_multiplyComplex, real part

Percentage Accurate: 99.2% → 99.6%

Time: 2.1s

Alternatives: 5

Speedup: 1.0×

• 23.3% 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

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma (- x.im) y.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(-x_46_im, y_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(-x_46_im), y_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[((-x$46$im) * y$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. distribute-rgt-neg-out 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-rgt-neg-out 99.6%

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

   5. distribute-lft-neg-in 99.6%

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

   6. fma-def 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma 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 fma(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 fma(x_46_re, y_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[(x$46$re * y$46$re + N[((-x$46$im) * y$46$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 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -8.5 \cdot 10^{+68}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq -1.7 \cdot 10^{+29} \lor \neg \left(x.re \cdot y.re \leq -2.3 \cdot 10^{-6}\right) \land x.re \cdot y.re \leq 6.5 \cdot 10^{+71}:\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= (* x.re y.re) -8.5e+68)
   (* x.re y.re)
   (if (or (<= (* x.re y.re) -1.7e+29)
           (and (not (<= (* x.re y.re) -2.3e-6)) (<= (* x.re y.re) 6.5e+71)))
     (* (- x.im) y.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 ((x_46_re * y_46_re) <= -8.5e+68) {
		tmp = x_46_re * y_46_re;
	} else if (((x_46_re * y_46_re) <= -1.7e+29) || (!((x_46_re * y_46_re) <= -2.3e-6) && ((x_46_re * y_46_re) <= 6.5e+71))) {
		tmp = -x_46_im * y_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 ((x_46re * y_46re) <= (-8.5d+68)) then
        tmp = x_46re * y_46re
    else if (((x_46re * y_46re) <= (-1.7d+29)) .or. (.not. ((x_46re * y_46re) <= (-2.3d-6))) .and. ((x_46re * y_46re) <= 6.5d+71)) then
        tmp = -x_46im * y_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 ((x_46_re * y_46_re) <= -8.5e+68) {
		tmp = x_46_re * y_46_re;
	} else if (((x_46_re * y_46_re) <= -1.7e+29) || (!((x_46_re * y_46_re) <= -2.3e-6) && ((x_46_re * y_46_re) <= 6.5e+71))) {
		tmp = -x_46_im * y_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 (x_46_re * y_46_re) <= -8.5e+68:
		tmp = x_46_re * y_46_re
	elif ((x_46_re * y_46_re) <= -1.7e+29) or (not ((x_46_re * y_46_re) <= -2.3e-6) and ((x_46_re * y_46_re) <= 6.5e+71)):
		tmp = -x_46_im * y_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 (Float64(x_46_re * y_46_re) <= -8.5e+68)
		tmp = Float64(x_46_re * y_46_re);
	elseif ((Float64(x_46_re * y_46_re) <= -1.7e+29) || (!(Float64(x_46_re * y_46_re) <= -2.3e-6) && (Float64(x_46_re * y_46_re) <= 6.5e+71)))
		tmp = Float64(Float64(-x_46_im) * y_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 ((x_46_re * y_46_re) <= -8.5e+68)
		tmp = x_46_re * y_46_re;
	elseif (((x_46_re * y_46_re) <= -1.7e+29) || (~(((x_46_re * y_46_re) <= -2.3e-6)) && ((x_46_re * y_46_re) <= 6.5e+71)))
		tmp = -x_46_im * y_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[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], -8.5e+68], N[(x$46$re * y$46$re), $MachinePrecision], If[Or[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], -1.7e+29], And[N[Not[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], -2.3e-6]], $MachinePrecision], LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 6.5e+71]]], N[((-x$46$im) * y$46$im), $MachinePrecision], N[(x$46$re * y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.re) < -8.49999999999999966e68 or -1.69999999999999991e29 < (*.f64 x.re y.re) < -2.3e-6 or 6.49999999999999954e71 < (*.f64 x.re y.re)

   1. Initial program 99.1%

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

   2. Taylor expanded in x.re around inf 88.2%

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

   if -8.49999999999999966e68 < (*.f64 x.re y.re) < -1.69999999999999991e29 or -2.3e-6 < (*.f64 x.re y.re) < 6.49999999999999954e71

   1. Initial program 100.0%

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

   2. Taylor expanded in x.re around 0 78.5%

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

      1. mul-1-neg 78.5%

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

      2. distribute-rgt-neg-in 78.5%

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

   4. Simplified 78.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -8.5 \cdot 10^{+68}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq -1.7 \cdot 10^{+29} \lor \neg \left(x.re \cdot y.re \leq -2.3 \cdot 10^{-6}\right) \land x.re \cdot y.re \leq 6.5 \cdot 10^{+71}:\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]

Alternative 4: 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]

Derivation

1. Initial program 99.6%

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

2. Final simplification 99.6%

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

Alternative 5: 52.4% 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 53.0%

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

3. Final simplification 53.0%

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

Reproduce

herbie shell --seed 2023230 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))