math.sqrt on complex, real part

Percentage Accurate: 41.5% → 72.1%

Time: 6.5s

Alternatives: 4

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: 41.5% 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: 72.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \sqrt{\left(\sqrt{im\_m \cdot im\_m + re \cdot re} + re\right) \cdot 2}\\ t_1 := \sqrt{\left(im\_m + re\right) \cdot 2} \cdot 0.5\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{-im\_m}{re} \cdot im\_m} \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (sqrt (* (+ (sqrt (+ (* im_m im_m) (* re re))) re) 2.0)))
        (t_1 (* (sqrt (* (+ im_m re) 2.0)) 0.5)))
   (if (<= t_0 0.0)
     (* (sqrt (* (/ (- im_m) re) im_m)) 0.5)
     (if (<= t_0 2e-79) t_1 (if (<= t_0 5e+75) (* 0.5 t_0) t_1)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = sqrt(((sqrt(((im_m * im_m) + (re * re))) + re) * 2.0));
	double t_1 = sqrt(((im_m + re) * 2.0)) * 0.5;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (t_0 <= 2e-79) {
		tmp = t_1;
	} else if (t_0 <= 5e+75) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((sqrt(((im_m * im_m) + (re * re))) + re) * 2.0d0))
    t_1 = sqrt(((im_m + re) * 2.0d0)) * 0.5d0
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((-im_m / re) * im_m)) * 0.5d0
    else if (t_0 <= 2d-79) then
        tmp = t_1
    else if (t_0 <= 5d+75) then
        tmp = 0.5d0 * t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = Math.sqrt(((Math.sqrt(((im_m * im_m) + (re * re))) + re) * 2.0));
	double t_1 = Math.sqrt(((im_m + re) * 2.0)) * 0.5;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (t_0 <= 2e-79) {
		tmp = t_1;
	} else if (t_0 <= 5e+75) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = math.sqrt(((math.sqrt(((im_m * im_m) + (re * re))) + re) * 2.0))
	t_1 = math.sqrt(((im_m + re) * 2.0)) * 0.5
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((-im_m / re) * im_m)) * 0.5
	elif t_0 <= 2e-79:
		tmp = t_1
	elif t_0 <= 5e+75:
		tmp = 0.5 * t_0
	else:
		tmp = t_1
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = sqrt(Float64(Float64(sqrt(Float64(Float64(im_m * im_m) + Float64(re * re))) + re) * 2.0))
	t_1 = Float64(sqrt(Float64(Float64(im_m + re) * 2.0)) * 0.5)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im_m) / re) * im_m)) * 0.5);
	elseif (t_0 <= 2e-79)
		tmp = t_1;
	elseif (t_0 <= 5e+75)
		tmp = Float64(0.5 * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = sqrt(((sqrt(((im_m * im_m) + (re * re))) + re) * 2.0));
	t_1 = sqrt(((im_m + re) * 2.0)) * 0.5;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	elseif (t_0 <= 2e-79)
		tmp = t_1;
	elseif (t_0 <= 5e+75)
		tmp = 0.5 * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(N[Sqrt[N[(N[(im$95$m * im$95$m), $MachinePrecision] + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(im$95$m + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[((-im$95$m) / re), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e-79], t$95$1, If[LessEqual[t$95$0, 5e+75], N[(0.5 * t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 56.9

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

   5. Applied rewrites 56.9%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 2e-79 or 5.0000000000000002e75 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 5.8%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 36.1

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

   5. Applied rewrites 36.1%

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

   if 2e-79 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 5.0000000000000002e75

   1. Initial program 100.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\sqrt{im \cdot im + re \cdot re} + re\right) \cdot 2} \leq 0:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;\sqrt{\left(\sqrt{im \cdot im + re \cdot re} + re\right) \cdot 2} \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;\sqrt{\left(\sqrt{im \cdot im + re \cdot re} + re\right) \cdot 2} \leq 5 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{im \cdot im + re \cdot re} + re\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\frac{-im\_m}{re} \cdot im\_m} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\left(im\_m + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -7.5e+73)
   (* (sqrt (* (/ (- im_m) re) im_m)) 0.5)
   (if (<= re 1.1e+45) (* (sqrt (* (+ im_m re) 2.0)) 0.5) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -7.5e+73) {
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= 1.1e+45) {
		tmp = sqrt(((im_m + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-7.5d+73)) then
        tmp = sqrt(((-im_m / re) * im_m)) * 0.5d0
    else if (re <= 1.1d+45) then
        tmp = sqrt(((im_m + re) * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -7.5e+73) {
		tmp = Math.sqrt(((-im_m / re) * im_m)) * 0.5;
	} else if (re <= 1.1e+45) {
		tmp = Math.sqrt(((im_m + re) * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -7.5e+73:
		tmp = math.sqrt(((-im_m / re) * im_m)) * 0.5
	elif re <= 1.1e+45:
		tmp = math.sqrt(((im_m + re) * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -7.5e+73)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im_m) / re) * im_m)) * 0.5);
	elseif (re <= 1.1e+45)
		tmp = Float64(sqrt(Float64(Float64(im_m + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -7.5e+73)
		tmp = sqrt(((-im_m / re) * im_m)) * 0.5;
	elseif (re <= 1.1e+45)
		tmp = sqrt(((im_m + re) * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -7.5e+73], N[(N[Sqrt[N[(N[((-im$95$m) / re), $MachinePrecision] * im$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 1.1e+45], N[(N[Sqrt[N[(N[(im$95$m + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.5e73

   1. Initial program 7.7%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 57.9

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

   5. Applied rewrites 57.9%

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

   if -7.5e73 < re < 1.1e45

   1. Initial program 54.5%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 41.4

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

   5. Applied rewrites 41.4%

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

   if 1.1e45 < re

   1. Initial program 54.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{im\_m \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 1.8e-73) (* (sqrt (* im_m 2.0)) 0.5) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 1.8e-73) {
		tmp = sqrt((im_m * 2.0)) * 0.5;
	} 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 <= 1.8d-73) then
        tmp = sqrt((im_m * 2.0d0)) * 0.5d0
    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 <= 1.8e-73) {
		tmp = Math.sqrt((im_m * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 1.8e-73:
		tmp = math.sqrt((im_m * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 1.8e-73)
		tmp = Float64(sqrt(Float64(im_m * 2.0)) * 0.5);
	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 <= 1.8e-73)
		tmp = sqrt((im_m * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 1.8e-73], N[(N[Sqrt[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.8e-73

   1. Initial program 39.2%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 35.8

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

   5. Applied rewrites 35.8%

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

   if 1.8e-73 < re

   1. Initial program 61.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 70.7

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

   5. Applied rewrites 70.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.2%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
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. lower-sqrt.f64 28.8

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

5. Applied rewrites 28.8%

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

Developer Target 1: 48.7% 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 2024283 
(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)))))