math.sqrt on complex, real part

Percentage Accurate: 41.7% → 88.3%

Time: 10.1s

Alternatives: 6

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.7% 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: 88.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{im\_m}{\sqrt{-re}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -8.8e-32)
   (* (/ 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 <= -8.8e-32) {
		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 <= -8.8e-32) {
		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 <= -8.8e-32:
		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 (re <= -8.8e-32)
		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 <= -8.8e-32)
		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[re, -8.8e-32], 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 re < -8.7999999999999999e-32

   1. Initial program 15.6%

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

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 41.8

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

   5. Applied rewrites 41.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sqrt-div N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-neg.f64 51.8

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

   7. Applied rewrites 51.8%

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

   if -8.7999999999999999e-32 < re

   1. Initial program 51.3%

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

      1. lower-hypot.f64 95.3

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

   4. Applied rewrites 95.3%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{im}{\sqrt{-re}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{im\_m}{\sqrt{-re}} \cdot 0.5\\ \mathbf{elif}\;re \leq 175000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.25e-38)
   (* (/ im_m (sqrt (- re))) 0.5)
   (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 <= -1.25e-38) {
		tmp = (im_m / sqrt(-re)) * 0.5;
	} 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 <= (-1.25d-38)) then
        tmp = (im_m / sqrt(-re)) * 0.5d0
    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 <= -1.25e-38) {
		tmp = (im_m / Math.sqrt(-re)) * 0.5;
	} 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 <= -1.25e-38:
		tmp = (im_m / math.sqrt(-re)) * 0.5
	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 <= -1.25e-38)
		tmp = Float64(Float64(im_m / sqrt(Float64(-re))) * 0.5);
	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 <= -1.25e-38)
		tmp = (im_m / sqrt(-re)) * 0.5;
	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, -1.25e-38], N[(N[(im$95$m / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision] * 0.5), $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 < -1.25000000000000008e-38

   1. Initial program 16.8%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 40.6

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

   5. Applied rewrites 40.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 40.6

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

      7. lift-sqrt.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. sqrt-div N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-neg.f64 51.9

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

   7. Applied rewrites 51.9%

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

   if -1.25000000000000008e-38 < re < 1.75e8

   1. Initial program 59.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + 2 \cdot re}} \]
   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. lower-*.f64 N/A

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

      3. lower-+.f64 43.5

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

   5. Applied rewrites 43.5%

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

   if 1.75e8 < re

   1. Initial program 37.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. lower-sqrt.f64 80.2

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

   5. Applied rewrites 80.2%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{im}{\sqrt{-re}} \cdot 0.5\\ \mathbf{elif}\;re \leq 175000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{+143}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 175000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -5.8e+143)
   0.0
   (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 <= -5.8e+143) {
		tmp = 0.0;
	} 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 <= (-5.8d+143)) then
        tmp = 0.0d0
    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 <= -5.8e+143) {
		tmp = 0.0;
	} 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 <= -5.8e+143:
		tmp = 0.0
	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 <= -5.8e+143)
		tmp = 0.0;
	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 <= -5.8e+143)
		tmp = 0.0;
	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, -5.8e+143], 0.0, 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 < -5.7999999999999996e143

   1. Initial program 3.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 3.8

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

   4. Applied rewrites 3.8%

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

   5. Taylor expanded in re around -inf

      \[\leadsto \sqrt{2 \cdot \left(re + \color{blue}{-1 \cdot re}\right)} \cdot \frac{1}{2} \]
   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. lower-neg.f64 27.8

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

   7. Applied rewrites 27.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 27.8

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

   9. Applied rewrites 27.8%

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

   if -5.7999999999999996e143 < re < 1.75e8

   1. Initial program 53.9%

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

   3. Taylor expanded in re around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 40.2

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

   5. Applied rewrites 40.2%

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

   if 1.75e8 < re

   1. Initial program 37.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. lower-sqrt.f64 80.2

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

   5. Applied rewrites 80.2%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{+143}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 175000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+177}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 175000000:\\ \;\;\;\;0.5 \cdot \sqrt{im\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.75e+177)
   0.0
   (if (<= re 175000000.0) (* 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 <= -1.75e+177) {
		tmp = 0.0;
	} else if (re <= 175000000.0) {
		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 <= (-1.75d+177)) then
        tmp = 0.0d0
    else if (re <= 175000000.0d0) 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 <= -1.75e+177) {
		tmp = 0.0;
	} else if (re <= 175000000.0) {
		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 <= -1.75e+177:
		tmp = 0.0
	elif re <= 175000000.0:
		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 <= -1.75e+177)
		tmp = 0.0;
	elseif (re <= 175000000.0)
		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 <= -1.75e+177)
		tmp = 0.0;
	elseif (re <= 175000000.0)
		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, -1.75e+177], 0.0, If[LessEqual[re, 175000000.0], 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 < -1.74999999999999996e177

   1. Initial program 2.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 2.1

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

   4. Applied rewrites 2.1%

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

   5. Taylor expanded in re around -inf

      \[\leadsto \sqrt{2 \cdot \left(re + \color{blue}{-1 \cdot re}\right)} \cdot \frac{1}{2} \]
   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. lower-neg.f64 31.4

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

   7. Applied rewrites 31.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 31.4

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

   9. Applied rewrites 31.4%

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

   if -1.74999999999999996e177 < re < 1.75e8

   1. Initial program 51.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. lower-*.f64 37.6

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

   5. Applied rewrites 37.6%

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

   if 1.75e8 < re

   1. Initial program 37.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. lower-sqrt.f64 80.2

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

   5. Applied rewrites 80.2%

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

Alternative 5: 30.9% 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 36.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 36.3

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

   4. Applied rewrites 36.3%

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

   5. Taylor expanded in re around -inf

      \[\leadsto \sqrt{2 \cdot \left(re + \color{blue}{-1 \cdot re}\right)} \cdot \frac{1}{2} \]
   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. lower-neg.f64 11.2

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

   7. Applied rewrites 11.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 11.2

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

   9. Applied rewrites 11.2%

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

   if -4.999999999999985e-310 < 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. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 54.9

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

   5. Applied rewrites 54.9%

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

Alternative 6: 6.0% 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 42.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 42.7

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

4. Applied rewrites 42.7%

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

5. Taylor expanded in re around -inf

   \[\leadsto \sqrt{2 \cdot \left(re + \color{blue}{-1 \cdot re}\right)} \cdot \frac{1}{2} \]
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. lower-neg.f64 6.9

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

7. Applied rewrites 6.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow1/2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. unsub-neg N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 6.9

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

9. Applied rewrites 6.9%

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

Developer Target 1: 48.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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)))))