_multiplyComplex, real part

Percentage Accurate: 99.1% → 99.5%

Time: 3.2s

Alternatives: 4

Speedup: 1.0×

• 23.5% 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, 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. fp-cancel-sign-sub-inv N/A

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

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

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

   6. unpow1 N/A

      \[\leadsto \color{blue}{{\left(x.re \cdot y.re\right)}^{1}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right)\right) \cdot y.im \]

   7. metadata-eval N/A

      \[\leadsto {\left(x.re \cdot y.re\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right)\right) \cdot y.im \]

   8. sqrt-pow1 N/A

      \[\leadsto \color{blue}{\sqrt{{\left(x.re \cdot y.re\right)}^{2}}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right)\right) \cdot y.im \]

   9. pow2 N/A

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

   10. rem-sqrt-square-rev N/A

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

   11. remove-double-neg N/A

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

   12. rem-sqrt-square-rev N/A

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

   13. pow2 N/A

       \[\leadsto \sqrt{\color{blue}{{\left(x.re \cdot y.re\right)}^{2}}} + \left(\mathsf{neg}\left(x.im\right)\right) \cdot y.im \]

   14. sqrt-pow1 N/A

       \[\leadsto \color{blue}{{\left(x.re \cdot y.re\right)}^{\left(\frac{2}{2}\right)}} + \left(\mathsf{neg}\left(x.im\right)\right) \cdot y.im \]

   15. metadata-eval N/A

       \[\leadsto {\left(x.re \cdot y.re\right)}^{\color{blue}{1}} + \left(\mathsf{neg}\left(x.im\right)\right) \cdot y.im \]

   16. unpow1 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. 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: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \cdot y.im \leq -1 \cdot 10^{+26} \lor \neg \left(x.im \cdot y.im \leq 2\right):\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.im y.im) -1e+26) (not (<= (* x.im y.im) 2.0)))
   (* (- 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 * y_46_im) <= -1e+26) || !((x_46_im * y_46_im) <= 2.0)) {
		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 * y_46im) <= (-1d+26)) .or. (.not. ((x_46im * y_46im) <= 2.0d0))) 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 * y_46_im) <= -1e+26) || !((x_46_im * y_46_im) <= 2.0)) {
		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 * y_46_im) <= -1e+26) or not ((x_46_im * y_46_im) <= 2.0):
		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(x_46_im * y_46_im) <= -1e+26) || !(Float64(x_46_im * y_46_im) <= 2.0))
		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 (((x_46_im * y_46_im) <= -1e+26) || ~(((x_46_im * y_46_im) <= 2.0)))
		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[N[(x$46$im * y$46$im), $MachinePrecision], -1e+26], N[Not[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], 2.0]], $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 (*.f64 x.im y.im) < -1.00000000000000005e26 or 2 < (*.f64 x.im y.im)

   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 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 84.8

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

   5. Applied rewrites 84.8%

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

   if -1.00000000000000005e26 < (*.f64 x.im y.im) < 2

   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 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 27.8

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.0

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

   8. Applied rewrites 79.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.im \leq -1 \cdot 10^{+26} \lor \neg \left(x.im \cdot y.im \leq 2\right):\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.re\\ \end{array} \]
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: 51.4% 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 99.6%

   \[x.re \cdot y.re - x.im \cdot y.im \]
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 55.4

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

5. Applied rewrites 55.4%

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

6. Taylor expanded in x.re around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 48.8

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

8. Applied rewrites 48.8%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))