_multiplyComplex, real part

Percentage Accurate: 99.3% → 99.6%

Time: 3.1s

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: 99.3% 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

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.re x.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 fma(y_46_re, x_46_re, (x_46_im * -y_46_im));
}

Julia

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$re * x$46$re + 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. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. fma-neg 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4.5 \cdot 10^{-25} \lor \neg \left(x.im \leq 5.2 \cdot 10^{-29}\right):\\ \;\;\;\;x.im \cdot \left(-y.im\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= x.im -4.5e-25) (not (<= x.im 5.2e-29)))
   (* x.im (- y.im))
   (* y.re 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 <= -4.5e-25) || !(x_46_im <= 5.2e-29)) {
		tmp = x_46_im * -y_46_im;
	} else {
		tmp = y_46_re * 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 <= (-4.5d-25)) .or. (.not. (x_46im <= 5.2d-29))) then
        tmp = x_46im * -y_46im
    else
        tmp = y_46re * 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 <= -4.5e-25) || !(x_46_im <= 5.2e-29)) {
		tmp = x_46_im * -y_46_im;
	} else {
		tmp = y_46_re * 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 <= -4.5e-25) or not (x_46_im <= 5.2e-29):
		tmp = x_46_im * -y_46_im
	else:
		tmp = y_46_re * x_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((x_46_im <= -4.5e-25) || !(x_46_im <= 5.2e-29))
		tmp = Float64(x_46_im * Float64(-y_46_im));
	else
		tmp = Float64(y_46_re * 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 <= -4.5e-25) || ~((x_46_im <= 5.2e-29)))
		tmp = x_46_im * -y_46_im;
	else
		tmp = y_46_re * 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[x$46$im, -4.5e-25], N[Not[LessEqual[x$46$im, 5.2e-29]], $MachinePrecision]], N[(x$46$im * (-y$46$im)), $MachinePrecision], N[(y$46$re * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.5000000000000001e-25 or 5.2000000000000004e-29 < x.im

   1. Initial program 98.3%

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

   3. Taylor expanded in x.re around 0 64.1%

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

      1. mul-1-neg 64.1%

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

      2. distribute-rgt-neg-out 64.1%

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

   5. Simplified 64.1%

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

   if -4.5000000000000001e-25 < x.im < 5.2000000000000004e-29

   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 inf 75.8%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.5 \cdot 10^{-25} \lor \neg \left(x.im \leq 5.2 \cdot 10^{-29}\right):\\ \;\;\;\;x.im \cdot \left(-y.im\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* y.re x.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 (y_46_re * x_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 = (y_46re * x_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 (y_46_re * x_46_re) - (x_46_im * y_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (y_46_re * x_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(y_46_re * x_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 = (y_46_re * x_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[(y$46$re * x$46$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%

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

Alternative 4: 52.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * 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_46re * 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_re * x_46_re;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$re * x$46$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 60.5%

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

4. Final simplification 60.5%

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

Reproduce

herbie shell --seed 2024050 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))