math.abs on complex

Percentage Accurate: 54.0% → 100.0%

Time: 1.7s

Alternatives: 4

Speedup: 1.1×

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: 100.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore modulus (re im) :precision binary64 (hypot re im))

C

double modulus(double re, double im) {
	return hypot(re, im);
}

Java

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

Python

def modulus(re, im):
	return math.hypot(re, im)

Julia

function modulus(re, im)
	return hypot(re, im)
end

MATLAB

function tmp = modulus(re, im)
	tmp = hypot(re, im);
end

Wolfram

modulus[re_, im_] := N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. lift-sqrt.f64 N/A

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

   2. sqrt-fabs-rev N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-sqrt-square-rev N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-+.f64 N/A

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

   9. add-flip N/A

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

   10. sub-flip N/A

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

   11. remove-double-neg N/A

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

   12. lift-*.f64 N/A

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

   13. sqr-abs-rev N/A

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

   14. sqr-neg-rev N/A

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

   15. fp-cancel-sign-sub-inv N/A

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

   16. fp-cancel-sub-sign-inv N/A

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

   17. lift-*.f64 N/A

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

   18. distribute-lft-neg-in N/A

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

   19. distribute-rgt-neg-out N/A

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

   20. sqr-neg-rev N/A

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

   21. sqr-neg-rev N/A

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

   22. sqr-abs-rev N/A

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

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(re, im\right)} \]
4. Add Preprocessing

Alternative 2: 54.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-flip N/A

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

   4. remove-double-neg N/A

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

   5. lift-*.f64 N/A

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

   6. sqr-abs-rev N/A

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

   7. sqr-neg-rev N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. fp-cancel-sub-sign-inv N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. distribute-lft-neg-in N/A

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

   13. sqr-neg-rev N/A

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

   14. sqr-abs-rev N/A

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

   15. lift-*.f64 N/A

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

   16. remove-double-neg N/A

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

   17. lower-fma.f64 54.0%

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

3. Applied rewrites 54.0%

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

Alternative 3: 50.6% accurate, 2.2× speedup

Math

\[-\mathsf{min}\left(re, im\right) \]

FPCore

(FPCore modulus (re im) :precision binary64 (- (fmin re im)))

C

double modulus(double re, double im) {
	return -fmin(re, 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 = -fmin(re, im)
end function

Java

public static double modulus(double re, double im) {
	return -fmin(re, im);
}

Python

def modulus(re, im):
	return -fmin(re, im)

Julia

function modulus(re, im)
	return Float64(-fmin(re, im))
end

MATLAB

function tmp = modulus(re, im)
	tmp = -min(re, im);
end

Wolfram

modulus[re_, im_] := (-N[Min[re, im], $MachinePrecision])

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in re around -inf

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

   1. lower-*.f64 26.3%

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

4. Applied rewrites 26.3%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 26.3%

      \[\leadsto -re \]

6. Applied rewrites 26.3%

   \[\leadsto -re \]
7. Add Preprocessing

Alternative 4: 26.6% accurate, 5.8× speedup

Math

\[-im \]

FPCore

(FPCore modulus (re im) :precision binary64 (- im))

C

double modulus(double re, double im) {
	return -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 = -im
end function

Java

public static double modulus(double re, double im) {
	return -im;
}

Python

def modulus(re, im):
	return -im

Julia

function modulus(re, im)
	return Float64(-im)
end

MATLAB

function tmp = modulus(re, im)
	tmp = -im;
end

Wolfram

modulus[re_, im_] := (-im)

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in im around -inf

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

   1. lower-*.f64 26.6%

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

4. Applied rewrites 26.6%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   3. lower-neg.f64 26.6%

      \[\leadsto -im \]

6. Applied rewrites 26.6%

   \[\leadsto -im \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))