_multiplyComplex, real part

Average Error: 0.0 → 0.0

Time: 6.2s

Precision: binary64

Cost: 13632

Math

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

↓

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

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

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.3

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

3. Applied egg-rr 0.3

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

4. Taylor expanded in y.im around 0 0.0

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

5. Simplified 0.0

   \[\leadsto \mathsf{fma}\left(y.re, x.re, -x.im \cdot y.im\right) + \left(\mathsf{fma}\left(-y.im, x.im, x.im \cdot y.im\right) + \color{blue}{0}\right) \]
   Proof
   0: 0 points increase in error, 0 points decrease in error
   (Rewrite<= mul0-lft_binary64 (*.f64 0 y.im)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 x.im)) y.im): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) x.im) y.im): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 -1 x.im) x.im)) y.im): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	6784

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

Alternative 2
Error	16.7
Cost	1296

\[\begin{array}{l} t_0 := x.im \cdot \left(-y.im\right)\\ \mathbf{if}\;y.re \cdot x.re \leq -1.8634962517508274 \cdot 10^{+69}:\\ \;\;\;\;y.re \cdot x.re\\ \mathbf{elif}\;y.re \cdot x.re \leq -8.514816003764764 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \cdot x.re \leq -3.6101526656525626 \cdot 10^{-128}:\\ \;\;\;\;y.re \cdot x.re\\ \mathbf{elif}\;y.re \cdot x.re \leq 1.1696838312395122 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.re\\ \end{array} \]

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	31.3
Cost	192

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

Reproduce

herbie shell --seed 2022296 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))