math.sqrt on complex, real part

Percentage Accurate: 41.8% → 89.9%

Time: 9.9s

Alternatives: 8

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: 41.8% 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: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im\_m \cdot im\_m}\right)} \leq 0:\\ \;\;\;\;\left(im\_m \cdot {\left(re \cdot re\right)}^{-0.25}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im_m im_m)))))) 0.0)
   (* (* im_m (pow (* re re) -0.25)) 0.5)
   (* 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 (sqrt((2.0 * (re + sqrt(((re * re) + (im_m * im_m)))))) <= 0.0) {
		tmp = (im_m * pow((re * re), -0.25)) * 0.5;
	} 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 (Math.sqrt((2.0 * (re + Math.sqrt(((re * re) + (im_m * im_m)))))) <= 0.0) {
		tmp = (im_m * Math.pow((re * re), -0.25)) * 0.5;
	} 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 math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im_m * im_m)))))) <= 0.0:
		tmp = (im_m * math.pow((re * re), -0.25)) * 0.5
	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 (sqrt(Float64(2.0 * Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))))) <= 0.0)
		tmp = Float64(Float64(im_m * (Float64(re * re) ^ -0.25)) * 0.5);
	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 (sqrt((2.0 * (re + sqrt(((re * re) + (im_m * im_m)))))) <= 0.0)
		tmp = (im_m * ((re * re) ^ -0.25)) * 0.5;
	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[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(im$95$m * N[Power[N[(re * re), $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision] * 0.5), $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 (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 4.7%

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

   3. Taylor expanded in re around -inf

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

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

      3. /-lowering-/.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. *-lowering-*.f64 59.3

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

   5. Simplified 59.3%

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

      1. *-commutative N/A

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

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

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

      3. distribute-neg-frac2 N/A

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

      4. sqrt-div N/A

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

      5. pow2 N/A

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

      6. sqrt-pow1 N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 61.0

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

   7. Applied egg-rr 61.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. metadata-eval N/A

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

      5. sqrt-div N/A

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

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

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. /-lowering-/.f64 61.2

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

   9. Applied egg-rr 61.2%

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

       1. frac-2neg N/A

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

       2. metadata-eval N/A

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

       3. sqrt-div N/A

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

       4. metadata-eval N/A

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

       5. unpow1/2 N/A

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

       6. metadata-eval N/A

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

       7. pow-pow N/A

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

       8. pow-flip N/A

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

       9. sqr-pow N/A

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

       10. pow-prod-down N/A

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

       11. pow-pow N/A

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

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

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

       13. sqr-neg N/A

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

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

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

       15. metadata-eval N/A

           \[\leadsto \left({\left(re \cdot re\right)}^{\left(\color{blue}{\frac{1}{8}} \cdot \left(\mathsf{neg}\left(2\right)\right)\right)} \cdot im\right) \cdot \frac{1}{2} \]

       16. metadata-eval N/A

           \[\leadsto \left({\left(re \cdot re\right)}^{\left(\frac{1}{8} \cdot \color{blue}{-2}\right)} \cdot im\right) \cdot \frac{1}{2} \]

       17. metadata-eval 61.1

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

   11. Applied egg-rr 61.1%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 43.3%

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

      1. accelerator-lowering-hypot.f64 89.3

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

   4. Applied egg-rr 89.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;\left(im \cdot {\left(re \cdot re\right)}^{-0.25}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -3000000000:\\ \;\;\;\;0.5 \cdot \frac{im\_m}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 2.25 \cdot 10^{+79}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(im\_m, \frac{im\_m}{re}, re \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -3000000000.0)
   (* 0.5 (/ im_m (sqrt (- re))))
   (if (<= re 2.25e+79)
     (* 0.5 (sqrt (* 2.0 (+ re im_m))))
     (* 0.5 (sqrt (fma im_m (/ im_m re) (* re 4.0)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -3000000000.0) {
		tmp = 0.5 * (im_m / sqrt(-re));
	} else if (re <= 2.25e+79) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} else {
		tmp = 0.5 * sqrt(fma(im_m, (im_m / re), (re * 4.0)));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -3000000000.0)
		tmp = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))));
	elseif (re <= 2.25e+79)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	else
		tmp = Float64(0.5 * sqrt(fma(im_m, Float64(im_m / re), Float64(re * 4.0))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -3000000000.0], N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.25e+79], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im$95$m * N[(im$95$m / re), $MachinePrecision] + N[(re * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3e9

   1. Initial program 7.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.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. *-lowering-*.f64 49.6

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

   5. Simplified 49.6%

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

      1. *-commutative N/A

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

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

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

      3. distribute-neg-frac2 N/A

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

      4. sqrt-div N/A

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

      5. pow2 N/A

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

      6. sqrt-pow1 N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 50.0

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

   7. Applied egg-rr 50.0%

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

   if -3e9 < re < 2.24999999999999997e79

   1. Initial program 56.7%

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

   3. Taylor expanded in re around 0

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 42.9

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

   5. Simplified 42.9%

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

   if 2.24999999999999997e79 < re

   1. Initial program 27.6%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 87.8

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

   5. Simplified 87.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(im, \frac{im}{re}, re \cdot 4\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3000000000:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 2.25 \cdot 10^{+79}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(im, \frac{im}{re}, re \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1650000000:\\ \;\;\;\;0.5 \cdot \frac{im\_m}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1650000000.0)
   (* 0.5 (/ im_m (sqrt (- re))))
   (if (<= re 3.8e+77) (* 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 <= -1650000000.0) {
		tmp = 0.5 * (im_m / sqrt(-re));
	} else if (re <= 3.8e+77) {
		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 <= (-1650000000.0d0)) then
        tmp = 0.5d0 * (im_m / sqrt(-re))
    else if (re <= 3.8d+77) 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 <= -1650000000.0) {
		tmp = 0.5 * (im_m / Math.sqrt(-re));
	} else if (re <= 3.8e+77) {
		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 <= -1650000000.0:
		tmp = 0.5 * (im_m / math.sqrt(-re))
	elif re <= 3.8e+77:
		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 <= -1650000000.0)
		tmp = Float64(0.5 * Float64(im_m / sqrt(Float64(-re))));
	elseif (re <= 3.8e+77)
		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 <= -1650000000.0)
		tmp = 0.5 * (im_m / sqrt(-re));
	elseif (re <= 3.8e+77)
		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, -1650000000.0], N[(0.5 * N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.8e+77], 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 < -1.65e9

   1. Initial program 7.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.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. *-lowering-*.f64 49.6

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

   5. Simplified 49.6%

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

      1. *-commutative N/A

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

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

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

      3. distribute-neg-frac2 N/A

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

      4. sqrt-div N/A

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

      5. pow2 N/A

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

      6. sqrt-pow1 N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 50.0

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

   7. Applied egg-rr 50.0%

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

   if -1.65e9 < re < 3.8000000000000001e77

   1. Initial program 56.7%

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

   3. Taylor expanded in re around 0

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 42.9

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

   5. Simplified 42.9%

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

   if 3.8000000000000001e77 < re

   1. Initial program 27.6%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 87.4

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

   5. Simplified 87.4%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1650000000:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1650000000:\\ \;\;\;\;im\_m \cdot \frac{0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1650000000.0)
   (* im_m (/ 0.5 (sqrt (- re))))
   (if (<= re 7.2e+82) (* 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 <= -1650000000.0) {
		tmp = im_m * (0.5 / sqrt(-re));
	} else if (re <= 7.2e+82) {
		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 <= (-1650000000.0d0)) then
        tmp = im_m * (0.5d0 / sqrt(-re))
    else if (re <= 7.2d+82) 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 <= -1650000000.0) {
		tmp = im_m * (0.5 / Math.sqrt(-re));
	} else if (re <= 7.2e+82) {
		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 <= -1650000000.0:
		tmp = im_m * (0.5 / math.sqrt(-re))
	elif re <= 7.2e+82:
		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 <= -1650000000.0)
		tmp = Float64(im_m * Float64(0.5 / sqrt(Float64(-re))));
	elseif (re <= 7.2e+82)
		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 <= -1650000000.0)
		tmp = im_m * (0.5 / sqrt(-re));
	elseif (re <= 7.2e+82)
		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, -1650000000.0], N[(im$95$m * N[(0.5 / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.2e+82], 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 < -1.65e9

   1. Initial program 7.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.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. *-lowering-*.f64 49.6

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

   5. Simplified 49.6%

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

      1. *-commutative N/A

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

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

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

      3. distribute-neg-frac2 N/A

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

      4. sqrt-div N/A

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

      5. pow2 N/A

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

      6. sqrt-pow1 N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 50.0

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

   7. Applied egg-rr 50.0%

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

      1. div-inv N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. neg-lowering-neg.f64 49.9

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

   9. Applied egg-rr 49.9%

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

   if -1.65e9 < re < 7.20000000000000028e82

   1. Initial program 56.7%

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

   3. Taylor expanded in re around 0

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 42.9

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

   5. Simplified 42.9%

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

   if 7.20000000000000028e82 < re

   1. Initial program 27.6%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 87.4

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

   5. Simplified 87.4%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1650000000:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+82}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.6 \cdot 10^{+198}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.6e+198)
   0.0
   (if (<= re 7e+83) (* 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 <= -2.6e+198) {
		tmp = 0.0;
	} else if (re <= 7e+83) {
		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 <= (-2.6d+198)) then
        tmp = 0.0d0
    else if (re <= 7d+83) 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 <= -2.6e+198) {
		tmp = 0.0;
	} else if (re <= 7e+83) {
		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 <= -2.6e+198:
		tmp = 0.0
	elif re <= 7e+83:
		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 <= -2.6e+198)
		tmp = 0.0;
	elseif (re <= 7e+83)
		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 <= -2.6e+198)
		tmp = 0.0;
	elseif (re <= 7e+83)
		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, -2.6e+198], 0.0, If[LessEqual[re, 7e+83], 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 < -2.59999999999999981e198

   1. Initial program 2.3%

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

   3. Taylor expanded in re around 0

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 2.1

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

   5. Simplified 2.1%

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

   6. Taylor expanded in re around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. /-lowering-/.f64 2.1

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

   8. Simplified 2.1%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 2.1

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

   10. Applied egg-rr 2.1%

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

   12. Applied egg-rr 33.2%

       \[\leadsto \color{blue}{0} \]

   if -2.59999999999999981e198 < re < 6.99999999999999954e83

   1. Initial program 47.9%

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

   3. Taylor expanded in re around 0

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 37.5

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

   5. Simplified 37.5%

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

   if 6.99999999999999954e83 < re

   1. Initial program 27.6%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 87.4

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

   5. Simplified 87.4%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.6 \cdot 10^{+198}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{+196}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -4.8e+196)
   0.0
   (if (<= re 4.5e+77) (* 0.5 (sqrt (* 2.0 im_m))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -4.8e+196) {
		tmp = 0.0;
	} else if (re <= 4.5e+77) {
		tmp = 0.5 * sqrt((2.0 * 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 <= (-4.8d+196)) then
        tmp = 0.0d0
    else if (re <= 4.5d+77) then
        tmp = 0.5d0 * sqrt((2.0d0 * 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 <= -4.8e+196) {
		tmp = 0.0;
	} else if (re <= 4.5e+77) {
		tmp = 0.5 * Math.sqrt((2.0 * im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -4.8e+196:
		tmp = 0.0
	elif re <= 4.5e+77:
		tmp = 0.5 * math.sqrt((2.0 * im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -4.8e+196)
		tmp = 0.0;
	elseif (re <= 4.5e+77)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 <= -4.8e+196)
		tmp = 0.0;
	elseif (re <= 4.5e+77)
		tmp = 0.5 * sqrt((2.0 * 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, -4.8e+196], 0.0, If[LessEqual[re, 4.5e+77], N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.8000000000000001e196

   1. Initial program 2.3%

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

   3. Taylor expanded in re around 0

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 2.1

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

   5. Simplified 2.1%

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

   6. Taylor expanded in re around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. /-lowering-/.f64 2.1

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

   8. Simplified 2.1%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 2.1

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

   10. Applied egg-rr 2.1%

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

   12. Applied egg-rr 33.2%

       \[\leadsto \color{blue}{0} \]

   if -4.8000000000000001e196 < re < 4.50000000000000024e77

   1. Initial program 47.9%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 36.3

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

   5. Simplified 36.3%

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

   if 4.50000000000000024e77 < re

   1. Initial program 27.6%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 87.4

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

   5. Simplified 87.4%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{+196}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (if (<= re -5e-310) 0.0 (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 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 <= (-5d-310)) then
        tmp = 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 <= -5e-310) {
		tmp = 0.0;
	} 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 = 0.0
	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 = 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 <= -5e-310)
		tmp = 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, -5e-310], 0.0, N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 25.8%

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

   3. Taylor expanded in re around 0

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

      1. distribute-lft-out N/A

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

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

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

      3. +-lowering-+.f64 23.6

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

   5. Simplified 23.6%

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

   6. Taylor expanded in re around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. /-lowering-/.f64 17.6

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

   8. Simplified 17.6%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. /-lowering-/.f64 17.6

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

   10. Applied egg-rr 17.6%

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

   12. Applied egg-rr 10.8%

       \[\leadsto \color{blue}{0} \]

   if -4.999999999999985e-310 < re

   1. Initial program 53.4%

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

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

      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. sqrt-lowering-sqrt.f64 46.5

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

   5. Simplified 46.5%

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

Alternative 8: 6.1% accurate, 47.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.0)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.0;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.0
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 0.0

Derivation

1. Initial program 39.5%

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

3. Taylor expanded in re around 0

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

   1. distribute-lft-out N/A

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

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

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

   3. +-lowering-+.f64 31.5

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

5. Simplified 31.5%

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

6. Taylor expanded in re around inf

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

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

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

   2. +-commutative N/A

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

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

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

   4. /-lowering-/.f64 24.7

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

8. Simplified 24.7%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

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

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

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

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

   6. /-lowering-/.f64 24.7

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

10. Applied egg-rr 24.7%

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

12. Applied egg-rr 6.9%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Developer Target 1: 48.8% 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 2024204 
(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)))))