_multiplyComplex, imaginary part

Average Error: 0.0 → 0.0

Time: 2.3s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.re x.im (* x.re 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_im) + (x_46_im * y_46_re);
}

↓

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_im, (x_46_re * 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_im) + Float64(x_46_im * y_46_re))
end

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x.re around 0 0.0

   \[\leadsto \color{blue}{x.re \cdot y.im + y.re \cdot x.im} \]

3. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(y.re, x.im, x.re \cdot y.im\right)} \]
   Proof
   (fma.f64 y.re x.im (*.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 y.re x.im) (*.f64 x.re y.im))): 2 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.im y.re)) (*.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 x.re y.im) (*.f64 x.im y.re))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 x.re y.im) (Rewrite=> *-commutative_binary64 (*.f64 y.re x.im))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	6720

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

Alternative 2
Error	17.1
Cost	1232

\[\begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -2.65 \cdot 10^{+70}:\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{elif}\;x.re \cdot y.im \leq -2.05 \cdot 10^{-54}:\\ \;\;\;\;y.re \cdot x.im\\ \mathbf{elif}\;x.re \cdot y.im \leq -4 \cdot 10^{-129}:\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{elif}\;x.re \cdot y.im \leq 3.9 \cdot 10^{-20}:\\ \;\;\;\;y.re \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.im\\ \end{array} \]

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	31.3
Cost	192

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

Reproduce

herbie shell --seed 2022329 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))