math.abs on complex

Percentage Accurate: 53.5% → 99.5%

Time: 4.3s

Alternatives: 4

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \sqrt{re \cdot re + im \cdot im} \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))

C

double modulus(double re, double im) {
	return sqrt(((re * re) + (im * im)));
}

Fortran

real(8) function modulus(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    modulus = sqrt(((re * re) + (im * im)))
end function

Java

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

Python

def modulus(re, im):
	return math.sqrt(((re * re) + (im * im)))

Julia

function modulus(re, im)
	return sqrt(Float64(Float64(re * re) + Float64(im * im)))
end

MATLAB

function tmp = modulus(re, im)
	tmp = sqrt(((re * re) + (im * im)));
end

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{re \cdot re + im \cdot im} \end{array} \]

FPCore

(FPCore modulus (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))

C

double modulus(double re, double im) {
	return sqrt(((re * re) + (im * im)));
}

Fortran

real(8) function modulus(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    modulus = sqrt(((re * re) + (im * im)))
end function

Java

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

Python

def modulus(re, im):
	return math.sqrt(((re * re) + (im * im)))

Julia

function modulus(re, im)
	return sqrt(Float64(Float64(re * re) + Float64(im * im)))
end

MATLAB

function tmp = modulus(re, im)
	tmp = sqrt(((re * re) + (im * im)));
end

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \mathsf{fma}\left(re\_m, \frac{re\_m \cdot 0.5}{im\_m}, im\_m\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore modulus (re_m im_m)
 :precision binary64
 (fma re_m (/ (* re_m 0.5) im_m) im_m))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double modulus(double re_m, double im_m) {
	return fma(re_m, ((re_m * 0.5) / im_m), im_m);
}

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function modulus(re_m, im_m)
	return fma(re_m, Float64(Float64(re_m * 0.5) / im_m), im_m)
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
modulus[re$95$m_, im$95$m_] := N[(re$95$m * N[(N[(re$95$m * 0.5), $MachinePrecision] / im$95$m), $MachinePrecision] + im$95$m), $MachinePrecision]

Derivation

1. Initial program 55.5%

   \[\sqrt{re \cdot re + im \cdot im} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 31.7

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

5. Applied rewrites 31.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{re \cdot 0.5}{im}, im\right)} \]
6. Add Preprocessing

Alternative 2: 53.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \sqrt{\mathsf{fma}\left(im\_m, im\_m, re\_m \cdot re\_m\right)} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore modulus (re_m im_m)
 :precision binary64
 (sqrt (fma im_m im_m (* re_m re_m))))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double modulus(double re_m, double im_m) {
	return sqrt(fma(im_m, im_m, (re_m * re_m)));
}

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function modulus(re_m, im_m)
	return sqrt(fma(im_m, im_m, Float64(re_m * re_m)))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
modulus[re$95$m_, im$95$m_] := N[Sqrt[N[(im$95$m * im$95$m + N[(re$95$m * re$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 55.5%

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

   1. lift-+.f64 N/A

      \[\leadsto \sqrt{\color{blue}{re \cdot re + im \cdot im}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{im \cdot im + re \cdot re}} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{im \cdot im} + re \cdot re} \]

   4. lower-fma.f64 55.5

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

4. Applied rewrites 55.5%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(im, im, re \cdot re\right)}} \]
5. Add Preprocessing

Alternative 3: 52.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ \sqrt{im\_m \cdot im\_m} \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore modulus (re_m im_m) :precision binary64 (sqrt (* im_m im_m)))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double modulus(double re_m, double im_m) {
	return sqrt((im_m * im_m));
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function modulus(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    modulus = sqrt((im_m * im_m))
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double modulus(double re_m, double im_m) {
	return Math.sqrt((im_m * im_m));
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def modulus(re_m, im_m):
	return math.sqrt((im_m * im_m))

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function modulus(re_m, im_m)
	return sqrt(Float64(im_m * im_m))
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = modulus(re_m, im_m)
	tmp = sqrt((im_m * im_m));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
modulus[re$95$m_, im$95$m_] := N[Sqrt[N[(im$95$m * im$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 55.5%

   \[\sqrt{re \cdot re + im \cdot im} \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 30.4

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

5. Applied rewrites 30.4%

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

Alternative 4: 1.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re_m = \left|re\right| \\ [re_m, im_m] = \mathsf{sort}([re_m, im_m])\\ \\ -re\_m \end{array} \]

FPCore

im_m = (fabs.f64 im)
re_m = (fabs.f64 re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
(FPCore modulus (re_m im_m) :precision binary64 (- re_m))

C

im_m = fabs(im);
re_m = fabs(re);
assert(re_m < im_m);
double modulus(double re_m, double im_m) {
	return -re_m;
}

Fortran

im_m = abs(im)
re_m = abs(re)
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
real(8) function modulus(re_m, im_m)
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    modulus = -re_m
end function

Java

im_m = Math.abs(im);
re_m = Math.abs(re);
assert re_m < im_m;
public static double modulus(double re_m, double im_m) {
	return -re_m;
}

Python

im_m = math.fabs(im)
re_m = math.fabs(re)
[re_m, im_m] = sort([re_m, im_m])
def modulus(re_m, im_m):
	return -re_m

Julia

im_m = abs(im)
re_m = abs(re)
re_m, im_m = sort([re_m, im_m])
function modulus(re_m, im_m)
	return Float64(-re_m)
end

MATLAB

im_m = abs(im);
re_m = abs(re);
re_m, im_m = num2cell(sort([re_m, im_m])){:}
function tmp = modulus(re_m, im_m)
	tmp = -re_m;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
re_m = N[Abs[re], $MachinePrecision]
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
modulus[re$95$m_, im$95$m_] := (-re$95$m)

Derivation

1. Initial program 55.5%

   \[\sqrt{re \cdot re + im \cdot im} \]
2. Add Preprocessing

3. Taylor expanded in re around -inf

   \[\leadsto \color{blue}{-1 \cdot re} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(re\right)} \]

   2. lower-neg.f64 27.2

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

5. Applied rewrites 27.2%

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

Reproduce

herbie shell --seed 2024219 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))