_multiplyComplex, imaginary part

Percentage Accurate: 99.3% → 99.7%
Time: 2.6s
Alternatives: 2
Speedup: 1.2×

Specification

?
\[\begin{array}{l} \\ x.re \cdot y.im + x.im \cdot y.re \end{array} \]
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (+ (* x.re y.im) (* x.im y.re)))
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);
}

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = (x_46re * y_46im) + (x_46im * y_46re)
end function

public static 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);
}

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (x_46_re * y_46_im) + (x_46_im * y_46_re)

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 tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (x_46_re * y_46_im) + (x_46_im * y_46_re);
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]

\begin{array}{l}

\\
x.re \cdot y.im + x.im \cdot y.re
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

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.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.3% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x.re \cdot y.im + x.im \cdot y.re \end{array} \]
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (+ (* x.re y.im) (* x.im y.re)))
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);
}

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = (x_46re * y_46im) + (x_46im * y_46re)
end function

public static 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);
}

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (x_46_re * y_46_im) + (x_46_im * y_46_re)

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 tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (x_46_re * y_46_im) + (x_46_im * y_46_re);
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]

\begin{array}{l}

\\
x.re \cdot y.im + x.im \cdot y.re
\end{array}

Alternative 1: 99.7% accurate, 1.2× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(y.im, x.re, x.im \cdot y.re\right) \end{array} \]
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.im x.re (* x.im y.re)))
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_re, (x_46_im * y_46_re));
}

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

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

\begin{array}{l}

\\
\mathsf{fma}\left(y.im, x.re, x.im \cdot y.re\right)
\end{array}
Derivation

  1. Initial program 99.6%

    \[x.re \cdot y.im + x.im \cdot y.re \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-+.f64N/A

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

    2. lift-*.f64N/A

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

    3. *-commutativeN/A

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

    4. lower-fma.f64100.0

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

    5. lift-*.f64N/A

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

    6. *-commutativeN/A

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

    7. lower-*.f64100.0

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

  4. Applied rewrites100.0%

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

  5. Final simplification100.0%

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

Alternative 2: 52.4% accurate, 2.3× speedup?

\[\begin{array}{l} \\ x.im \cdot y.re \end{array} \]
(FPCore (x.re x.im y.re y.im) :precision binary64 (* x.im y.re))
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im * y_46_re;
}

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = x_46im * y_46re
end function

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im * y_46_re;
}

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return x_46_im * y_46_re

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

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = x_46_im * y_46_re;
end

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

\begin{array}{l}

\\
x.im \cdot y.re
\end{array}
Derivation

  1. Initial program 99.6%

    \[x.re \cdot y.im + x.im \cdot y.re \]
  2. Add Preprocessing

  3. Taylor expanded in x.re around 0

    \[\leadsto \color{blue}{x.im \cdot y.re} \]
  4. Step-by-step derivation

    1. *-commutativeN/A

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

    2. lower-*.f6448.1

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

  5. Applied rewrites48.1%

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

  6. Final simplification48.1%

    \[\leadsto x.im \cdot y.re \]
  7. Add Preprocessing

Reproduce

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herbie shell --seed 2024332 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))