_multiplyComplex, imaginary part

Percentage Accurate: 99.3% → 99.7%
Time: 1.7s
Alternatives: 4
Speedup: 1.0×

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 4 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, 0.1× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right) \end{array} \]
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma 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 fma(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 fma(x_46_re, y_46_im, Float64(x_46_im * y_46_re))
end

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

\begin{array}{l}

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

  1. Initial program 98.8%

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

    1. fma-def100.0%

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

  3. Simplified100.0%

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

  4. Final simplification100.0%

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

Alternative 2: 67.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.8 \cdot 10^{+51}:\\ \;\;\;\;x.im \cdot y.re\\ \mathbf{elif}\;x.im \leq 2.3 \cdot 10^{-106}:\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \end{array} \]
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -1.8e+51)
   (* x.im y.re)
   (if (<= x.im 2.3e-106) (* 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) {
	double tmp;
	if (x_46_im <= -1.8e+51) {
		tmp = x_46_im * y_46_re;
	} else if (x_46_im <= 2.3e-106) {
		tmp = x_46_re * y_46_im;
	} else {
		tmp = x_46_im * y_46_re;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x_46im <= (-1.8d+51)) then
        tmp = x_46im * y_46re
    else if (x_46im <= 2.3d-106) then
        tmp = x_46re * y_46im
    else
        tmp = x_46im * y_46re
    end if
    code = tmp
end function

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -1.8e+51) {
		tmp = x_46_im * y_46_re;
	} else if (x_46_im <= 2.3e-106) {
		tmp = x_46_re * y_46_im;
	} else {
		tmp = x_46_im * y_46_re;
	}
	return tmp;
}

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -1.8e+51:
		tmp = x_46_im * y_46_re
	elif x_46_im <= 2.3e-106:
		tmp = x_46_re * y_46_im
	else:
		tmp = x_46_im * y_46_re
	return tmp

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -1.8e+51)
		tmp = Float64(x_46_im * y_46_re);
	elseif (x_46_im <= 2.3e-106)
		tmp = Float64(x_46_re * y_46_im);
	else
		tmp = Float64(x_46_im * y_46_re);
	end
	return tmp
end

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -1.8e+51)
		tmp = x_46_im * y_46_re;
	elseif (x_46_im <= 2.3e-106)
		tmp = x_46_re * y_46_im;
	else
		tmp = x_46_im * y_46_re;
	end
	tmp_2 = tmp;
end

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -1.8e+51], N[(x$46$im * y$46$re), $MachinePrecision], If[LessEqual[x$46$im, 2.3e-106], N[(x$46$re * y$46$im), $MachinePrecision], N[(x$46$im * y$46$re), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x.im \leq -1.8 \cdot 10^{+51}:\\
\;\;\;\;x.im \cdot y.re\\

\mathbf{elif}\;x.im \leq 2.3 \cdot 10^{-106}:\\
\;\;\;\;x.re \cdot y.im\\

\mathbf{else}:\\
\;\;\;\;x.im \cdot y.re\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if x.im < -1.80000000000000005e51 or 2.3000000000000001e-106 < x.im

    1. Initial program 97.8%

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

    2. Taylor expanded in x.re around 0 61.6%

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

    if -1.80000000000000005e51 < x.im < 2.3000000000000001e-106

    1. Initial program 100.0%

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

    2. Taylor expanded in x.re around inf 69.3%

      \[\leadsto \color{blue}{x.re \cdot y.im} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification65.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.8 \cdot 10^{+51}:\\ \;\;\;\;x.im \cdot y.re\\ \mathbf{elif}\;x.im \leq 2.3 \cdot 10^{-106}:\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \]

Alternative 3: 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}
Derivation

  1. Initial program 98.8%

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

  2. Final simplification98.8%

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

Alternative 4: 51.9% accurate, 2.3× speedup?

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

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
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;
}

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

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

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

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

\begin{array}{l}

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

  1. Initial program 98.8%

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

  2. Taylor expanded in x.re around inf 54.4%

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

  3. Final simplification54.4%

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

Reproduce

?
herbie shell --seed 2023199 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))