_multiplyComplex, real part

Percentage Accurate: 99.3% → 99.6%

Time: 2.8s

Alternatives: 4

Speedup: 1.0×

• 24.0% 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: 99.3% 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: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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_re, (y_46_im * -x_46_im));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. fma-neg 99.2%

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

   2. distribute-lft-neg-out 99.2%

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

   3. *-commutative 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 75.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -0.0026 \lor \neg \left(x.re \cdot y.re \leq 3.6 \cdot 10^{+28}\right):\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(-x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.re y.re) -0.0026) (not (<= (* x.re y.re) 3.6e+28)))
   (* x.re y.re)
   (* y.im (- x.im))))

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) <= -0.0026) || !((x_46_re * y_46_re) <= 3.6e+28)) {
		tmp = x_46_re * y_46_re;
	} else {
		tmp = y_46_im * -x_46_im;
	}
	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) <= (-0.0026d0)) .or. (.not. ((x_46re * y_46re) <= 3.6d+28))) then
        tmp = x_46re * y_46re
    else
        tmp = y_46im * -x_46im
    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) <= -0.0026) || !((x_46_re * y_46_re) <= 3.6e+28)) {
		tmp = x_46_re * y_46_re;
	} else {
		tmp = y_46_im * -x_46_im;
	}
	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) <= -0.0026) or not ((x_46_re * y_46_re) <= 3.6e+28):
		tmp = x_46_re * y_46_re
	else:
		tmp = y_46_im * -x_46_im
	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) <= -0.0026) || !(Float64(x_46_re * y_46_re) <= 3.6e+28))
		tmp = Float64(x_46_re * y_46_re);
	else
		tmp = Float64(y_46_im * Float64(-x_46_im));
	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) <= -0.0026) || ~(((x_46_re * y_46_re) <= 3.6e+28)))
		tmp = x_46_re * y_46_re;
	else
		tmp = y_46_im * -x_46_im;
	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$46$re), $MachinePrecision], -0.0026], N[Not[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 3.6e+28]], $MachinePrecision]], N[(x$46$re * y$46$re), $MachinePrecision], N[(y$46$im * (-x$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.re) < -0.0025999999999999999 or 3.5999999999999999e28 < (*.f64 x.re y.re)

   1. Initial program 97.7%

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

   2. Taylor expanded in x.re around inf 75.9%

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

   if -0.0025999999999999999 < (*.f64 x.re y.re) < 3.5999999999999999e28

   1. Initial program 100.0%

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

   2. Taylor expanded in x.re around 0 73.9%

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

      1. mul-1-neg 73.9%

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

      2. distribute-lft-neg-in 73.9%

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

      3. *-commutative 73.9%

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

   4. Simplified 73.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -0.0026 \lor \neg \left(x.re \cdot y.re \leq 3.6 \cdot 10^{+28}\right):\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* x.re y.re) (* y.im x.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) - (y_46_im * x_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) - (y_46im * x_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) - (y_46_im * x_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (x_46_re * y_46_re) - (y_46_im * x_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(y_46_im * x_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) - (y_46_im * x_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[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

2. Final simplification 98.8%

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

Alternative 4: 53.2% accurate, 2.3× speedup

Math

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

FPCore

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

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

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

3. Final simplification 54.4%

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

Reproduce

herbie shell --seed 2023305 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))