math.abs on complex

Percentage Accurate: 54.2% → 98.9%

Time: 2.6s

Alternatives: 2

Speedup: 11.4×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function modulus(re, im)
use fmin_fmax_functions
    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: 54.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function modulus(re, im)
use fmin_fmax_functions
    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: 98.9% accurate, 11.4× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
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

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

Fortran

re_m =     private
im_m =     private
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function modulus(re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    modulus = im_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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 54.2%

   \[\sqrt{re \cdot re + im \cdot im} \]

2. Taylor expanded in im around inf

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

4. Applied rewrites 88.1%

   \[\leadsto \color{blue}{im \cdot \left(1 + 0.5 \cdot \frac{{re}^{2}}{{im}^{2}}\right)} \]

5. Taylor expanded in re around 0

   \[\leadsto im \]

7. Applied rewrites 98.9%

   \[\leadsto im \]
8. Add Preprocessing

Alternative 2: 5.0% accurate, 11.4× speedup

Math

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

FPCore

re_m = (fabs.f64 re)
im_m = (fabs.f64 im)
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

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

Fortran

re_m =     private
im_m =     private
NOTE: re_m and im_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function modulus(re_m, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im_m
    modulus = re_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

re_m = N[Abs[re], $MachinePrecision]
im_m = N[Abs[im], $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 54.2%

   \[\sqrt{re \cdot re + im \cdot im} \]

2. Taylor expanded in re around -inf

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

4. Applied rewrites 1.8%

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

6. Applied rewrites 5.0%

   \[\leadsto re \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))