math.abs on complex

Percentage Accurate: 53.6% → 100.0%

Time: 1.6s

Alternatives: 2

Speedup: 1.9×

• Could not converge on a ground truth

  1. re = 4.614906735260092e-21
  2. im = +nan.0

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

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: 53.6% 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

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, 0.7× 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 53.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lift-*.f64 N/A

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

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

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

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

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

   10. sqr-neg-rev N/A

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

   11. sqr-neg-rev N/A

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

   12. lower-hypot.f64 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 2: 98.9% accurate, 1.9× 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

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 53.6%

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

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. Taylor expanded in undef-var around zero

   \[\leadsto -1 \cdot 0 \]
6. Step-by-step derivation

7. Applied rewrites 2.7%

   \[\leadsto -1 \cdot 0 \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{0} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(0\right) \]

   3. lower-neg.f64 2.7%

      \[\leadsto -0 \]

9. Applied rewrites 2.7%

   \[\leadsto -0 \]

10. Taylor expanded in im around inf

    \[\leadsto \color{blue}{im} \]
11. Step-by-step derivation

12. Applied rewrites 27.5%

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

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))