math.sqrt on complex, real part

Percentage Accurate: 40.9% → 89.7%

Time: 9.4s

Alternatives: 10

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.7% accurate, 0.3× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im_m im_m)))) 0.0)
   (* (/ im_m (sqrt (- re))) 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 ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = (im_m / sqrt(-re)) * 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 ((re + Math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = (im_m / Math.sqrt(-re)) * 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 (re + math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0:
		tmp = (im_m / math.sqrt(-re)) * 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 (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))) <= 0.0)
		tmp = Float64(Float64(im_m / sqrt(Float64(-re))) * 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 ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0)
		tmp = (im_m / sqrt(-re)) * 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[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im$95$m / N[Sqrt[(-re)], $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 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 7.6%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. 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 40.0

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

   5. Simplified 40.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{im \cdot im}{re}}} \]
   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 51.4

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

   7. Applied egg-rr 51.4%

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

   if 0.0 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 50.8%

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

      1. accelerator-lowering-hypot.f64 90.4

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

   4. Applied egg-rr 90.4%

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

4. Final simplification 84.9%

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

Alternative 2: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{-115}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;re \leq 4200000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(im\_m, \frac{im\_m}{re}, re \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.35e-115)
   (* 0.5 (* im_m (sqrt (/ -1.0 re))))
   (if (<= re 4200000.0)
     (* 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 <= -1.35e-115) {
		tmp = 0.5 * (im_m * sqrt((-1.0 / re)));
	} else if (re <= 4200000.0) {
		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 <= -1.35e-115)
		tmp = Float64(0.5 * Float64(im_m * sqrt(Float64(-1.0 / re))));
	elseif (re <= 4200000.0)
		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, -1.35e-115], N[(0.5 * N[(im$95$m * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4200000.0], 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 < -1.35e-115

   1. Initial program 18.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 38.4

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

   5. Simplified 38.4%

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

      1. distribute-neg-frac2 N/A

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

      2. div-inv N/A

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

      3. sqrt-prod N/A

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

      4. pow2 N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

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

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

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

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

      10. distribute-frac-neg2 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 41.2

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

   7. Applied egg-rr 41.2%

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

   if -1.35e-115 < re < 4.2e6

   1. Initial program 61.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 43.3

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

   5. Simplified 43.3%

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

   if 4.2e6 < re

   1. Initial program 48.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 78.8

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

   5. Simplified 78.8%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{-115}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;re \leq 4200000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(im, \frac{im}{re}, re \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;re \leq 30000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.5e-116)
   (* 0.5 (* im_m (sqrt (/ -1.0 re))))
   (if (<= re 30000000.0) (* 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 <= -1.5e-116) {
		tmp = 0.5 * (im_m * sqrt((-1.0 / re)));
	} else if (re <= 30000000.0) {
		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 <= (-1.5d-116)) then
        tmp = 0.5d0 * (im_m * sqrt(((-1.0d0) / re)))
    else if (re <= 30000000.0d0) 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 <= -1.5e-116) {
		tmp = 0.5 * (im_m * Math.sqrt((-1.0 / re)));
	} else if (re <= 30000000.0) {
		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 <= -1.5e-116:
		tmp = 0.5 * (im_m * math.sqrt((-1.0 / re)))
	elif re <= 30000000.0:
		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 <= -1.5e-116)
		tmp = Float64(0.5 * Float64(im_m * sqrt(Float64(-1.0 / re))));
	elseif (re <= 30000000.0)
		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 <= -1.5e-116)
		tmp = 0.5 * (im_m * sqrt((-1.0 / re)));
	elseif (re <= 30000000.0)
		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, -1.5e-116], N[(0.5 * N[(im$95$m * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 30000000.0], 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 < -1.50000000000000013e-116

   1. Initial program 18.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 38.4

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

   5. Simplified 38.4%

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

      1. distribute-neg-frac2 N/A

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

      2. div-inv N/A

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

      3. sqrt-prod N/A

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

      4. pow2 N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

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

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

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

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

      10. distribute-frac-neg2 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 41.2

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

   7. Applied egg-rr 41.2%

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

   if -1.50000000000000013e-116 < re < 3e7

   1. Initial program 61.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 43.3

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

   5. Simplified 43.3%

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

   if 3e7 < re

   1. Initial program 48.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 78.5

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

   5. Simplified 78.5%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;re \leq 30000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-115}:\\ \;\;\;\;\frac{im\_m \cdot 0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 48000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5e-115)
   (/ (* im_m 0.5) (sqrt (- re)))
   (if (<= re 48000.0) (* 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 <= -5e-115) {
		tmp = (im_m * 0.5) / sqrt(-re);
	} else if (re <= 48000.0) {
		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 <= (-5d-115)) then
        tmp = (im_m * 0.5d0) / sqrt(-re)
    else if (re <= 48000.0d0) 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 <= -5e-115) {
		tmp = (im_m * 0.5) / Math.sqrt(-re);
	} else if (re <= 48000.0) {
		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 <= -5e-115:
		tmp = (im_m * 0.5) / math.sqrt(-re)
	elif re <= 48000.0:
		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 <= -5e-115)
		tmp = Float64(Float64(im_m * 0.5) / sqrt(Float64(-re)));
	elseif (re <= 48000.0)
		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 <= -5e-115)
		tmp = (im_m * 0.5) / sqrt(-re);
	elseif (re <= 48000.0)
		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, -5e-115], N[(N[(im$95$m * 0.5), $MachinePrecision] / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 48000.0], 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 < -5.0000000000000003e-115

   1. Initial program 18.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 38.4

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

   5. Simplified 38.4%

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

      1. distribute-neg-frac2 N/A

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

      2. div-inv N/A

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

      3. sqrt-prod N/A

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

      4. pow2 N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

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

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

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

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

      10. distribute-frac-neg2 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 41.2

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

   7. Applied egg-rr 41.2%

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

      1. associate-*r* N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. neg-lowering-neg.f64 41.2

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

   9. Applied egg-rr 41.2%

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

   if -5.0000000000000003e-115 < re < 48000

   1. Initial program 61.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 43.3

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

   5. Simplified 43.3%

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

   if 48000 < re

   1. Initial program 48.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 78.5

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

   5. Simplified 78.5%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-115}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 48000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{-115}:\\ \;\;\;\;\frac{im\_m}{\sqrt{-re}} \cdot 0.5\\ \mathbf{elif}\;re \leq 760:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.35e-115)
   (* (/ im_m (sqrt (- re))) 0.5)
   (if (<= re 760.0) (* 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 <= -1.35e-115) {
		tmp = (im_m / sqrt(-re)) * 0.5;
	} else if (re <= 760.0) {
		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 <= (-1.35d-115)) then
        tmp = (im_m / sqrt(-re)) * 0.5d0
    else if (re <= 760.0d0) 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 <= -1.35e-115) {
		tmp = (im_m / Math.sqrt(-re)) * 0.5;
	} else if (re <= 760.0) {
		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 <= -1.35e-115:
		tmp = (im_m / math.sqrt(-re)) * 0.5
	elif re <= 760.0:
		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 <= -1.35e-115)
		tmp = Float64(Float64(im_m / sqrt(Float64(-re))) * 0.5);
	elseif (re <= 760.0)
		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 <= -1.35e-115)
		tmp = (im_m / sqrt(-re)) * 0.5;
	elseif (re <= 760.0)
		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, -1.35e-115], N[(N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 760.0], 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 < -1.35e-115

   1. Initial program 18.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 38.4

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

   5. Simplified 38.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{im \cdot im}{re}}} \]
   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 41.2

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

   7. Applied egg-rr 41.2%

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

   if -1.35e-115 < re < 760

   1. Initial program 61.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 43.3

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

   5. Simplified 43.3%

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

   if 760 < re

   1. Initial program 48.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 78.5

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

   5. Simplified 78.5%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{-115}:\\ \;\;\;\;\frac{im}{\sqrt{-re}} \cdot 0.5\\ \mathbf{elif}\;re \leq 760:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-115}:\\ \;\;\;\;im\_m \cdot \frac{0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 23000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5e-115)
   (* im_m (/ 0.5 (sqrt (- re))))
   (if (<= re 23000.0) (* 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 <= -5e-115) {
		tmp = im_m * (0.5 / sqrt(-re));
	} else if (re <= 23000.0) {
		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 <= (-5d-115)) then
        tmp = im_m * (0.5d0 / sqrt(-re))
    else if (re <= 23000.0d0) 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 <= -5e-115) {
		tmp = im_m * (0.5 / Math.sqrt(-re));
	} else if (re <= 23000.0) {
		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 <= -5e-115:
		tmp = im_m * (0.5 / math.sqrt(-re))
	elif re <= 23000.0:
		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 <= -5e-115)
		tmp = Float64(im_m * Float64(0.5 / sqrt(Float64(-re))));
	elseif (re <= 23000.0)
		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 <= -5e-115)
		tmp = im_m * (0.5 / sqrt(-re));
	elseif (re <= 23000.0)
		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, -5e-115], N[(im$95$m * N[(0.5 / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 23000.0], 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 < -5.0000000000000003e-115

   1. Initial program 18.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 38.4

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

   5. Simplified 38.4%

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

      1. distribute-neg-frac2 N/A

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

      2. div-inv N/A

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

      3. sqrt-prod N/A

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

      4. pow2 N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

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

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

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

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

      10. distribute-frac-neg2 N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 41.2

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

   7. Applied egg-rr 41.2%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

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

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

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

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

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

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

      11. neg-lowering-neg.f64 41.2

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

   9. Applied egg-rr 41.2%

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

   if -5.0000000000000003e-115 < re < 23000

   1. Initial program 61.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 43.3

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

   5. Simplified 43.3%

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

   if 23000 < re

   1. Initial program 48.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 78.5

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

   5. Simplified 78.5%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-115}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 23000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{+122}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 550000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.1e+122)
   0.0
   (if (<= re 550000.0) (* 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 <= -2.1e+122) {
		tmp = 0.0;
	} else if (re <= 550000.0) {
		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 <= (-2.1d+122)) then
        tmp = 0.0d0
    else if (re <= 550000.0d0) 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 <= -2.1e+122) {
		tmp = 0.0;
	} else if (re <= 550000.0) {
		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 <= -2.1e+122:
		tmp = 0.0
	elif re <= 550000.0:
		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 <= -2.1e+122)
		tmp = 0.0;
	elseif (re <= 550000.0)
		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 <= -2.1e+122)
		tmp = 0.0;
	elseif (re <= 550000.0)
		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, -2.1e+122], 0.0, If[LessEqual[re, 550000.0], 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 < -2.10000000000000016e122

   1. Initial program 6.3%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 6.3

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

   4. Applied egg-rr 6.3%

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

   5. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 24.1

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

   7. Simplified 24.1%

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

      1. unsub-neg N/A

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

      2. +-inverses N/A

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

      3. metadata-eval N/A

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

      4. +-inverses N/A

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

      5. unsub-neg N/A

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

      6. pow1/2 N/A

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

      7. unsub-neg N/A

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

      8. +-inverses N/A

         \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \cdot \frac{1}{2} \]

      9. metadata-eval N/A

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

      10. metadata-eval 24.1

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

   9. Applied egg-rr 24.1%

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

   if -2.10000000000000016e122 < re < 5.5e5

   1. Initial program 52.6%

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

   3. Taylor expanded in re around 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 36.7

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

   5. Simplified 36.7%

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

   if 5.5e5 < re

   1. Initial program 48.8%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 78.5

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

   5. Simplified 78.5%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{+122}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 550000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{+152}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1.8:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5.8e+152)
   0.0
   (if (<= re 1.8) (* 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 <= -5.8e+152) {
		tmp = 0.0;
	} else if (re <= 1.8) {
		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 <= (-5.8d+152)) then
        tmp = 0.0d0
    else if (re <= 1.8d0) 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 <= -5.8e+152) {
		tmp = 0.0;
	} else if (re <= 1.8) {
		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 <= -5.8e+152:
		tmp = 0.0
	elif re <= 1.8:
		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 <= -5.8e+152)
		tmp = 0.0;
	elseif (re <= 1.8)
		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 <= -5.8e+152)
		tmp = 0.0;
	elseif (re <= 1.8)
		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, -5.8e+152], 0.0, If[LessEqual[re, 1.8], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.7999999999999997e152

   1. Initial program 5.5%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 5.5

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

   4. Applied egg-rr 5.5%

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

   5. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 27.8

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

   7. Simplified 27.8%

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

      1. unsub-neg N/A

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

      2. +-inverses N/A

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

      3. metadata-eval N/A

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

      4. +-inverses N/A

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

      5. unsub-neg N/A

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

      6. pow1/2 N/A

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

      7. unsub-neg N/A

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

      8. +-inverses N/A

         \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \cdot \frac{1}{2} \]

      9. metadata-eval N/A

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

      10. metadata-eval 27.8

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

   9. Applied egg-rr 27.8%

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

   if -5.7999999999999997e152 < re < 1.80000000000000004

   1. Initial program 51.1%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 34.6

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

   5. Simplified 34.6%

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

   if 1.80000000000000004 < re

   1. Initial program 48.9%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 77.6

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

   5. Simplified 77.6%

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

Alternative 9: 31.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-309}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (if (<= re -1e-309) 0.0 (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1e-309) {
		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 <= (-1d-309)) 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 <= -1e-309) {
		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 <= -1e-309:
		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 <= -1e-309)
		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 <= -1e-309)
		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, -1e-309], 0.0, N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.000000000000002e-309

   1. Initial program 30.7%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 30.7

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

   4. Applied egg-rr 30.7%

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

   5. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 10.1

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

   7. Simplified 10.1%

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

      1. unsub-neg N/A

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

      2. +-inverses N/A

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

      3. metadata-eval N/A

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

      4. +-inverses N/A

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

      5. unsub-neg N/A

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

      6. pow1/2 N/A

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

      7. unsub-neg N/A

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

      8. +-inverses N/A

         \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \cdot \frac{1}{2} \]

      9. metadata-eval N/A

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

      10. metadata-eval 10.1

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

   9. Applied egg-rr 10.1%

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

   if -1.000000000000002e-309 < re

   1. Initial program 58.1%

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

   3. Taylor expanded in re around 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 47.7

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

   5. Simplified 47.7%

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

Alternative 10: 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 44.7%

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

   1. *-commutative N/A

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

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

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

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

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

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

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

   5. +-commutative N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f64 44.7

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

4. Applied egg-rr 44.7%

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

5. Taylor expanded in re around -inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 6.4

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

7. Simplified 6.4%

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

   1. unsub-neg N/A

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

   2. +-inverses N/A

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

   3. metadata-eval N/A

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

   4. +-inverses N/A

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

   5. unsub-neg N/A

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

   6. pow1/2 N/A

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

   7. unsub-neg N/A

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

   8. +-inverses N/A

      \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \cdot \frac{1}{2} \]

   9. metadata-eval N/A

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

   10. metadata-eval 6.4

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

9. Applied egg-rr 6.4%

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

Developer Target 1: 47.9% 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 2024199 
(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)))))