_multiplyComplex, real part

Percentage Accurate: 99.1% → 99.5%

Time: 3.3s

Alternatives: 4

Speedup: 1.0×

• 23.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x.re \cdot y.re - x.im \cdot y.im \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* x.re y.re) (* 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);
}

Fortran

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_46re) - (x_46im * y_46im)
end function

Java

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_re) - (x_46_im * y_46_im);
}

Python

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

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (x_46_re * y_46_re) - (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]

Initial Program: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x.re \cdot y.re - x.im \cdot y.im \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* x.re y.re) (* 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);
}

Fortran

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_46re) - (x_46im * y_46im)
end function

Java

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_re) - (x_46_im * y_46_im);
}

Python

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

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (x_46_re * y_46_re) - (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]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-y.im, x.im, x.re \cdot y.re\right) \end{array} \]

FPCore

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

C

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_im, (x_46_re * y_46_re));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.2

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -5 \cdot 10^{-81}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= (* x.re y.re) -5e-81)
   (* x.re y.re)
   (if (<= (* x.re y.re) 5e-113) (* (- x.im) y.im) (* x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((x_46_re * y_46_re) <= -5e-81) {
		tmp = x_46_re * y_46_re;
	} else if ((x_46_re * y_46_re) <= 5e-113) {
		tmp = -x_46_im * y_46_im;
	} else {
		tmp = x_46_re * y_46_re;
	}
	return tmp;
}

Fortran

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_46re * y_46re) <= (-5d-81)) then
        tmp = x_46re * y_46re
    else if ((x_46re * y_46re) <= 5d-113) then
        tmp = -x_46im * y_46im
    else
        tmp = x_46re * y_46re
    end if
    code = tmp
end function

Java

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_re * y_46_re) <= -5e-81) {
		tmp = x_46_re * y_46_re;
	} else if ((x_46_re * y_46_re) <= 5e-113) {
		tmp = -x_46_im * y_46_im;
	} else {
		tmp = x_46_re * y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (x_46_re * y_46_re) <= -5e-81:
		tmp = x_46_re * y_46_re
	elif (x_46_re * y_46_re) <= 5e-113:
		tmp = -x_46_im * y_46_im
	else:
		tmp = x_46_re * y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(x_46_re * y_46_re) <= -5e-81)
		tmp = Float64(x_46_re * y_46_re);
	elseif (Float64(x_46_re * y_46_re) <= 5e-113)
		tmp = Float64(Float64(-x_46_im) * y_46_im);
	else
		tmp = Float64(x_46_re * y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((x_46_re * y_46_re) <= -5e-81)
		tmp = x_46_re * y_46_re;
	elseif ((x_46_re * y_46_re) <= 5e-113)
		tmp = -x_46_im * y_46_im;
	else
		tmp = x_46_re * y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], -5e-81], N[(x$46$re * y$46$re), $MachinePrecision], If[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 5e-113], N[((-x$46$im) * y$46$im), $MachinePrecision], N[(x$46$re * y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.re) < -4.99999999999999981e-81 or 4.9999999999999997e-113 < (*.f64 x.re y.re)

   1. Initial program 98.6%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if -4.99999999999999981e-81 < (*.f64 x.re y.re) < 4.9999999999999997e-113

   1. Initial program 100.0%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot \left(x.im \cdot y.im\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x.im\right) \cdot y.im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x.im\right) \cdot y.im} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 83.6

         \[\leadsto \color{blue}{\left(-x.im\right)} \cdot y.im \]

   5. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(-x.im\right) \cdot y.im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -5 \cdot 10^{-81}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))