_multiplyComplex, imaginary part

Percentage Accurate: 99.3% → 99.6%

Time: 3.2s

Alternatives: 3

Speedup: 1.2×

• 23.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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);
}

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_46im) + (x_46im * y_46re)
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_im) + (x_46_im * y_46_re);
}

Python

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)

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

MATLAB

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

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]

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

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);
}

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_46im) + (x_46im * y_46re)
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_im) + (x_46_im * y_46_re);
}

Python

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)

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

MATLAB

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

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]

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 100.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \cdot y.re \leq -2 \cdot 10^{-30}:\\ \;\;\;\;x.im \cdot y.re\\ \mathbf{elif}\;x.im \cdot y.re \leq 2 \cdot 10^{-37}:\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= (* x.im y.re) -2e-30)
   (* x.im y.re)
   (if (<= (* x.im y.re) 2e-37) (* x.re y.im) (* x.im 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_im * y_46_re) <= -2e-30) {
		tmp = x_46_im * y_46_re;
	} else if ((x_46_im * y_46_re) <= 2e-37) {
		tmp = x_46_re * y_46_im;
	} else {
		tmp = x_46_im * 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_46im * y_46re) <= (-2d-30)) then
        tmp = x_46im * y_46re
    else if ((x_46im * y_46re) <= 2d-37) then
        tmp = x_46re * y_46im
    else
        tmp = x_46im * 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_im * y_46_re) <= -2e-30) {
		tmp = x_46_im * y_46_re;
	} else if ((x_46_im * y_46_re) <= 2e-37) {
		tmp = x_46_re * y_46_im;
	} else {
		tmp = x_46_im * 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_im * y_46_re) <= -2e-30:
		tmp = x_46_im * y_46_re
	elif (x_46_im * y_46_re) <= 2e-37:
		tmp = x_46_re * y_46_im
	else:
		tmp = x_46_im * 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_im * y_46_re) <= -2e-30)
		tmp = Float64(x_46_im * y_46_re);
	elseif (Float64(x_46_im * y_46_re) <= 2e-37)
		tmp = Float64(x_46_re * y_46_im);
	else
		tmp = Float64(x_46_im * 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_im * y_46_re) <= -2e-30)
		tmp = x_46_im * y_46_re;
	elseif ((x_46_im * y_46_re) <= 2e-37)
		tmp = x_46_re * y_46_im;
	else
		tmp = x_46_im * 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$46$im * y$46$re), $MachinePrecision], -2e-30], N[(x$46$im * y$46$re), $MachinePrecision], If[LessEqual[N[(x$46$im * y$46$re), $MachinePrecision], 2e-37], N[(x$46$re * y$46$im), $MachinePrecision], N[(x$46$im * y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.im y.re) < -2e-30 or 2.00000000000000013e-37 < (*.f64 x.im y.re)

   1. Initial program 100.0%

      \[x.re \cdot y.im + x.im \cdot y.re \]
   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. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if -2e-30 < (*.f64 x.im y.re) < 2.00000000000000013e-37

   1. Initial program 100.0%

      \[x.re \cdot y.im + x.im \cdot y.re \]
   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. *-commutative N/A

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

      2. lower-*.f64 25.5

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

   5. Applied rewrites 25.5%

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

   6. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.4

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

   8. Applied rewrites 80.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.re \leq -2 \cdot 10^{-30}:\\ \;\;\;\;x.im \cdot y.re\\ \mathbf{elif}\;x.im \cdot y.re \leq 2 \cdot 10^{-37}:\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.3% accurate, 2.3× speedup

Math

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

FPCore

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

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_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_im;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x.re \cdot y.im + x.im \cdot y.re \]
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. *-commutative N/A

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

   2. lower-*.f64 55.6

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

5. Applied rewrites 55.6%

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

6. Taylor expanded in x.re around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 49.4

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

8. Applied rewrites 49.4%

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

9. Final simplification 49.4%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))