math.abs on complex

Percentage Accurate: 55.0% → 99.6%

Time: 5.7s

Alternatives: 2

Speedup: 24.0×

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: 55.0% 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: 99.6% accurate, 1.0× speedup

Math

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

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 im_m) re_m) 0.5 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 / im_m) * re_m), 0.5, 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(Float64(Float64(re_m / im_m) * re_m), 0.5, 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[(N[(N[(re$95$m / im$95$m), $MachinePrecision] * re$95$m), $MachinePrecision] * 0.5 + im$95$m), $MachinePrecision]

Derivation

1. Initial program 57.6%

   \[\sqrt{re \cdot re + im \cdot im} \]
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. lower-fma.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 28.7

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

5. Applied rewrites 28.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{im}, re \cdot re, im\right)} \]
6. Step-by-step derivation

7. Applied rewrites 29.1%

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

Alternative 2: 99.0% accurate, 24.0× speedup

Math

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

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 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 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 = 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 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 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 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 = 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_] := im$95$m

Derivation

1. Initial program 57.6%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-/.f64 N/A

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

4. Applied rewrites 41.3%

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

5. Taylor expanded in re around 0

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

   1. *-lft-identity N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r/ N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. mul0-lft N/A

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

   11. metadata-eval N/A

       \[\leadsto 1 \cdot im + \frac{re \cdot \color{blue}{0}}{im} \]

   12. *-commutative N/A

       \[\leadsto 1 \cdot im + \frac{\color{blue}{0 \cdot re}}{im} \]

   13. associate-*r/ N/A

       \[\leadsto 1 \cdot im + \color{blue}{0 \cdot \frac{re}{im}} \]

   14. mul0-lft N/A

       \[\leadsto 1 \cdot im + \color{blue}{0} \]

   15. mul0-lft N/A

       \[\leadsto 1 \cdot im + \color{blue}{0 \cdot \frac{re}{im}} \]

   16. associate-*r/ N/A

       \[\leadsto 1 \cdot im + \color{blue}{\frac{0 \cdot re}{im}} \]

   17. mul0-lft N/A

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

   18. mul0-lft N/A

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

   19. metadata-eval N/A

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

   20. mul0-lft N/A

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

   21. metadata-eval N/A

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

   22. distribute-rgt1-in N/A

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

7. Applied rewrites 28.9%

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

Reproduce

herbie shell --seed 2024249 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))