_multiplyComplex, imaginary part

?

Percentage Accurate: 99.3% → 99.7%
Time: 1.8s
Precision: binary64
Cost: 6720

?

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

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

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]

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

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Derivation?

  1. Initial program 98.0%

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

  2. Taylor expanded in x.re around 0 98.0%

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

  3. Simplified100.0%

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

    [Start]98.0

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

    +-commutative [=>]98.0

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

    fma-def [=>]100.0

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

  4. Final simplification100.0%

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

Alternatives

Alternative 1
Accuracy99.7%
Cost964
\[\begin{array}{l} t_0 := x.re \cdot y.im + y.re \cdot x.im\\ \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.im\\ \end{array} \]
Alternative 2
Accuracy76.7%
Cost712
\[\begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{elif}\;x.re \cdot y.im \leq 310000000000:\\ \;\;\;\;y.re \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.im\\ \end{array} \]
Alternative 3
Accuracy51.7%
Cost192
\[x.re \cdot y.im \]

Error

Reproduce?

herbie shell --seed 2023159 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))