_multiplyComplex, real part

Percentage Accurate: 99.2% → 99.6%

Time: 3.0s

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: 99.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. sub-neg 98.4%

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

   2. +-commutative 98.4%

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

   3. *-commutative 98.4%

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

   4. distribute-lft-neg-in 98.4%

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

   5. fma-define 99.2%

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 67.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.3 \cdot 10^{-94} \lor \neg \left(y.im \leq 2 \cdot 10^{+24}\right):\\ \;\;\;\;y.im \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.3e-94) (not (<= y.im 2e+24)))
   (* y.im (- x.im))
   (* x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.3e-94) || !(y_46_im <= 2e+24)) {
		tmp = y_46_im * -x_46_im;
	} else {
		tmp = x_46_re * 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 ((y_46im <= (-4.3d-94)) .or. (.not. (y_46im <= 2d+24))) then
        tmp = y_46im * -x_46im
    else
        tmp = x_46re * 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 ((y_46_im <= -4.3e-94) || !(y_46_im <= 2e+24)) {
		tmp = y_46_im * -x_46_im;
	} else {
		tmp = x_46_re * y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.3e-94) or not (y_46_im <= 2e+24):
		tmp = y_46_im * -x_46_im
	else:
		tmp = x_46_re * y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.3e-94) || !(y_46_im <= 2e+24))
		tmp = Float64(y_46_im * Float64(-x_46_im));
	else
		tmp = Float64(x_46_re * 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 ((y_46_im <= -4.3e-94) || ~((y_46_im <= 2e+24)))
		tmp = y_46_im * -x_46_im;
	else
		tmp = x_46_re * 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[y$46$im, -4.3e-94], N[Not[LessEqual[y$46$im, 2e+24]], $MachinePrecision]], N[(y$46$im * (-x$46$im)), $MachinePrecision], N[(x$46$re * y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.2999999999999998e-94 or 2e24 < y.im

   1. Initial program 97.0%

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

   3. Taylor expanded in x.re around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. distribute-rgt-neg-out 67.2%

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

   5. Simplified 67.2%

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

   if -4.2999999999999998e-94 < y.im < 2e24

   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 73.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.3 \cdot 10^{-94} \lor \neg \left(y.im \leq 2 \cdot 10^{+24}\right):\\ \;\;\;\;y.im \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% 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 98.4%

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

3. Final simplification 98.4%

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

Alternative 4: 52.0% 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 98.4%

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

3. Taylor expanded in x.re around inf 53.4%

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

4. Final simplification 53.4%

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

Reproduce

herbie shell --seed 2024053 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))