_multiplyComplex, real part

Percentage Accurate: 99.1% → 99.5%

Time: 8.4s

Alternatives: 4

Speedup: 1.0×

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

Specification

Math

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

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, 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 99.6%

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

   1. fma-neg 100.0%

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

   2. distribute-lft-neg-out 100.0%

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

   3. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -12500000000000 \lor \neg \left(x.re \cdot y.re \leq -3.7 \cdot 10^{-19}\right) \land \left(x.re \cdot y.re \leq -1.55 \cdot 10^{-69} \lor \neg \left(x.re \cdot y.re \leq 62\right)\right):\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(-x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.re y.re) -12500000000000.0)
         (and (not (<= (* x.re y.re) -3.7e-19))
              (or (<= (* x.re y.re) -1.55e-69) (not (<= (* x.re y.re) 62.0)))))
   (* 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) <= -12500000000000.0) || (!((x_46_re * y_46_re) <= -3.7e-19) && (((x_46_re * y_46_re) <= -1.55e-69) || !((x_46_re * y_46_re) <= 62.0)))) {
		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) <= (-12500000000000.0d0)) .or. (.not. ((x_46re * y_46re) <= (-3.7d-19))) .and. ((x_46re * y_46re) <= (-1.55d-69)) .or. (.not. ((x_46re * y_46re) <= 62.0d0))) 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) <= -12500000000000.0) || (!((x_46_re * y_46_re) <= -3.7e-19) && (((x_46_re * y_46_re) <= -1.55e-69) || !((x_46_re * y_46_re) <= 62.0)))) {
		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) <= -12500000000000.0) or (not ((x_46_re * y_46_re) <= -3.7e-19) and (((x_46_re * y_46_re) <= -1.55e-69) or not ((x_46_re * y_46_re) <= 62.0))):
		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) <= -12500000000000.0) || (!(Float64(x_46_re * y_46_re) <= -3.7e-19) && ((Float64(x_46_re * y_46_re) <= -1.55e-69) || !(Float64(x_46_re * y_46_re) <= 62.0))))
		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) <= -12500000000000.0) || (~(((x_46_re * y_46_re) <= -3.7e-19)) && (((x_46_re * y_46_re) <= -1.55e-69) || ~(((x_46_re * y_46_re) <= 62.0)))))
		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], -12500000000000.0], And[N[Not[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], -3.7e-19]], $MachinePrecision], Or[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], -1.55e-69], N[Not[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 62.0]], $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) < -1.25e13 or -3.70000000000000005e-19 < (*.f64 x.re y.re) < -1.55e-69 or 62 < (*.f64 x.re y.re)

   1. Initial program 99.2%

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

   2. Taylor expanded in x.re around inf 81.1%

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

   if -1.25e13 < (*.f64 x.re y.re) < -3.70000000000000005e-19 or -1.55e-69 < (*.f64 x.re y.re) < 62

   1. Initial program 100.0%

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

   2. Taylor expanded in x.re around 0 81.5%

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

      1. mul-1-neg 81.5%

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

      2. distribute-lft-neg-in 81.5%

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

      3. *-commutative 81.5%

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

   4. Simplified 81.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -12500000000000 \lor \neg \left(x.re \cdot y.re \leq -3.7 \cdot 10^{-19}\right) \land \left(x.re \cdot y.re \leq -1.55 \cdot 10^{-69} \lor \neg \left(x.re \cdot y.re \leq 62\right)\right):\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 3: 99.1% 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 99.6%

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

2. Final simplification 99.6%

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

Alternative 4: 51.9% 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 99.6%

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

2. Taylor expanded in x.re around inf 53.8%

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

3. Final simplification 53.8%

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

Reproduce

herbie shell --seed 2023326 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))