math.abs on complex

Average Error: 31.5 → 8.0

Time: 4.4s

Precision: binary64

Cost: 7376

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-179}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{-165}:\\ \;\;\;\;im\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{-116}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

FPCore

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

↓

(FPCore modulus (re im)
 :precision binary64
 (if (<= im 2.8e-179)
   (- re)
   (if (<= im 3.6e-165)
     im
     (if (<= im 1.8e-116)
       (- re)
       (if (<= im 5.6e+144) (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 <= 2.8e-179) {
		tmp = -re;
	} else if (im <= 3.6e-165) {
		tmp = im;
	} else if (im <= 1.8e-116) {
		tmp = -re;
	} else if (im <= 5.6e+144) {
		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 <= 2.8d-179) then
        tmp = -re
    else if (im <= 3.6d-165) then
        tmp = im
    else if (im <= 1.8d-116) then
        tmp = -re
    else if (im <= 5.6d+144) 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 <= 2.8e-179) {
		tmp = -re;
	} else if (im <= 3.6e-165) {
		tmp = im;
	} else if (im <= 1.8e-116) {
		tmp = -re;
	} else if (im <= 5.6e+144) {
		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 <= 2.8e-179:
		tmp = -re
	elif im <= 3.6e-165:
		tmp = im
	elif im <= 1.8e-116:
		tmp = -re
	elif im <= 5.6e+144:
		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 <= 2.8e-179)
		tmp = Float64(-re);
	elseif (im <= 3.6e-165)
		tmp = im;
	elseif (im <= 1.8e-116)
		tmp = Float64(-re);
	elseif (im <= 5.6e+144)
		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 <= 2.8e-179)
		tmp = -re;
	elseif (im <= 3.6e-165)
		tmp = im;
	elseif (im <= 1.8e-116)
		tmp = -re;
	elseif (im <= 5.6e+144)
		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, 2.8e-179], (-re), If[LessEqual[im, 3.6e-165], im, If[LessEqual[im, 1.8e-116], (-re), If[LessEqual[im, 5.6e+144], N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], im]]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.8000000000000001e-179 or 3.59999999999999984e-165 < im < 1.79999999999999988e-116

   1. Initial program 30.3

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

   2. Taylor expanded in re around -inf 7.2

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

   3. Simplified 7.2

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

      [Start] 7.2	\[ -1 \cdot re \]
      rational.json-simplify-2 [=>] 7.2	\[ \color{blue}{re \cdot -1} \]
      rational.json-simplify-8 [<=] 7.2	\[ \color{blue}{-re} \]

   if 2.8000000000000001e-179 < im < 3.59999999999999984e-165 or 5.60000000000000013e144 < im

   1. Initial program 59.9

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

   2. Taylor expanded in re around 0 6.8

      \[\leadsto \color{blue}{im} \]

   if 1.79999999999999988e-116 < im < 5.60000000000000013e144

   1. Initial program 10.1

      \[\sqrt{re \cdot re + im \cdot im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-179}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{-165}:\\ \;\;\;\;im\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{-116}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternatives

Alternative 1
Error	11.1
Cost	524

\[\begin{array}{l} \mathbf{if}\;im \leq 2 \cdot 10^{-179}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{-166}:\\ \;\;\;\;im\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{-113}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 2
Error	31.3
Cost	64

\[im \]

Reproduce

herbie shell --seed 2023077 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))