_multiplyComplex, real part

Average Error: 0.0 → 0.0

Time: 4.4s

Precision: binary64

Cost: 7296

Math

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

↓

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

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

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	16.3
Cost	1297

\[\begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -3000000:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq 1.34 \cdot 10^{-79} \lor \neg \left(x.re \cdot y.re \leq 3.65 \cdot 10^{-17}\right) \land x.re \cdot y.re \leq 5.5 \cdot 10^{+51}:\\ \;\;\;\;x.im \cdot \left(-y.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]

Alternative 2
Error	0.0
Cost	832

\[x.im \cdot y.im - \left(x.im \cdot \left(y.im + y.im\right) - x.re \cdot y.re\right) \]

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	30.8
Cost	192

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

Reproduce

herbie shell --seed 2023066 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))