_multiplyComplex, real part

Average Error: 0.0 → 0.0

Time: 2.6s

Precision: binary64

Cost: 6784

Math

\[x.re \cdot y.re - x.im \cdot y.im \]

↓

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* x.re y.re) (* x.im y.im)))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.im (- x.im) (* x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (x_46_re * y_46_re) - (x_46_im * y_46_im);
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return fma(y_46_im, -x_46_im, (x_46_re * y_46_re));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(x_46_re * y_46_re) - Float64(x_46_im * y_46_im))
end

↓

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return fma(y_46_im, Float64(-x_46_im), Float64(x_46_re * y_46_re))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(x$46$re * y$46$re), $MachinePrecision] - N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$im * (-x$46$im) + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[x.re \cdot y.re - x.im \cdot y.im \]

2. Taylor expanded in x.re around 0 0.0

   \[\leadsto \color{blue}{-1 \cdot \left(y.im \cdot x.im\right) + x.re \cdot y.re} \]

3. Simplified 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	6784

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

Alternative 2
Error	15.2
Cost	1296

\[\begin{array}{l} t_0 := y.im \cdot \left(-x.im\right)\\ \mathbf{if}\;x.re \cdot y.re \leq -2.5678844387930983 \cdot 10^{-37}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq 8.52868702875946 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \cdot y.re \leq 1.3479656974096894 \cdot 10^{-24}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq 3129268840.642778:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]

Alternative 3
Error	0.0
Cost	448

\[x.re \cdot y.re - y.im \cdot x.im \]

Alternative 4
Error	30.8
Cost	192

\[x.re \cdot y.re \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))