math.abs on complex

Average Accuracy: 50.7% → 100.0%

Time: 1.9s

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 50.7%

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

2. Simplified 100.0%

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

   [Start] 50.7	\[ \sqrt{re \cdot re + im \cdot im} \]
   hypot-def [=>] 100.0	\[ \color{blue}{\mathsf{hypot}\left(re, im\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	42.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;re \leq -1.6 \cdot 10^{-175}:\\ \;\;\;\;\frac{im}{\frac{re}{im}} \cdot -0.5 - re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 2
Accuracy	42.3%
Cost	260

\[\begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{-173}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]

Alternative 3
Accuracy	26.9%
Cost	64

\[im \]

Reproduce

herbie shell --seed 2023125 
(FPCore modulus (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))