_multiplyComplex, real part

Percentage Accurate: 99.2% → 99.6%

Time: 3.9s

Alternatives: 5

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: 99.2% 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, 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 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. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x.im\right) \cdot y.im\\ \mathbf{if}\;x.im \cdot y.im \leq -1 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x.im \cdot y.im \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (- x.im) y.im)))
   (if (<= (* x.im y.im) -1e-73)
     t_0
     (if (<= (* x.im y.im) 5e+78) (fma y.im x.im (* x.re y.re)) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_im * y_46_im;
	double tmp;
	if ((x_46_im * y_46_im) <= -1e-73) {
		tmp = t_0;
	} else if ((x_46_im * y_46_im) <= 5e+78) {
		tmp = fma(y_46_im, x_46_im, (x_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_im) * y_46_im)
	tmp = 0.0
	if (Float64(x_46_im * y_46_im) <= -1e-73)
		tmp = t_0;
	elseif (Float64(x_46_im * y_46_im) <= 5e+78)
		tmp = fma(y_46_im, x_46_im, Float64(x_46_re * y_46_re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$im) * y$46$im), $MachinePrecision]}, If[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], -1e-73], t$95$0, If[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], 5e+78], N[(y$46$im * x$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.im y.im) < -9.99999999999999997e-74 or 4.99999999999999984e78 < (*.f64 x.im y.im)

   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 78.2

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

   5. Applied rewrites 78.2%

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

   if -9.99999999999999997e-74 < (*.f64 x.im y.im) < 4.99999999999999984e78

   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. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 100.0

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

   6. Applied rewrites 80.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.im \leq -1 \cdot 10^{-73}:\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \mathbf{elif}\;x.im \cdot y.im \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 100.0%

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

Alternative 4: 52.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* (- 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_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_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_im * y_46_im;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return -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_im) * y_46_im)
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = -x_46_im * y_46_im;
end

Wolfram

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

Derivation

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 51.4

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

5. Applied rewrites 51.4%

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

Alternative 5: 3.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ x.im \cdot y.im \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* 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_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_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_im * y_46_im;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return x_46_im * y_46_im

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 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_im * y_46_im;
end

Wolfram

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

Derivation

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 51.4

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

5. Applied rewrites 51.4%

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

7. Applied rewrites 3.5%

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

8. Final simplification 3.5%

   \[\leadsto x.im \cdot y.im \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))