math.square on complex, real part

Average Error: 0.0 → 0.0

Time: 1.3s

Precision: binary64

Cost: 6784

Math

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

↓

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

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

↓

(FPCore re_sqr (re im) :precision binary64 (fma re re (* im (- im))))

C

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));
}

Julia

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

↓

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

Wolfram

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]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	14.5
Cost	1556

\[\begin{array}{l} t_0 := im \cdot \left(-im\right)\\ \mathbf{if}\;re \cdot re \leq 1.3 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \cdot re \leq 2.4 \cdot 10^{-226}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;re \cdot re \leq 2.6 \cdot 10^{-168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \cdot re \leq 2.2 \cdot 10^{-120}:\\ \;\;\;\;re \cdot re\\ \mathbf{elif}\;re \cdot re \leq 6.8 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

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

Alternative 3
Error	27.8
Cost	192

\[re \cdot re \]

Reproduce

herbie shell --seed 2022325 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))