math.abs on complex

Average Error: 31.4 → 7.2

Time: 2.8s

Precision: binary64

Cost: 7112

\[ \begin{array}{c}[re, im] = \mathsf{sort}([re, im])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-168}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

FPCore

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

↓

(FPCore modulus (re im)
 :precision binary64
 (if (<= im 2.5e-168)
   (- re)
   (if (<= im 4.1e+151) (sqrt (+ (* re re) (* im im))) im)))

C

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

↓

double modulus(double re, double im) {
	double tmp;
	if (im <= 2.5e-168) {
		tmp = -re;
	} else if (im <= 4.1e+151) {
		tmp = sqrt(((re * re) + (im * im)));
	} else {
		tmp = im;
	}
	return tmp;
}

Fortran

real(8) function modulus(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    modulus = sqrt(((re * re) + (im * im)))
end function

↓

real(8) function modulus(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.5d-168) then
        tmp = -re
    else if (im <= 4.1d+151) then
        tmp = sqrt(((re * re) + (im * im)))
    else
        tmp = im
    end if
    modulus = tmp
end function

Java

public static double modulus(double re, double im) {
	return Math.sqrt(((re * re) + (im * im)));
}

↓

public static double modulus(double re, double im) {
	double tmp;
	if (im <= 2.5e-168) {
		tmp = -re;
	} else if (im <= 4.1e+151) {
		tmp = Math.sqrt(((re * re) + (im * im)));
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def modulus(re, im):
	return math.sqrt(((re * re) + (im * im)))

↓

def modulus(re, im):
	tmp = 0
	if im <= 2.5e-168:
		tmp = -re
	elif im <= 4.1e+151:
		tmp = math.sqrt(((re * re) + (im * im)))
	else:
		tmp = im
	return tmp

Julia

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

↓

function modulus(re, im)
	tmp = 0.0
	if (im <= 2.5e-168)
		tmp = Float64(-re);
	elseif (im <= 4.1e+151)
		tmp = sqrt(Float64(Float64(re * re) + Float64(im * im)));
	else
		tmp = im;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = modulus(re, im)
	tmp = 0.0;
	if (im <= 2.5e-168)
		tmp = -re;
	elseif (im <= 4.1e+151)
		tmp = sqrt(((re * re) + (im * im)));
	else
		tmp = im;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

modulus[re_, im_] := If[LessEqual[im, 2.5e-168], (-re), If[LessEqual[im, 4.1e+151], N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], im]]

Derivation

1. Split input into 3 regimes

2. if im < 2.50000000000000001e-168

   1. Initial program 32.1

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

   2. Taylor expanded in re around -inf 4.8

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

   3. Simplified 4.8

      \[\leadsto \color{blue}{-re} \]
      Proof

      [Start] 4.8	\[ -1 \cdot re \]
      rational_best-simplify-1 [=>] 4.8	\[ \color{blue}{re \cdot -1} \]
      rational_best-simplify-10 [=>] 4.8	\[ \color{blue}{-re} \]

   if 2.50000000000000001e-168 < im < 4.0999999999999998e151

   1. Initial program 11.2

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

   if 4.0999999999999998e151 < im

   1. Initial program 63.2

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

   2. Taylor expanded in re around 0 4.6

      \[\leadsto \color{blue}{im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \cdot 10^{-168}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternatives

Alternative 1
Error	10.9
Cost	260

\[\begin{array}{l} \mathbf{if}\;im \leq 1.4 \cdot 10^{-27}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 2
Error	31.0
Cost	64

\[im \]

Reproduce

herbie shell --seed 2023100 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))