math.abs on complex

Percentage Accurate: 54.0% → 99.5%

Time: 1.3s

Alternatives: 2

Speedup: 0.2×

Specification

Math

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

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: 54.0% accurate, 1.0× speedup

Math

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

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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|re\right|, \left|im\right|\right)\\ t_1 := \mathsf{min}\left(\left|re\right|, \left|im\right|\right)\\ t\_0 - \frac{-1}{2} \cdot \left(\frac{t\_1}{t\_0} \cdot t\_1\right) \end{array} \]

FPCore

(FPCore modulus (re im)
  :precision binary64
  (let* ((t_0 (fmax (fabs re) (fabs im)))
       (t_1 (fmin (fabs re) (fabs im))))
  (- t_0 (* -1/2 (* (/ t_1 t_0) t_1)))))

C

double modulus(double re, double im) {
	double t_0 = fmax(fabs(re), fabs(im));
	double t_1 = fmin(fabs(re), fabs(im));
	return t_0 - (-0.5 * ((t_1 / t_0) * t_1));
}

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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = fmax(abs(re), abs(im))
    t_1 = fmin(abs(re), abs(im))
    modulus = t_0 - ((-0.5d0) * ((t_1 / t_0) * t_1))
end function

Java

public static double modulus(double re, double im) {
	double t_0 = fmax(Math.abs(re), Math.abs(im));
	double t_1 = fmin(Math.abs(re), Math.abs(im));
	return t_0 - (-0.5 * ((t_1 / t_0) * t_1));
}

Python

def modulus(re, im):
	t_0 = fmax(math.fabs(re), math.fabs(im))
	t_1 = fmin(math.fabs(re), math.fabs(im))
	return t_0 - (-0.5 * ((t_1 / t_0) * t_1))

Julia

function modulus(re, im)
	t_0 = fmax(abs(re), abs(im))
	t_1 = fmin(abs(re), abs(im))
	return Float64(t_0 - Float64(-0.5 * Float64(Float64(t_1 / t_0) * t_1)))
end

MATLAB

function tmp = modulus(re, im)
	t_0 = max(abs(re), abs(im));
	t_1 = min(abs(re), abs(im));
	tmp = t_0 - (-0.5 * ((t_1 / t_0) * t_1));
end

Wolfram

modulus[re_, im_] := Block[{t$95$0 = N[Max[N[Abs[re], $MachinePrecision], N[Abs[im], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[re], $MachinePrecision], N[Abs[im], $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 - N[(-1/2 * N[(N[(t$95$1 / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 23.8%

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

4. Applied rewrites 23.8%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. lift-pow.f64 N/A

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

   6. pow2 N/A

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

   7. sqr-neg-rev N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. frac-2neg-rev N/A

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

   11. lower-/.f64 N/A

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

   12. frac-2neg-rev N/A

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

   13. lower-/.f64 27.0%

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

6. Applied rewrites 27.0%

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

8. Applied rewrites 27.1%

   \[\leadsto im - \color{blue}{\frac{-1}{2} \cdot \left(\frac{re}{im} \cdot re\right)} \]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 0.2× speedup

Math

\[\mathsf{max}\left(\left|re\right|, \left|im\right|\right) \]

FPCore

(FPCore modulus (re im)
  :precision binary64
  (fmax (fabs re) (fabs im)))

C

double modulus(double re, double im) {
	return fmax(fabs(re), fabs(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 = fmax(abs(re), abs(im))
end function

Java

public static double modulus(double re, double im) {
	return fmax(Math.abs(re), Math.abs(im));
}

Python

def modulus(re, im):
	return fmax(math.fabs(re), math.fabs(im))

Julia

function modulus(re, im)
	return fmax(abs(re), abs(im))
end

MATLAB

function tmp = modulus(re, im)
	tmp = max(abs(re), abs(im));
end

Wolfram

modulus[re_, im_] := N[Max[N[Abs[re], $MachinePrecision], N[Abs[im], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 23.8%

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

4. Applied rewrites 23.8%

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

5. Taylor expanded in re around 0

   \[\leadsto im \]
6. Step-by-step derivation

7. Applied rewrites 26.8%

   \[\leadsto im \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))