math.abs on complex

Average Error: 32.1 → 7.6

Time: 3.7s

Precision: binary64

Cost: 7112

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+81}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

FPCore

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

↓

(FPCore modulus (re im)
 :precision binary64
 (if (<= re -1.1e+81)
   (- re)
   (if (<= re -9.2e-164) (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 <= -1.1e+81) {
		tmp = -re;
	} else if (re <= -9.2e-164) {
		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 <= (-1.1d+81)) then
        tmp = -re
    else if (re <= (-9.2d-164)) 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 <= -1.1e+81) {
		tmp = -re;
	} else if (re <= -9.2e-164) {
		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 <= -1.1e+81:
		tmp = -re
	elif re <= -9.2e-164:
		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 <= -1.1e+81)
		tmp = Float64(-re);
	elseif (re <= -9.2e-164)
		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 <= -1.1e+81)
		tmp = -re;
	elseif (re <= -9.2e-164)
		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, -1.1e+81], (-re), If[LessEqual[re, -9.2e-164], N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], im]]

Derivation

1. Split input into 3 regimes

2. if re < -1.09999999999999993e81

   1. Initial program 49.1

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

   2. Taylor expanded in re around -inf 6.9

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

   3. Simplified 6.9

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

      [Start] 6.9	\[ -1 \cdot re \]
      rational.json-simplify-2 [=>] 6.9	\[ \color{blue}{re \cdot -1} \]
      rational.json-simplify-9 [=>] 6.9	\[ \color{blue}{-re} \]

   if -1.09999999999999993e81 < re < -9.19999999999999942e-164

   1. Initial program 12.0

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

   if -9.19999999999999942e-164 < re

   1. Initial program 31.7

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

   2. Taylor expanded in re around 0 5.1

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

4. Final simplification 7.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+81}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -9.2 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternatives

Alternative 1
Error	10.8
Cost	524

\[\begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{-73}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -3.8 \cdot 10^{-86}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq -6 \cdot 10^{-134}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 2
Error	31.1
Cost	64

\[im \]

Reproduce

herbie shell --seed 2023073 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))