Average Error: 0.0 → 0.0
Time: 2.2s
Precision: binary64
Cost: 6720
\[ \begin{array}{c}[re, im] = \mathsf{sort}([re, im])\\ \end{array} \]
\[re \cdot re + im \cdot im \]
↓
\[\mathsf{fma}\left(re, re, im \cdot im\right) \]
(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))
↓
(FPCore modulus_sqr (re im) :precision binary64 (fma re re (* im im)))
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));
}

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

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]

re \cdot re + im \cdot im
↓
\mathsf{fma}\left(re, re, im \cdot im\right)

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, im \cdot im\right)} \]
    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))): 2 points increase in error, 1 points decrease in error

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost448
\[im \cdot im + re \cdot re \]
Alternative 2
Error7.6
Cost324
\[\begin{array}{l} \mathbf{if}\;im \leq 6.169623707202513 \cdot 10^{-111}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot im\\ \end{array} \]
Alternative 3
Error28.2
Cost192
\[im \cdot im \]

Error

Reproduce

herbie shell --seed 2022291 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))