math.abs on complex (squared)

Average Error: 0.01% → 0%

Time: 1.1s

Precision: binary64

Cost: 6720

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.01

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

2. Taylor expanded in re around 0 0.01

   \[\leadsto \color{blue}{{re}^{2} + {im}^{2}} \]

3. Simplified 0

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

   [Start] 0.01	\[ {re}^{2} + {im}^{2} \]
   +-commutative [=>] 0.01	\[ \color{blue}{{im}^{2} + {re}^{2}} \]
   unpow2 [=>] 0.01	\[ \color{blue}{im \cdot im} + {re}^{2} \]
   unpow2 [=>] 0.01	\[ im \cdot im + \color{blue}{re \cdot re} \]
   fma-udef [<=] 0	\[ \color{blue}{\mathsf{fma}\left(im, im, re \cdot re\right)} \]

4. Final simplification 0

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

Alternatives

Alternative 1
Error	0.01%
Cost	448

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

Alternative 2
Error	12.14%
Cost	324

\[\begin{array}{l} \mathbf{if}\;re \leq -6.6 \cdot 10^{-136}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot im\\ \end{array} \]

Alternative 3
Error	43.49%
Cost	192

\[im \cdot im \]

Reproduce

herbie shell --seed 2023088 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))