_multiplyComplex, imaginary part

Percentage Accurate: 99.2% → 99.7%

Time: 4.7s

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: 99.2% 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: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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, (y_46_im * x_46_re));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 98.4

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \cdot y.re \leq -2 \cdot 10^{+132} \lor \neg \left(x.im \cdot y.re \leq 2 \cdot 10^{-104}\right):\\ \;\;\;\;y.re \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.im y.re) -2e+132) (not (<= (* x.im y.re) 2e-104)))
   (* y.re x.im)
   (* y.im x.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+132) || !((x_46_im * y_46_re) <= 2e-104)) {
		tmp = y_46_re * x_46_im;
	} else {
		tmp = y_46_im * x_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+132)) .or. (.not. ((x_46im * y_46re) <= 2d-104))) then
        tmp = y_46re * x_46im
    else
        tmp = y_46im * x_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+132) || !((x_46_im * y_46_re) <= 2e-104)) {
		tmp = y_46_re * x_46_im;
	} else {
		tmp = y_46_im * x_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+132) or not ((x_46_im * y_46_re) <= 2e-104):
		tmp = y_46_re * x_46_im
	else:
		tmp = y_46_im * x_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+132) || !(Float64(x_46_im * y_46_re) <= 2e-104))
		tmp = Float64(y_46_re * x_46_im);
	else
		tmp = Float64(y_46_im * x_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+132) || ~(((x_46_im * y_46_re) <= 2e-104)))
		tmp = y_46_re * x_46_im;
	else
		tmp = y_46_im * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[N[(x$46$im * y$46$re), $MachinePrecision], -2e+132], N[Not[LessEqual[N[(x$46$im * y$46$re), $MachinePrecision], 2e-104]], $MachinePrecision]], N[(y$46$re * x$46$im), $MachinePrecision], N[(y$46$im * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.im y.re) < -1.99999999999999998e132 or 1.99999999999999985e-104 < (*.f64 x.im y.re)

   1. Initial program 96.2%

      \[x.re \cdot y.im + x.im \cdot y.re \]
   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 78.3

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

   5. Applied rewrites 78.3%

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

   if -1.99999999999999998e132 < (*.f64 x.im y.re) < 1.99999999999999985e-104

   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 23.4

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

   5. Applied rewrites 23.4%

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

   7. Applied rewrites 9.1%

      \[\leadsto \left|y.re \cdot x.im\right| \]

   8. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.2

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

   10. Applied rewrites 80.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.re \leq -2 \cdot 10^{+132} \lor \neg \left(x.im \cdot y.re \leq 2 \cdot 10^{-104}\right):\\ \;\;\;\;y.re \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_im * x_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 = y_46im * x_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 y_46_im * x_46_re;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

   \[x.re \cdot y.im + x.im \cdot y.re \]
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 51.9

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

5. Applied rewrites 51.9%

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

7. Applied rewrites 31.0%

   \[\leadsto \left|y.re \cdot x.im\right| \]

8. Taylor expanded in x.re around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 51.9

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

10. Applied rewrites 51.9%

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))