Average Error: 0.0 → 0.0
Time: 1.1s
Precision: binary64
\[re \cdot re + im \cdot im \]
↓
\[\mathsf{fma}\left(im, im, re \cdot re\right) \]
(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

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Taylor expanded in re around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022160 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))