_multiplyComplex, real part

Percentage Accurate: 99.3% → 99.3%

Time: 5.6s

Alternatives: 3

Speedup: 1.0×

• 23.9% 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: 99.3% 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: 99.3% 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 2: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -1.65 \cdot 10^{+75}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq 1.9 \cdot 10^{-49}:\\ \;\;\;\;0 - x.im \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= (* x.re y.re) -1.65e+75)
   (* x.re y.re)
   (if (<= (* x.re y.re) 1.9e-49) (- 0.0 (* 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) <= -1.65e+75) {
		tmp = x_46_re * y_46_re;
	} else if ((x_46_re * y_46_re) <= 1.9e-49) {
		tmp = 0.0 - (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) <= (-1.65d+75)) then
        tmp = x_46re * y_46re
    else if ((x_46re * y_46re) <= 1.9d-49) then
        tmp = 0.0d0 - (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) <= -1.65e+75) {
		tmp = x_46_re * y_46_re;
	} else if ((x_46_re * y_46_re) <= 1.9e-49) {
		tmp = 0.0 - (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) <= -1.65e+75:
		tmp = x_46_re * y_46_re
	elif (x_46_re * y_46_re) <= 1.9e-49:
		tmp = 0.0 - (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) <= -1.65e+75)
		tmp = Float64(x_46_re * y_46_re);
	elseif (Float64(x_46_re * y_46_re) <= 1.9e-49)
		tmp = Float64(0.0 - 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) <= -1.65e+75)
		tmp = x_46_re * y_46_re;
	elseif ((x_46_re * y_46_re) <= 1.9e-49)
		tmp = 0.0 - (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], -1.65e+75], N[(x$46$re * y$46$re), $MachinePrecision], If[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 1.9e-49], N[(0.0 - N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re * y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.re) < -1.64999999999999999e75 or 1.8999999999999999e-49 < (*.f64 x.re y.re)

   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 inf

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

      1. *-lowering-*.f64 79.1%

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{y.re}\right) \]

   5. Simplified 79.1%

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

   if -1.64999999999999999e75 < (*.f64 x.re y.re) < 1.8999999999999999e-49

   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. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x.im \cdot y.im} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(x.im \cdot y.im\right)}\right) \]

      4. *-lowering-*.f64 77.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{y.im}\right)\right) \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{0 - x.im \cdot y.im} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\left(0 + x.im \cdot y.im\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(x.im \cdot y.im\right)}^{3}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(x.im \cdot y.im\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(x.im \cdot y.im\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      5. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      6. sqr-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(\left(\mathsf{neg}\left(x.im \cdot y.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im \cdot y.im\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      7. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(\left(0 - x.im \cdot y.im\right) \cdot \left(\mathsf{neg}\left(x.im \cdot y.im\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      8. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(\left(0 - x.im \cdot y.im\right) \cdot \left(0 - x.im \cdot y.im\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(0 - x.im \cdot y.im\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - x.im \cdot y.im\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      10. sqr-pow N/A

          \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(0 - x.im \cdot y.im\right)}^{3}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\frac{0 + {\left(0 - x.im \cdot y.im\right)}^{3}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      12. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{0 + {\left(\mathsf{neg}\left(x.im \cdot y.im\right)\right)}^{3}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      13. cube-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{0 + \left(\mathsf{neg}\left({\left(x.im \cdot y.im\right)}^{3}\right)\right)}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{0 - {\left(x.im \cdot y.im\right)}^{3}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} - {\left(x.im \cdot y.im\right)}^{3}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      16. mul0-lft N/A

          \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} - {\left(x.im \cdot y.im\right)}^{3}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) - 0\right)}\right) \]

      17. fmm-def N/A

          \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} - {\left(x.im \cdot y.im\right)}^{3}}{0 \cdot 0 + \mathsf{fma}\left(x.im \cdot y.im, x.im \cdot y.im, \mathsf{neg}\left(0\right)\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} - {\left(x.im \cdot y.im\right)}^{3}}{0 \cdot 0 + \mathsf{fma}\left(x.im \cdot y.im, x.im \cdot y.im, 0\right)}\right) \]

      19. mul0-lft N/A

          \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} - {\left(x.im \cdot y.im\right)}^{3}}{0 \cdot 0 + \mathsf{fma}\left(x.im \cdot y.im, x.im \cdot y.im, 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

      20. fma-define N/A

          \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} - {\left(x.im \cdot y.im\right)}^{3}}{0 \cdot 0 + \left(\left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right) + 0 \cdot \left(x.im \cdot y.im\right)\right)}\right) \]

   7. Applied egg-rr 77.4%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -1.65 \cdot 10^{+75}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq 1.9 \cdot 10^{-49}:\\ \;\;\;\;0 - x.im \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.2% 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. Taylor expanded in x.re around inf

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

   1. *-lowering-*.f64 53.1%

      \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{y.re}\right) \]

5. Simplified 53.1%

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

Reproduce

herbie shell --seed 2024156 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))