math.abs on complex (squared)

Percentage Accurate: 100.0% → 100.0%

Time: 1.5s

Alternatives: 2

Speedup: 1.2×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))

C

double modulus_sqr(double re, double im) {
	return (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_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    modulus_sqr = (re * re) + (im * im)
end function

Java

public static double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}

Python

def modulus_sqr(re, im):
	return (re * re) + (im * im)

Julia

function modulus_sqr(re, im)
	return Float64(Float64(re * re) + Float64(im * im))
end

MATLAB

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

Wolfram

modulus$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))

C

double modulus_sqr(double re, double im) {
	return (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_sqr(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    modulus_sqr = (re * re) + (im * im)
end function

Java

public static double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}

Python

def modulus_sqr(re, im):
	return (re * re) + (im * im)

Julia

function modulus_sqr(re, im)
	return Float64(Float64(re * re) + Float64(im * im))
end

MATLAB

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

Wolfram

modulus$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 78.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

re_m =     private
NOTE: re_m and im 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_sqr(re_m, im)
use fmin_fmax_functions
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im
    modulus_sqr = im * im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[re \cdot re + im \cdot im \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 53.7

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

5. Applied rewrites 53.7%

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

Reproduce

herbie shell --seed 2025017 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))