_multiplyComplex, real part

Percentage Accurate: 99.3% → 99.6%

Time: 2.1s

Precision: binary64

Cost: 6784

• Unsound rule application detected in e-graph. Results may not be sound.

• 23.3% of points produce a very large (infinite) output. You may want to add a precondition.

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(Float64(-y_46_im), 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 99.6%

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

2. Applied egg-rr 100.0%

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

   [Start] 99.6	\[ x.re \cdot y.re - x.im \cdot y.im \]
   sub-neg [=>] 99.6	\[ \color{blue}{x.re \cdot y.re + \left(-x.im \cdot y.im\right)} \]
   +-commutative [=>] 99.6	\[ \color{blue}{\left(-x.im \cdot y.im\right) + x.re \cdot y.re} \]
   *-commutative [=>] 99.6	\[ \left(-\color{blue}{y.im \cdot x.im}\right) + x.re \cdot y.re \]
   distribute-lft-neg-in [=>] 99.6	\[ \color{blue}{\left(-y.im\right) \cdot x.im} + x.re \cdot y.re \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(-y.im, x.im, x.re \cdot y.re\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	6784

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

Alternative 2
Accuracy	76.4%
Cost	776

\[\begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -3.2 \cdot 10^{-55}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;y.im \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	448

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

Alternative 4
Accuracy	51.0%
Cost	192

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

Reproduce

herbie shell --seed 2023160 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))