Average Error: 0.0 → 0.0
Time: 1.4s
Precision: binary64
Cost: 6720
\[ \begin{array}{c}[re, im] = \mathsf{sort}([re, im])\\ \end{array} \]
\[re \cdot re + im \cdot im \]
↓
\[\mathsf{fma}\left(im, im, re \cdot re\right) \]
(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))
↓
(FPCore modulus_sqr (re im) :precision binary64 (fma im im (* re re)))
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));
}

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

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]

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

Error

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded in re around 0 0.0

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

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, re \cdot re\right)} \]
    Proof
    (fma.f64 im im (*.f64 re re)): 0 points increase in error, 0 points decrease in error
    (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 im im) (*.f64 re re))): 1 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 re re) (*.f64 im im))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 re 2)) (*.f64 im im)): 0 points increase in error, 0 points decrease in error
    (+.f64 (pow.f64 re 2) (Rewrite<= unpow2_binary64 (pow.f64 im 2))): 0 points increase in error, 0 points decrease in error

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error8.1
Cost588
\[\begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-145}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;im \cdot im\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{-73}:\\ \;\;\;\;re \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot im\\ \end{array} \]
Alternative 2
Error0.0
Cost448
\[re \cdot re + im \cdot im \]
Alternative 3
Error27.2
Cost192
\[im \cdot im \]

Error

Reproduce

herbie shell --seed 2022329 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))