_multiplyComplex, real part

Percentage Accurate: 99.1% → 99.5%

Time: 2.1s

Alternatives: 4

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 24.1% 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

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

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(-Float64(x_46_im * 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 98.4%

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

   1. *-commutative 98.4%

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

   2. fma-neg 99.2%

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

   3. distribute-rgt-neg-in 99.2%

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

3. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 73.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \cdot x.re \leq -2.45 \cdot 10^{+157}:\\ \;\;\;\;y.re \cdot x.re\\ \mathbf{elif}\;y.re \cdot x.re \leq -0.008 \lor \neg \left(y.re \cdot x.re \leq -4.3 \cdot 10^{-103}\right) \land y.re \cdot x.re \leq 6.2 \cdot 10^{+35}:\\ \;\;\;\;-x.im \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= (* y.re x.re) -2.45e+157)
   (* y.re x.re)
   (if (or (<= (* y.re x.re) -0.008)
           (and (not (<= (* y.re x.re) -4.3e-103)) (<= (* y.re x.re) 6.2e+35)))
     (- (* 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 ((y_46_re * x_46_re) <= -2.45e+157) {
		tmp = y_46_re * x_46_re;
	} else if (((y_46_re * x_46_re) <= -0.008) || (!((y_46_re * x_46_re) <= -4.3e-103) && ((y_46_re * x_46_re) <= 6.2e+35))) {
		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 ((y_46re * x_46re) <= (-2.45d+157)) then
        tmp = y_46re * x_46re
    else if (((y_46re * x_46re) <= (-0.008d0)) .or. (.not. ((y_46re * x_46re) <= (-4.3d-103))) .and. ((y_46re * x_46re) <= 6.2d+35)) 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 ((y_46_re * x_46_re) <= -2.45e+157) {
		tmp = y_46_re * x_46_re;
	} else if (((y_46_re * x_46_re) <= -0.008) || (!((y_46_re * x_46_re) <= -4.3e-103) && ((y_46_re * x_46_re) <= 6.2e+35))) {
		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 (y_46_re * x_46_re) <= -2.45e+157:
		tmp = y_46_re * x_46_re
	elif ((y_46_re * x_46_re) <= -0.008) or (not ((y_46_re * x_46_re) <= -4.3e-103) and ((y_46_re * x_46_re) <= 6.2e+35)):
		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 (Float64(y_46_re * x_46_re) <= -2.45e+157)
		tmp = Float64(y_46_re * x_46_re);
	elseif ((Float64(y_46_re * x_46_re) <= -0.008) || (!(Float64(y_46_re * x_46_re) <= -4.3e-103) && (Float64(y_46_re * x_46_re) <= 6.2e+35)))
		tmp = Float64(-Float64(x_46_im * 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 ((y_46_re * x_46_re) <= -2.45e+157)
		tmp = y_46_re * x_46_re;
	elseif (((y_46_re * x_46_re) <= -0.008) || (~(((y_46_re * x_46_re) <= -4.3e-103)) && ((y_46_re * x_46_re) <= 6.2e+35)))
		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[LessEqual[N[(y$46$re * x$46$re), $MachinePrecision], -2.45e+157], N[(y$46$re * x$46$re), $MachinePrecision], If[Or[LessEqual[N[(y$46$re * x$46$re), $MachinePrecision], -0.008], And[N[Not[LessEqual[N[(y$46$re * x$46$re), $MachinePrecision], -4.3e-103]], $MachinePrecision], LessEqual[N[(y$46$re * x$46$re), $MachinePrecision], 6.2e+35]]], (-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 (*.f64 x.re y.re) < -2.4500000000000001e157 or -0.0080000000000000002 < (*.f64 x.re y.re) < -4.30000000000000023e-103 or 6.19999999999999973e35 < (*.f64 x.re y.re)

   1. Initial program 96.7%

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

   2. Taylor expanded in x.re around inf 84.0%

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

   if -2.4500000000000001e157 < (*.f64 x.re y.re) < -0.0080000000000000002 or -4.30000000000000023e-103 < (*.f64 x.re y.re) < 6.19999999999999973e35

   1. Initial program 100.0%

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

   2. Taylor expanded in x.re around 0 80.6%

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

      1. mul-1-neg 80.6%

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

      2. distribute-rgt-neg-in 80.6%

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

   4. Simplified 80.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \cdot x.re \leq -2.45 \cdot 10^{+157}:\\ \;\;\;\;y.re \cdot x.re\\ \mathbf{elif}\;y.re \cdot x.re \leq -0.008 \lor \neg \left(y.re \cdot x.re \leq -4.3 \cdot 10^{-103}\right) \land y.re \cdot x.re \leq 6.2 \cdot 10^{+35}:\\ \;\;\;\;-x.im \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.re\\ \end{array} \]

Alternative 3: 99.1% 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 98.4%

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

2. Final simplification 98.4%

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

Alternative 4: 52.0% 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 98.4%

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

2. Taylor expanded in x.re around inf 52.0%

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

3. Final simplification 52.0%

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

Reproduce

herbie shell --seed 2023174 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))