Initial program 37.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 35.5

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

   5. Applied rewrites 35.5%

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

   if 2e-131 < re

   1. Initial program 46.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 73.9

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

   5. Applied rewrites 73.9%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{+223}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{elif}\;re \leq 2 \cdot 10^{-131}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-308}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2e-308) (* 0.5 (sqrt 0.0)) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2e-308) {
		tmp = 0.5 * sqrt(0.0);
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-2d-308)) then
        tmp = 0.5d0 * sqrt(0.0d0)
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -2e-308) {
		tmp = 0.5 * Math.sqrt(0.0);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -2e-308:
		tmp = 0.5 * math.sqrt(0.0)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2e-308)
		tmp = Float64(0.5 * sqrt(0.0));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -2e-308)
		tmp = 0.5 * sqrt(0.0);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2e-308], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.9999999999999998e-308

   1. Initial program 23.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 22.0%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \left(\frac{\color{blue}{0} \cdot im}{im} \cdot \frac{{re}^{2}}{im}\right)} \]

      6. mul0-lft N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \left(\frac{\color{blue}{0}}{im} \cdot \frac{{re}^{2}}{im}\right)} \]

      7. times-frac N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \color{blue}{\frac{0 \cdot {re}^{2}}{im \cdot im}}} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \frac{\color{blue}{0}}{im \cdot im}} \]

      9. mul0-lft N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \frac{\color{blue}{0 \cdot re}}{im \cdot im}} \]

      10. mul0-lft N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{4 \cdot \frac{\color{blue}{\left(0 \cdot {re}^{2}\right)} \cdot re}{im \cdot im}} \]

      11. metadata-eval N/A

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

      12. distribute-lft1-in N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

   7. Applied rewrites 10.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0}} \]

   if -1.9999999999999998e-308 < re

   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 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 51.6

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

   5. Applied rewrites 51.6%

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

Alternative 7: 26.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sqrt{re} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (sqrt re))

C

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

Fortran

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

Java

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

Python

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

Julia

im_m = abs(im)
function code(re, im_m)
	return sqrt(re)
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 37.8%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
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 26.4

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

5. Applied rewrites 26.4%

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

Developer Target 1: 48.0% 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 2024214 
(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)))))