_multiplyComplex, real part

Average Accuracy: 76.4% → 76.5%

Time: 2.7s

Precision: binary64

Cost: 6784

Math

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

↓

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

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 x.re y.re (* y.im (- x.im))))

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(x_46_re, y_46_re, (y_46_im * -x_46_im));
}

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(x_46_re, y_46_re, Float64(y_46_im * Float64(-x_46_im)))
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[(x$46$re * y$46$re + N[(y$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

2. Simplified 76.9%

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

   [Start] 76.9	\[ x.re \cdot y.re - x.im \cdot y.im \]
   fma-neg [=>] 76.9	\[ \color{blue}{\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)} \]
   distribute-rgt-neg-in [=>] 76.9	\[ \mathsf{fma}\left(x.re, y.re, \color{blue}{x.im \cdot \left(-y.im\right)}\right) \]

3. Final simplification 76.9%

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

Alternatives

Alternative 1
Accuracy	51.3%
Cost	521

\[\begin{array}{l} \mathbf{if}\;x.im \leq -6.9 \cdot 10^{-99} \lor \neg \left(x.im \leq 7 \cdot 10^{-44}\right):\\ \;\;\;\;y.im \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]

Alternative 2
Accuracy	76.4%
Cost	448

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

Alternative 3
Accuracy	39.6%
Cost	192

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

Reproduce

herbie shell --seed 2023157 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))