_multiplyComplex, imaginary part

Percentage Accurate: 99.2% → 99.6%

Time: 3.1s

Alternatives: 4

Speedup: 1.0×

• 23.6% 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: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. +-commutative 98.8%

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

   2. fma-define 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 76.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -4.4 \cdot 10^{+16} \lor \neg \left(x.re \cdot y.im \leq 0.76\right):\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.re y.im) -4.4e+16) (not (<= (* x.re y.im) 0.76)))
   (* 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_re * y_46_im) <= -4.4e+16) || !((x_46_re * y_46_im) <= 0.76)) {
		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_46re * y_46im) <= (-4.4d+16)) .or. (.not. ((x_46re * y_46im) <= 0.76d0))) 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_re * y_46_im) <= -4.4e+16) || !((x_46_re * y_46_im) <= 0.76)) {
		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_re * y_46_im) <= -4.4e+16) or not ((x_46_re * y_46_im) <= 0.76):
		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_re * y_46_im) <= -4.4e+16) || !(Float64(x_46_re * y_46_im) <= 0.76))
		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_re * y_46_im) <= -4.4e+16) || ~(((x_46_re * y_46_im) <= 0.76)))
		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[Or[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], -4.4e+16], N[Not[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], 0.76]], $MachinePrecision]], 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.re y.im) < -4.4e16 or 0.76000000000000001 < (*.f64 x.re y.im)

   1. Initial program 97.5%

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

   3. Taylor expanded in x.re around inf 78.1%

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

   if -4.4e16 < (*.f64 x.re y.im) < 0.76000000000000001

   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 81.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -4.4 \cdot 10^{+16} \lor \neg \left(x.re \cdot y.im \leq 0.76\right):\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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]

Derivation

1. Initial program 98.8%

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

Alternative 4: 52.8% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* 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_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_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_im * y_46_re;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in x.re around 0 53.3%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))