math.sqrt on complex, real part

Percentage Accurate: 42.2% → 87.9%

Time: 11.0s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 42.2% 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: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -6.5 \cdot 10^{-47}:\\ \;\;\;\;{2}^{0.25} \cdot \left(\left(0.5 \cdot \left(im\_m \cdot {\left(\frac{-0.5}{re}\right)}^{0.5}\right)\right) \cdot {2}^{0.25}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -6.5e-47)
   (* (pow 2.0 0.25) (* (* 0.5 (* im_m (pow (/ -0.5 re) 0.5))) (pow 2.0 0.25)))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -6.5e-47) {
		tmp = pow(2.0, 0.25) * ((0.5 * (im_m * pow((-0.5 / re), 0.5))) * pow(2.0, 0.25));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -6.5e-47) {
		tmp = Math.pow(2.0, 0.25) * ((0.5 * (im_m * Math.pow((-0.5 / re), 0.5))) * Math.pow(2.0, 0.25));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -6.5e-47:
		tmp = math.pow(2.0, 0.25) * ((0.5 * (im_m * math.pow((-0.5 / re), 0.5))) * math.pow(2.0, 0.25))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -6.5e-47)
		tmp = Float64((2.0 ^ 0.25) * Float64(Float64(0.5 * Float64(im_m * (Float64(-0.5 / re) ^ 0.5))) * (2.0 ^ 0.25)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im_m)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -6.5e-47)
		tmp = (2.0 ^ 0.25) * ((0.5 * (im_m * ((-0.5 / re) ^ 0.5))) * (2.0 ^ 0.25));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -6.5e-47], N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(N[(0.5 * N[(im$95$m * N[Power[N[(-0.5 / re), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.5000000000000004e-47

   1. Initial program 10.3%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} + re\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)}\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} + re\right), \frac{1}{2}\right), \left(\color{blue}{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(re + \sqrt{re \cdot re + im \cdot im}\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      9. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{2}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 34.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   4. Applied egg-rr 34.1%

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

   5. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{{im}^{2}}{re}\right)}, \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{{im}^{2}}{re}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. *-lowering-*.f64 52.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   7. Simplified 52.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{im \cdot im}{re} \cdot \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{\frac{re}{im \cdot im}} \cdot \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(\frac{1}{re} \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{re} \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{re} \cdot \left(\frac{-1}{2} \cdot \left(im \cdot im\right)\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{re}\right), \left(\frac{-1}{2} \cdot \left(im \cdot im\right)\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, re\right), \left(\frac{-1}{2} \cdot \left(im \cdot im\right)\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, re\right), \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, re\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{2}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      10. *-lowering-*.f64 52.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   9. Applied egg-rr 52.1%

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

       1. associate-*r* N/A

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

       2. pow1/2 N/A

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

       3. sqr-pow N/A

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

       4. associate-*r* N/A

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

       5. *-lowering-*.f64 N/A

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

   11. Applied egg-rr 54.9%

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

   if -6.5000000000000004e-47 < re

   1. Initial program 51.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} + re\right), 2\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(re + \sqrt{re \cdot re + im \cdot im}\right), 2\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right), 2\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 93.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), 2\right)\right)\right) \]

   4. Applied egg-rr 93.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.5 \cdot 10^{-47}:\\ \;\;\;\;{2}^{0.25} \cdot \left(\left(0.5 \cdot \left(im \cdot {\left(\frac{-0.5}{re}\right)}^{0.5}\right)\right) \cdot {2}^{0.25}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{-51}:\\ \;\;\;\;\left(im\_m \cdot \sqrt{\frac{-0.5}{re}}\right) \cdot \left(0.5 \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -8e-51)
   (* (* im_m (sqrt (/ -0.5 re))) (* 0.5 (sqrt 2.0)))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -8e-51) {
		tmp = (im_m * sqrt((-0.5 / re))) * (0.5 * sqrt(2.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -8e-51) {
		tmp = (im_m * Math.sqrt((-0.5 / re))) * (0.5 * Math.sqrt(2.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -8e-51:
		tmp = (im_m * math.sqrt((-0.5 / re))) * (0.5 * math.sqrt(2.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -8e-51)
		tmp = Float64(Float64(im_m * sqrt(Float64(-0.5 / re))) * Float64(0.5 * sqrt(2.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im_m)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -8e-51)
		tmp = (im_m * sqrt((-0.5 / re))) * (0.5 * sqrt(2.0));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -8e-51], N[(N[(im$95$m * N[Sqrt[N[(-0.5 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -8.0000000000000001e-51

   1. Initial program 10.3%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} + re\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)}\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} + re\right), \frac{1}{2}\right), \left(\color{blue}{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(re + \sqrt{re \cdot re + im \cdot im}\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      9. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{2}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 34.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   4. Applied egg-rr 34.1%

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

   5. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{{im}^{2}}{re}\right)}, \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{{im}^{2}}{re}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. *-lowering-*.f64 52.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   7. Simplified 52.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{im \cdot im}{re} \cdot \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{\frac{re}{im \cdot im}} \cdot \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(\frac{1}{re} \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2}\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{re} \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{re} \cdot \left(\frac{-1}{2} \cdot \left(im \cdot im\right)\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{re}\right), \left(\frac{-1}{2} \cdot \left(im \cdot im\right)\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, re\right), \left(\frac{-1}{2} \cdot \left(im \cdot im\right)\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, re\right), \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, re\right), \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{2}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      10. *-lowering-*.f64 52.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2}\right)\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   9. Applied egg-rr 52.1%

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

       1. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re} \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right)}\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\left(\left(im \cdot im\right) \cdot \frac{-1}{2}\right) \cdot \frac{1}{re}}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\left(im \cdot im\right) \cdot \left(\frac{-1}{2} \cdot \frac{1}{re}\right)}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       4. sqrt-prod N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{im \cdot im} \cdot \sqrt{\frac{-1}{2} \cdot \frac{1}{re}}\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       5. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(im \cdot im\right)}^{\frac{1}{2}} \cdot \sqrt{\frac{-1}{2} \cdot \frac{1}{re}}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left({im}^{2}\right)}^{\frac{1}{2}} \cdot \sqrt{\frac{-1}{2} \cdot \frac{1}{re}}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       7. pow-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot \frac{1}{2}\right)} \cdot \sqrt{\frac{-1}{2} \cdot \frac{1}{re}}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{1} \cdot \sqrt{\frac{-1}{2} \cdot \frac{1}{re}}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       9. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(im \cdot \sqrt{\frac{-1}{2} \cdot \frac{1}{re}}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \left(\sqrt{\frac{-1}{2} \cdot \frac{1}{re}}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       11. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\left(\frac{-1}{2} \cdot \frac{1}{re}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       12. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{2}}{re}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       13. /-lowering-/.f64 54.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, re\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   11. Applied egg-rr 54.8%

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

   if -8.0000000000000001e-51 < re

   1. Initial program 51.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} + re\right), 2\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(re + \sqrt{re \cdot re + im \cdot im}\right), 2\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right), 2\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 93.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), 2\right)\right)\right) \]

   4. Applied egg-rr 93.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8 \cdot 10^{-51}:\\ \;\;\;\;\left(im \cdot \sqrt{\frac{-0.5}{re}}\right) \cdot \left(0.5 \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{0 - \frac{im\_m \cdot im\_m}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.6e-39)
   (* 0.5 (sqrt (- 0.0 (/ (* im_m im_m) re))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.6e-39) {
		tmp = 0.5 * sqrt((0.0 - ((im_m * im_m) / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -3.6e-39) {
		tmp = 0.5 * Math.sqrt((0.0 - ((im_m * im_m) / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.6e-39:
		tmp = 0.5 * math.sqrt((0.0 - ((im_m * im_m) / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.6e-39)
		tmp = Float64(0.5 * sqrt(Float64(0.0 - Float64(Float64(im_m * im_m) / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im_m)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -3.6e-39)
		tmp = 0.5 * sqrt((0.0 - ((im_m * im_m) / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.6e-39], N[(0.5 * N[Sqrt[N[(0.0 - N[(N[(im$95$m * im$95$m), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.6000000000000001e-39

   1. Initial program 10.3%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \frac{{im}^{2}}{re}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(0 - \frac{{im}^{2}}{re}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right)\right) \]

      6. *-lowering-*.f64 52.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right)\right) \]

   5. Simplified 52.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - \frac{im \cdot im}{re}}} \]

   if -3.6000000000000001e-39 < re

   1. Initial program 51.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} + re\right), 2\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(re + \sqrt{re \cdot re + im \cdot im}\right), 2\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \left(\sqrt{re \cdot re + im \cdot im}\right)\right), 2\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 93.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(re, \mathsf{hypot.f64}\left(re, im\right)\right), 2\right)\right)\right) \]

   4. Applied egg-rr 93.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \sqrt{0 - \frac{im \cdot im}{re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{-41}:\\ \;\;\;\;0.5 \cdot \sqrt{0 - \frac{im\_m \cdot im\_m}{re}}\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+140}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3.9e-41)
   (* 0.5 (sqrt (- 0.0 (/ (* im_m im_m) re))))
   (if (<= re 1.85e+140) (* 0.5 (sqrt (* 2.0 (+ re im_m)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3.9e-41) {
		tmp = 0.5 * sqrt((0.0 - ((im_m * im_m) / re)));
	} else if (re <= 1.85e+140) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} 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 <= (-3.9d-41)) then
        tmp = 0.5d0 * sqrt((0.0d0 - ((im_m * im_m) / re)))
    else if (re <= 1.85d+140) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    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 <= -3.9e-41) {
		tmp = 0.5 * Math.sqrt((0.0 - ((im_m * im_m) / re)));
	} else if (re <= 1.85e+140) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -3.9e-41:
		tmp = 0.5 * math.sqrt((0.0 - ((im_m * im_m) / re)))
	elif re <= 1.85e+140:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3.9e-41)
		tmp = Float64(0.5 * sqrt(Float64(0.0 - Float64(Float64(im_m * im_m) / re))));
	elseif (re <= 1.85e+140)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	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 <= -3.9e-41)
		tmp = 0.5 * sqrt((0.0 - ((im_m * im_m) / re)));
	elseif (re <= 1.85e+140)
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3.9e-41], N[(0.5 * N[Sqrt[N[(0.0 - N[(N[(im$95$m * im$95$m), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.85e+140], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.89999999999999991e-41

   1. Initial program 10.3%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \frac{{im}^{2}}{re}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(0 - \frac{{im}^{2}}{re}\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{{im}^{2}}{re}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({im}^{2}\right), re\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(im \cdot im\right), re\right)\right)\right)\right) \]

      6. *-lowering-*.f64 52.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(im, im\right), re\right)\right)\right)\right) \]

   5. Simplified 52.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - \frac{im \cdot im}{re}}} \]

   if -3.89999999999999991e-41 < re < 1.85000000000000001e140

   1. Initial program 58.8%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\color{blue}{im}, re\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 41.7%

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

   if 1.85000000000000001e140 < re

   1. Initial program 18.8%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \sqrt{re} \]

      7. sqrt-lowering-sqrt.f64 85.1%

         \[\leadsto \mathsf{sqrt.f64}\left(re\right) \]

   5. Simplified 85.1%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{-41}:\\ \;\;\;\;0.5 \cdot \sqrt{0 - \frac{im \cdot im}{re}}\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+140}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{+159}:\\ \;\;\;\;0.5 \cdot {\left(\left(im\_m \cdot im\_m\right) \cdot 4\right)}^{0.25}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+140}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -7.5e+159)
   (* 0.5 (pow (* (* im_m im_m) 4.0) 0.25))
   (if (<= re 2.1e+140) (* 0.5 (sqrt (* 2.0 (+ re im_m)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -7.5e+159) {
		tmp = 0.5 * pow(((im_m * im_m) * 4.0), 0.25);
	} else if (re <= 2.1e+140) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} 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 <= (-7.5d+159)) then
        tmp = 0.5d0 * (((im_m * im_m) * 4.0d0) ** 0.25d0)
    else if (re <= 2.1d+140) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    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 <= -7.5e+159) {
		tmp = 0.5 * Math.pow(((im_m * im_m) * 4.0), 0.25);
	} else if (re <= 2.1e+140) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -7.5e+159:
		tmp = 0.5 * math.pow(((im_m * im_m) * 4.0), 0.25)
	elif re <= 2.1e+140:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -7.5e+159)
		tmp = Float64(0.5 * (Float64(Float64(im_m * im_m) * 4.0) ^ 0.25));
	elseif (re <= 2.1e+140)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	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 <= -7.5e+159)
		tmp = 0.5 * (((im_m * im_m) * 4.0) ^ 0.25);
	elseif (re <= 2.1e+140)
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -7.5e+159], N[(0.5 * N[Power[N[(N[(im$95$m * im$95$m), $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+140], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.4999999999999997e159

   1. Initial program 2.8%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 5.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 5.7%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(im \cdot 2\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\left(\left(im \cdot 2\right) \cdot \left(im \cdot 2\right)\right), \color{blue}{\frac{1}{4}}\right)\right) \]

      6. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\left(\left(im \cdot im\right) \cdot \left(2 \cdot 2\right)\right), \frac{1}{4}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\left(\left(im \cdot im\right) \cdot 4\right), \frac{1}{4}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\left(\left(im \cdot im\right) \cdot \left(2 + 2\right)\right), \frac{1}{4}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left(2 + 2\right)\right), \frac{1}{4}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(2 + 2\right)\right), \frac{1}{4}\right)\right) \]

      11. metadata-eval 19.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), 4\right), \frac{1}{4}\right)\right) \]

   7. Applied egg-rr 19.7%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(im \cdot im\right) \cdot 4\right)}^{0.25}} \]

   if -7.4999999999999997e159 < re < 2.1000000000000002e140

   1. Initial program 50.7%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\color{blue}{im}, re\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 36.5%

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

   if 2.1000000000000002e140 < re

   1. Initial program 18.8%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \sqrt{re} \]

      7. sqrt-lowering-sqrt.f64 85.1%

         \[\leadsto \mathsf{sqrt.f64}\left(re\right) \]

   5. Simplified 85.1%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{+159}:\\ \;\;\;\;0.5 \cdot {\left(\left(im \cdot im\right) \cdot 4\right)}^{0.25}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+140}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.25 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+140}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.25e-135)
   (* 0.5 (sqrt (* im_m 2.0)))
   (if (<= re 1.85e+140) (* 0.5 (sqrt (* 2.0 (+ re im_m)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.25e-135) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} else if (re <= 1.85e+140) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} 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 <= (-2.25d-135)) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    else if (re <= 1.85d+140) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    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 <= -2.25e-135) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else if (re <= 1.85e+140) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -2.25e-135:
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	elif re <= 1.85e+140:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.25e-135)
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	elseif (re <= 1.85e+140)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	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 <= -2.25e-135)
		tmp = 0.5 * sqrt((im_m * 2.0));
	elseif (re <= 1.85e+140)
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.25e-135], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.85e+140], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.24999999999999994e-135

   1. Initial program 16.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 15.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 15.5%

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

   if -2.24999999999999994e-135 < re < 1.85000000000000001e140

   1. Initial program 60.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\color{blue}{im}, re\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 43.0%

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

   if 1.85000000000000001e140 < re

   1. Initial program 18.8%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \sqrt{re} \]

      7. sqrt-lowering-sqrt.f64 85.1%

         \[\leadsto \mathsf{sqrt.f64}\left(re\right) \]

   5. Simplified 85.1%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.25 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+140}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 2.4 \cdot 10^{+124}:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 2.4e+124) (* 0.5 (sqrt (* im_m 2.0))) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 2.4e+124) {
		tmp = 0.5 * sqrt((im_m * 2.0));
	} 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 <= 2.4d+124) then
        tmp = 0.5d0 * sqrt((im_m * 2.0d0))
    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 <= 2.4e+124) {
		tmp = 0.5 * Math.sqrt((im_m * 2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 2.4e+124:
		tmp = 0.5 * math.sqrt((im_m * 2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 2.4e+124)
		tmp = Float64(0.5 * sqrt(Float64(im_m * 2.0)));
	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 <= 2.4e+124)
		tmp = 0.5 * sqrt((im_m * 2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 2.4e+124], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.40000000000000006e124

   1. Initial program 41.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 28.5%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 28.5%

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

   if 2.40000000000000006e124 < re

   1. Initial program 22.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \sqrt{re} \]

      7. sqrt-lowering-sqrt.f64 81.8%

         \[\leadsto \mathsf{sqrt.f64}\left(re\right) \]

   5. Simplified 81.8%

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

Alternative 8: 28.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;{\left(re \cdot re\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5e-310) (pow (* re re) 0.25) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5e-310) {
		tmp = pow((re * re), 0.25);
	} 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 <= (-5d-310)) then
        tmp = (re * re) ** 0.25d0
    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 <= -5e-310) {
		tmp = Math.pow((re * re), 0.25);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -5e-310:
		tmp = math.pow((re * re), 0.25)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = Float64(re * re) ^ 0.25;
	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 <= -5e-310)
		tmp = (re * re) ^ 0.25;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -5e-310], N[Power[N[(re * re), $MachinePrecision], 0.25], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 25.3%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \sqrt{re} \]

      7. sqrt-lowering-sqrt.f64 0.0%

         \[\leadsto \mathsf{sqrt.f64}\left(re\right) \]

   5. Simplified 0.0%

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

      1. pow1/2 N/A

         \[\leadsto {re}^{\color{blue}{\frac{1}{2}}} \]

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(re \cdot re\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

      6. metadata-eval 4.4%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{4}\right) \]

   7. Applied egg-rr 4.4%

      \[\leadsto \color{blue}{{\left(re \cdot re\right)}^{0.25}} \]

   if -4.999999999999985e-310 < re

   1. Initial program 51.4%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

         \[\leadsto \sqrt{re} \]

      7. sqrt-lowering-sqrt.f64 45.0%

         \[\leadsto \mathsf{sqrt.f64}\left(re\right) \]

   5. Simplified 45.0%

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

Alternative 9: 26.2% accurate, 2.1× 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 38.7%

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

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

      \[\leadsto \sqrt{re} \]

   7. sqrt-lowering-sqrt.f64 23.0%

      \[\leadsto \mathsf{sqrt.f64}\left(re\right) \]

5. Simplified 23.0%

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

Developer Target 1: 49.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t\_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t\_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024192 
(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)))))