math.abs on complex

Average Error: 31.5 → 7.2

Time: 1.8s

Precision: binary64

Cost: 7112

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+66}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -2.55 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

FPCore

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

↓

(FPCore modulus (re im)
 :precision binary64
 (if (<= re -1e+66)
   (- re)
   (if (<= re -2.55e-152) (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 <= -1e+66) {
		tmp = -re;
	} else if (re <= -2.55e-152) {
		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 <= (-1d+66)) then
        tmp = -re
    else if (re <= (-2.55d-152)) 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 <= -1e+66) {
		tmp = -re;
	} else if (re <= -2.55e-152) {
		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 <= -1e+66:
		tmp = -re
	elif re <= -2.55e-152:
		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 <= -1e+66)
		tmp = Float64(-re);
	elseif (re <= -2.55e-152)
		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 <= -1e+66)
		tmp = -re;
	elseif (re <= -2.55e-152)
		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, -1e+66], (-re), If[LessEqual[re, -2.55e-152], N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], im]]

Derivation

1. Split input into 3 regimes

2. if re < -9.99999999999999945e65

   1. Initial program 45.5

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

   2. Taylor expanded in re around -inf 6.4

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

   3. Simplified 6.4

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

      [Start] 6.4	\[ -1 \cdot re \]
      rational_best-simplify-2 [=>] 6.4	\[ \color{blue}{re \cdot -1} \]
      rational_best-simplify-12 [=>] 6.4	\[ \color{blue}{-re} \]

   if -9.99999999999999945e65 < re < -2.5500000000000002e-152

   1. Initial program 11.4

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

   if -2.5500000000000002e-152 < re

   1. Initial program 31.7

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

   2. Taylor expanded in re around 0 5.2

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

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{+66}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -2.55 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	260

\[\begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{-72}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 2
Error	31.3
Cost	64

\[im \]

Reproduce

herbie shell --seed 2023096 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))