_multiplyComplex, real part

Percentage Accurate: 99.1% → 99.6%

Time: 4.4s

Alternatives: 4

Speedup: 1.0×

• 23.6% 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: 99.6% accurate, 1.0× speedup

Math

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

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 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \cdot y.im \leq -5 \cdot 10^{-48} \lor \neg \left(x.im \cdot y.im \leq 10^{+31}\right):\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.im y.im) -5e-48) (not (<= (* x.im y.im) 1e+31)))
   (* (- 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_im * y_46_im) <= -5e-48) || !((x_46_im * y_46_im) <= 1e+31)) {
		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_46im * y_46im) <= (-5d-48)) .or. (.not. ((x_46im * y_46im) <= 1d+31))) 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_im * y_46_im) <= -5e-48) || !((x_46_im * y_46_im) <= 1e+31)) {
		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_im * y_46_im) <= -5e-48) or not ((x_46_im * y_46_im) <= 1e+31):
		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_im * y_46_im) <= -5e-48) || !(Float64(x_46_im * y_46_im) <= 1e+31))
		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_im * y_46_im) <= -5e-48) || ~(((x_46_im * y_46_im) <= 1e+31)))
		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[Or[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], -5e-48], N[Not[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], 1e+31]], $MachinePrecision]], 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.im y.im) < -4.9999999999999999e-48 or 9.9999999999999996e30 < (*.f64 x.im y.im)

   1. Initial program 99.1%

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

   3. Taylor expanded in x.re around 0

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x.im\right) \cdot y.im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x.im\right) \cdot y.im} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 78.3

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

   5. Applied rewrites 78.3%

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

   if -4.9999999999999999e-48 < (*.f64 x.im y.im) < 9.9999999999999996e30

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 83.1%

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

   7. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{x.re \cdot y.re} \]
   8. Step-by-step derivation

      1. lower-*.f64 83.5

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

   9. Applied rewrites 83.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.im \leq -5 \cdot 10^{-48} \lor \neg \left(x.im \cdot y.im \leq 10^{+31}\right):\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.6%

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

Alternative 4: 51.5% 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. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 53.0%

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

7. Taylor expanded in x.re around inf

   \[\leadsto \color{blue}{x.re \cdot y.re} \]
8. Step-by-step derivation

   1. lower-*.f64 54.9

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

9. Applied rewrites 54.9%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))