math.sqrt on complex, real part

Percentage Accurate: 41.9% → 90.0%

Time: 31.9s

Alternatives: 8

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.9% 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: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im\_m \cdot im\_m} \leq 0:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im_m im_m)))) 0.0)
   (* 0.5 (* im_m (sqrt (/ -1.0 re))))
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * (im_m * sqrt((-1.0 / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if ((re + Math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * (im_m * Math.sqrt((-1.0 / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (re + math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0:
		tmp = 0.5 * (im_m * math.sqrt((-1.0 / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))) <= 0.0)
		tmp = Float64(0.5 * Float64(im_m * sqrt(Float64(-1.0 / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im_m)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0)
		tmp = 0.5 * (im_m * sqrt((-1.0 / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(im$95$m * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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-sub0 N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 42.2

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

   5. Simplified 42.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - \frac{im \cdot im}{re}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. div-inv N/A

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

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

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

      4. sqrt-prod N/A

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. distribute-frac-neg2 N/A

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

      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 60.2

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

   7. Applied egg-rr 60.2%

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

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

   1. Initial program 48.7%

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

      1. accelerator-lowering-hypot.f64 91.2

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

   4. Applied egg-rr 91.2%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;re \leq 265000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -4.8e-72)
   (* 0.5 (* im_m (sqrt (/ -1.0 re))))
   (if (<= re 265000000.0) (* 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 <= -4.8e-72) {
		tmp = 0.5 * (im_m * sqrt((-1.0 / re)));
	} else if (re <= 265000000.0) {
		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 <= (-4.8d-72)) then
        tmp = 0.5d0 * (im_m * sqrt(((-1.0d0) / re)))
    else if (re <= 265000000.0d0) 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 <= -4.8e-72) {
		tmp = 0.5 * (im_m * Math.sqrt((-1.0 / re)));
	} else if (re <= 265000000.0) {
		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 <= -4.8e-72:
		tmp = 0.5 * (im_m * math.sqrt((-1.0 / re)))
	elif re <= 265000000.0:
		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 <= -4.8e-72)
		tmp = Float64(0.5 * Float64(im_m * sqrt(Float64(-1.0 / re))));
	elseif (re <= 265000000.0)
		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 <= -4.8e-72)
		tmp = 0.5 * (im_m * sqrt((-1.0 / re)));
	elseif (re <= 265000000.0)
		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, -4.8e-72], N[(0.5 * N[(im$95$m * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 265000000.0], 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 < -4.8e-72

   1. Initial program 19.0%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 39.1

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

   5. Simplified 39.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - \frac{im \cdot im}{re}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. div-inv N/A

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

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

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

      4. sqrt-prod N/A

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. distribute-frac-neg2 N/A

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

      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 41.4

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

   7. Applied egg-rr 41.4%

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

   if -4.8e-72 < re < 2.65e8

   1. Initial program 60.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + 2 \cdot re}} \]
   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 39.6

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

   5. Simplified 39.6%

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

   if 2.65e8 < re

   1. Initial program 43.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 82.8

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

   5. Simplified 82.8%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;re \leq 265000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5.1 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \frac{im\_m}{\sqrt{0 - re}}\\ \mathbf{elif}\;re \leq 175000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5.1e-72)
   (* 0.5 (/ im_m (sqrt (- 0.0 re))))
   (if (<= re 175000000.0) (* 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 <= -5.1e-72) {
		tmp = 0.5 * (im_m / sqrt((0.0 - re)));
	} else if (re <= 175000000.0) {
		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 <= (-5.1d-72)) then
        tmp = 0.5d0 * (im_m / sqrt((0.0d0 - re)))
    else if (re <= 175000000.0d0) 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 <= -5.1e-72) {
		tmp = 0.5 * (im_m / Math.sqrt((0.0 - re)));
	} else if (re <= 175000000.0) {
		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 <= -5.1e-72:
		tmp = 0.5 * (im_m / math.sqrt((0.0 - re)))
	elif re <= 175000000.0:
		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 <= -5.1e-72)
		tmp = Float64(0.5 * Float64(im_m / sqrt(Float64(0.0 - re))));
	elseif (re <= 175000000.0)
		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 <= -5.1e-72)
		tmp = 0.5 * (im_m / sqrt((0.0 - re)));
	elseif (re <= 175000000.0)
		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, -5.1e-72], N[(0.5 * N[(im$95$m / N[Sqrt[N[(0.0 - re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 175000000.0], 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 < -5.1000000000000003e-72

   1. Initial program 19.0%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 39.1

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

   5. Simplified 39.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - \frac{im \cdot im}{re}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

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

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

      3. sub0-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. sqrt-div N/A

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

      6. sqrt-unprod N/A

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

      7. rem-square-sqrt N/A

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

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

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

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

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

      10. neg-sub0 N/A

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

      11. --lowering--.f64 41.4

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

   7. Applied egg-rr 41.4%

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

   if -5.1000000000000003e-72 < re < 1.75e8

   1. Initial program 60.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + 2 \cdot re}} \]
   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 39.6

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

   5. Simplified 39.6%

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

   if 1.75e8 < re

   1. Initial program 43.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 82.8

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

   5. Simplified 82.8%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.1 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{0 - re}}\\ \mathbf{elif}\;re \leq 175000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 80000000:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 80000000.0) (* 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 <= 80000000.0) {
		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 <= 80000000.0d0) 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 <= 80000000.0) {
		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 <= 80000000.0:
		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 <= 80000000.0)
		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 <= 80000000.0)
		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, 80000000.0], N[(0.5 * N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8e7

   1. Initial program 41.7%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 29.2

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

   5. Simplified 29.2%

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

   if 8e7 < re

   1. Initial program 43.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 82.8

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

   5. Simplified 82.8%

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

Alternative 5: 31.4% 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 33.5%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

      3. --lowering--.f64 7.8

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

   5. Simplified 7.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(0 - re\right) + re\right)} \cdot \frac{1}{2}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(0 - re\right) + re\right)\right)}^{\frac{1}{2}}} \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto {\left(2 \cdot \color{blue}{\left(re + \left(0 - re\right)\right)}\right)}^{\frac{1}{2}} \cdot \frac{1}{2} \]

      4. sub0-neg N/A

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

      5. sub-neg N/A

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

      6. +-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 7.8

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

   7. Applied egg-rr 7.8%

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

   if -4.999999999999985e-310 < re

   1. Initial program 52.2%

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

   3. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. sqrt-lowering-sqrt.f64 57.8

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

   5. Simplified 57.8%

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

Alternative 6: 8.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;re \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (if (<= re -5e-310) 0.0 (* re 0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = re * 0.5;
	}
	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 = re * 0.5d0
    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 = re * 0.5;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -5e-310:
		tmp = 0.0
	else:
		tmp = re * 0.5
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = 0.0;
	else
		tmp = Float64(re * 0.5);
	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 = re * 0.5;
	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[(re * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 33.5%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

      3. --lowering--.f64 7.8

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

   5. Simplified 7.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(0 - re\right) + re\right)} \cdot \frac{1}{2}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(0 - re\right) + re\right)\right)}^{\frac{1}{2}}} \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto {\left(2 \cdot \color{blue}{\left(re + \left(0 - re\right)\right)}\right)}^{\frac{1}{2}} \cdot \frac{1}{2} \]

      4. sub0-neg N/A

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

      5. sub-neg N/A

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

      6. +-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 7.8

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

   7. Applied egg-rr 7.8%

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

   if -4.999999999999985e-310 < re

   1. Initial program 52.2%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   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-sub0 N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 1.0

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

   5. Simplified 1.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - \frac{im \cdot im}{re}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

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

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

      3. sub0-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. sqrt-div N/A

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

      6. sqrt-unprod N/A

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

      7. rem-square-sqrt N/A

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

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

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

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

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

      10. neg-sub0 N/A

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

      11. --lowering--.f64 0.0

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

   7. Applied egg-rr 0.0%

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

   9. Applied egg-rr 6.3%

      \[\leadsto \color{blue}{re} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 8.0% accurate, 6.7× speedup

Math

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

FPCore

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

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = 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 = 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 = 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 = 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 = 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 = 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, re]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 33.5%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

      3. --lowering--.f64 7.8

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

   5. Simplified 7.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(0 - re\right) + re\right)} \cdot \frac{1}{2}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(0 - re\right) + re\right)\right)}^{\frac{1}{2}}} \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto {\left(2 \cdot \color{blue}{\left(re + \left(0 - re\right)\right)}\right)}^{\frac{1}{2}} \cdot \frac{1}{2} \]

      4. sub0-neg N/A

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

      5. sub-neg N/A

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

      6. +-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 7.8

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

   7. Applied egg-rr 7.8%

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

   if -4.999999999999985e-310 < re

   1. Initial program 52.2%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

      3. --lowering--.f64 2.8

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

   5. Simplified 2.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

   7. Applied egg-rr 6.3%

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

Alternative 8: 6.3% 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 42.3%

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

3. Taylor expanded in re around -inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

   3. --lowering--.f64 5.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]

5. Simplified 5.4%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 - re\right)} + re\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(0 - re\right) + re\right)} \cdot \frac{1}{2}} \]

   2. pow1/2 N/A

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(0 - re\right) + re\right)\right)}^{\frac{1}{2}}} \cdot \frac{1}{2} \]

   3. +-commutative N/A

      \[\leadsto {\left(2 \cdot \color{blue}{\left(re + \left(0 - re\right)\right)}\right)}^{\frac{1}{2}} \cdot \frac{1}{2} \]

   4. sub0-neg N/A

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

   5. sub-neg N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval 5.4

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

7. Applied egg-rr 5.4%

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

Developer Target 1: 49.2% 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 2024198 
(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)))))