_multiplyComplex, real part

Percentage Accurate: 99.1% → 99.1%

Time: 3.4s

Alternatives: 3

Speedup: 1.0×

• 23.7% 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: 99.1% 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]

Derivation

1. Initial program 99.2%

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

3. Final simplification 99.2%

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

Alternative 2: 75.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -1.06 \cdot 10^{-86} \lor \neg \left(x.re \cdot y.re \leq 1.06 \cdot 10^{-41}\right) \land \left(x.re \cdot y.re \leq 9.6 \cdot 10^{+24} \lor \neg \left(x.re \cdot y.re \leq 5.2 \cdot 10^{+57}\right)\right):\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-y.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.re y.re) -1.06e-86)
         (and (not (<= (* x.re y.re) 1.06e-41))
              (or (<= (* x.re y.re) 9.6e+24)
                  (not (<= (* x.re y.re) 5.2e+57)))))
   (* 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) {
	double tmp;
	if (((x_46_re * y_46_re) <= -1.06e-86) || (!((x_46_re * y_46_re) <= 1.06e-41) && (((x_46_re * y_46_re) <= 9.6e+24) || !((x_46_re * y_46_re) <= 5.2e+57)))) {
		tmp = x_46_re * y_46_re;
	} else {
		tmp = x_46_im * -y_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) <= (-1.06d-86)) .or. (.not. ((x_46re * y_46re) <= 1.06d-41)) .and. ((x_46re * y_46re) <= 9.6d+24) .or. (.not. ((x_46re * y_46re) <= 5.2d+57))) then
        tmp = x_46re * y_46re
    else
        tmp = x_46im * -y_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) <= -1.06e-86) || (!((x_46_re * y_46_re) <= 1.06e-41) && (((x_46_re * y_46_re) <= 9.6e+24) || !((x_46_re * y_46_re) <= 5.2e+57)))) {
		tmp = x_46_re * y_46_re;
	} else {
		tmp = x_46_im * -y_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) <= -1.06e-86) or (not ((x_46_re * y_46_re) <= 1.06e-41) and (((x_46_re * y_46_re) <= 9.6e+24) or not ((x_46_re * y_46_re) <= 5.2e+57))):
		tmp = x_46_re * y_46_re
	else:
		tmp = x_46_im * -y_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) <= -1.06e-86) || (!(Float64(x_46_re * y_46_re) <= 1.06e-41) && ((Float64(x_46_re * y_46_re) <= 9.6e+24) || !(Float64(x_46_re * y_46_re) <= 5.2e+57))))
		tmp = Float64(x_46_re * y_46_re);
	else
		tmp = Float64(x_46_im * Float64(-y_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) <= -1.06e-86) || (~(((x_46_re * y_46_re) <= 1.06e-41)) && (((x_46_re * y_46_re) <= 9.6e+24) || ~(((x_46_re * y_46_re) <= 5.2e+57)))))
		tmp = x_46_re * y_46_re;
	else
		tmp = x_46_im * -y_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], -1.06e-86], And[N[Not[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 1.06e-41]], $MachinePrecision], Or[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 9.6e+24], N[Not[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 5.2e+57]], $MachinePrecision]]]], N[(x$46$re * y$46$re), $MachinePrecision], N[(x$46$im * (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.re) < -1.05999999999999997e-86 or 1.06e-41 < (*.f64 x.re y.re) < 9.6000000000000003e24 or 5.2e57 < (*.f64 x.re y.re)

   1. Initial program 98.6%

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

   3. Taylor expanded in x.re around inf 77.8%

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

   if -1.05999999999999997e-86 < (*.f64 x.re y.re) < 1.06e-41 or 9.6000000000000003e24 < (*.f64 x.re y.re) < 5.2e57

   1. Initial program 100.0%

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

   3. Taylor expanded in x.re around 0 84.8%

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

      1. mul-1-neg 84.8%

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

      2. distribute-rgt-neg-out 84.8%

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

   5. Simplified 84.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -1.06 \cdot 10^{-86} \lor \neg \left(x.re \cdot y.re \leq 1.06 \cdot 10^{-41}\right) \land \left(x.re \cdot y.re \leq 9.6 \cdot 10^{+24} \lor \neg \left(x.re \cdot y.re \leq 5.2 \cdot 10^{+57}\right)\right):\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-y.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.2%

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

3. Taylor expanded in x.re around inf 53.7%

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

4. Final simplification 53.7%

   \[\leadsto x.re \cdot y.re \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))