math.abs on complex

Average Error: 31.0 → 7.1

Time: 2.0s

Precision: binary64

Cost: 7112

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{+106}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -7.2 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

FPCore

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

↓

(FPCore modulus (re im)
 :precision binary64
 (if (<= re -4.8e+106)
   (- re)
   (if (<= re -7.2e-141) (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 (re <= -4.8e+106) {
		tmp = -re;
	} else if (re <= -7.2e-141) {
		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 (re <= (-4.8d+106)) then
        tmp = -re
    else if (re <= (-7.2d-141)) 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 (re <= -4.8e+106) {
		tmp = -re;
	} else if (re <= -7.2e-141) {
		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 re <= -4.8e+106:
		tmp = -re
	elif re <= -7.2e-141:
		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 (re <= -4.8e+106)
		tmp = Float64(-re);
	elseif (re <= -7.2e-141)
		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 (re <= -4.8e+106)
		tmp = -re;
	elseif (re <= -7.2e-141)
		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[re, -4.8e+106], (-re), If[LessEqual[re, -7.2e-141], N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], im]]

Derivation

1. Split input into 3 regimes

2. if re < -4.8000000000000001e106

   1. Initial program 51.5

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

   2. Taylor expanded in re around -inf 5.6

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

   3. Simplified 5.6

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

      [Start] 5.6	\[ -1 \cdot re \]
      rational.json-simplify-2 [=>] 5.6	\[ \color{blue}{re \cdot -1} \]
      rational.json-simplify-9 [=>] 5.6	\[ \color{blue}{-re} \]

   if -4.8000000000000001e106 < re < -7.2000000000000003e-141

   1. Initial program 10.5

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

   if -7.2000000000000003e-141 < re

   1. Initial program 31.4

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

   2. Taylor expanded in re around 0 5.6

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

4. Final simplification 7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{+106}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -7.2 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternatives

Alternative 1
Error	10.8
Cost	260

\[\begin{array}{l} \mathbf{if}\;im \leq 1.95 \cdot 10^{-148}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 2
Error	30.8
Cost	64

\[im \]

Reproduce

herbie shell --seed 2023065 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))