math.abs on complex

Average Error: 31.2 → 0.0

Time: 3.5s

Precision: binary64

Cost: 6528

Math

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

↓

\[\mathsf{hypot}\left(re, im\right) \]

FPCore

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

↓

(FPCore modulus (re im) :precision binary64 (hypot re im))

C

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

↓

double modulus(double re, double im) {
	return hypot(re, im);
}

Java

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

↓

public static double modulus(double re, double im) {
	return Math.hypot(re, im);
}

Python

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

↓

def modulus(re, im):
	return math.hypot(re, im)

Julia

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

↓

function modulus(re, im)
	return hypot(re, im)
end

MATLAB

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

↓

function tmp = modulus(re, im)
	tmp = hypot(re, im);
end

Wolfram

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

↓

modulus[re_, im_] := N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]

Derivation

1. Initial program 31.2

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{hypot}\left(re, im\right)} \]
   Proof
   (hypot.f64 re im): 0 points increase in error, 0 points decrease in error
   (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im)))): 143 points increase in error, 1 points decrease in error

3. Final simplification 0.0

   \[\leadsto \mathsf{hypot}\left(re, im\right) \]

Alternatives

Alternative 1
Error	36.6
Cost	788

\[\begin{array}{l} \mathbf{if}\;re \leq -1.0005096622161097 \cdot 10^{-35}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -3.295502009220733 \cdot 10^{-49}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq -5.962936472060393 \cdot 10^{-101}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -3.4550455850610005 \cdot 10^{-177}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq -1.618986003622812 \cdot 10^{-205}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 2
Error	36.7
Cost	788

\[\begin{array}{l} \mathbf{if}\;re \leq -1.0005096622161097 \cdot 10^{-35}:\\ \;\;\;\;\frac{im}{\frac{re}{im}} \cdot -0.5 - re\\ \mathbf{elif}\;re \leq -3.295502009220733 \cdot 10^{-49}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq -5.962936472060393 \cdot 10^{-101}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -3.4550455850610005 \cdot 10^{-177}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq -1.618986003622812 \cdot 10^{-205}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 3
Error	46.7
Cost	64

\[im \]

Reproduce

herbie shell --seed 2022300 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))