math.abs on complex (squared)

Average Error: 0.0 → 0.0

Time: 1.6s

Precision: binary64

Cost: 6720

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

Math

\[re \cdot re + im \cdot im \]

↓

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

FPCore

(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))

↓

(FPCore modulus_sqr (re im) :precision binary64 (fma re re (* im im)))

C

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

↓

double modulus_sqr(double re, double im) {
	return fma(re, re, (im * im));
}

Julia

function modulus_sqr(re, im)
	return Float64(Float64(re * re) + Float64(im * im))
end

↓

function modulus_sqr(re, im)
	return fma(re, re, Float64(im * im))
end

Wolfram

modulus$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]

↓

modulus$95$sqr[re_, im_] := N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[re \cdot re + im \cdot im \]

2. Simplified 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	9.1
Cost	588

\[\begin{array}{l} \mathbf{if}\;re \leq -1.8773351392824087 \cdot 10^{+21}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;re \leq -14988.653602232162:\\ \;\;\;\;im \cdot im\\ \mathbf{elif}\;re \leq -8.293314044445579 \cdot 10^{-162}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot im\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

\[im \cdot im + re \cdot re \]

Alternative 3
Error	27.5
Cost	192

\[re \cdot re \]

Reproduce

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