_multiplyComplex, imaginary part

Percentage Accurate: 99.1% → 99.6%

Time: 3.9s

Alternatives: 3

Speedup: 1.2×

• 23.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (+ (* x.re y.im) (* x.im 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_im) + (x_46_im * 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_46im) + (x_46im * 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_im) + (x_46_im * y_46_re);
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(x_46_re * y_46_im) + Float64(x_46_im * 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_im) + (x_46_im * y_46_re);
end

Wolfram

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

Initial Program: 99.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (+ (* x.re y.im) (* x.im 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_im) + (x_46_im * 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_46im) + (x_46im * 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_im) + (x_46_im * y_46_re);
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(x_46_re * y_46_im) + Float64(x_46_im * 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_im) + (x_46_im * y_46_re);
end

Wolfram

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.im x.re (* x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return fma(y_46_im, x_46_re, (x_46_im * y_46_re));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return fma(y_46_im, x_46_re, Float64(x_46_im * y_46_re))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. *-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-lowering-*.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \cdot y.re \leq -1 \cdot 10^{-15}:\\ \;\;\;\;x.im \cdot y.re\\ \mathbf{elif}\;x.im \cdot y.re \leq 5 \cdot 10^{+75}:\\ \;\;\;\;y.im \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= (* x.im y.re) -1e-15)
   (* x.im y.re)
   (if (<= (* x.im y.re) 5e+75) (* y.im x.re) (* x.im 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_re) <= -1e-15) {
		tmp = x_46_im * y_46_re;
	} else if ((x_46_im * y_46_re) <= 5e+75) {
		tmp = y_46_im * x_46_re;
	} else {
		tmp = x_46_im * 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_46re) <= (-1d-15)) then
        tmp = x_46im * y_46re
    else if ((x_46im * y_46re) <= 5d+75) then
        tmp = y_46im * x_46re
    else
        tmp = x_46im * 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_re) <= -1e-15) {
		tmp = x_46_im * y_46_re;
	} else if ((x_46_im * y_46_re) <= 5e+75) {
		tmp = y_46_im * x_46_re;
	} else {
		tmp = x_46_im * 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_re) <= -1e-15:
		tmp = x_46_im * y_46_re
	elif (x_46_im * y_46_re) <= 5e+75:
		tmp = y_46_im * x_46_re
	else:
		tmp = x_46_im * 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_re) <= -1e-15)
		tmp = Float64(x_46_im * y_46_re);
	elseif (Float64(x_46_im * y_46_re) <= 5e+75)
		tmp = Float64(y_46_im * x_46_re);
	else
		tmp = Float64(x_46_im * 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_re) <= -1e-15)
		tmp = x_46_im * y_46_re;
	elseif ((x_46_im * y_46_re) <= 5e+75)
		tmp = y_46_im * x_46_re;
	else
		tmp = x_46_im * 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$46$im * y$46$re), $MachinePrecision], -1e-15], N[(x$46$im * y$46$re), $MachinePrecision], If[LessEqual[N[(x$46$im * y$46$re), $MachinePrecision], 5e+75], N[(y$46$im * x$46$re), $MachinePrecision], N[(x$46$im * y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.im y.re) < -1.0000000000000001e-15 or 5.0000000000000002e75 < (*.f64 x.im y.re)

   1. Initial program 98.4%

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

   3. Taylor expanded in x.re around 0

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 80.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, y.re, 0\right)} \]

   5. Simplified 80.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, y.re, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 80.7

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

   7. Applied egg-rr 80.7%

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

   if -1.0000000000000001e-15 < (*.f64 x.im y.re) < 5.0000000000000002e75

   1. Initial program 100.0%

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

   3. Taylor expanded in x.re around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(x.re \cdot y.im\right) + 0\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(x.re \cdot y.im\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)} \]

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 77.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.im, 0\right)} \]

   5. Simplified 77.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.im, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 77.3

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

   7. Applied egg-rr 77.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.re \leq -1 \cdot 10^{-15}:\\ \;\;\;\;x.im \cdot y.re\\ \mathbf{elif}\;x.im \cdot y.re \leq 5 \cdot 10^{+75}:\\ \;\;\;\;y.im \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.7% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* y.im x.re))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_im * 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_46im * 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_im * x_46_re;
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(y_46_im * x_46_re)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x.re around inf

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

   1. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. +-rgt-identity N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(x.re \cdot y.im\right) + 0\right)}\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(x.re \cdot y.im\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)} \]

   5. mul-1-neg N/A

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

   6. remove-double-neg N/A

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

   7. metadata-eval N/A

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

   8. accelerator-lowering-fma.f64 51.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.im, 0\right)} \]

5. Simplified 51.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.im, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 51.1

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

7. Applied egg-rr 51.1%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))