Average Error: 0.0 → 0.0
Time: 1.5s
Precision: binary64
\[re \cdot re - im \cdot im \]
↓
\[\mathsf{fma}\left(re, re, -im \cdot im\right) \]
(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))
↓
(FPCore re_sqr (re im) :precision binary64 (fma re re (- (* im im))))
double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}
↓
double re_sqr(double re, double im) {
	return fma(re, re, -(im * im));
}

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end
↓
function re_sqr(re, im)
	return fma(re, re, Float64(-Float64(im * im)))
end

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

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[re \cdot re - im \cdot im \]

  2. Applied fma-neg_binary640.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2022131 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))