_multiplyComplex, real part

Percentage Accurate: 99.1% → 99.6%

Time: 5.7s

Alternatives: 4

Speedup: 1.0×

• 23.7% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y.re, x.re, 0 - x.im \cdot y.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.re x.re (- 0.0 (* 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, (0.0 - (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(0.0 - 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[(0.0 - N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. neg-sub0 N/A

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

   5. --lowering--.f64 N/A

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

   6. *-lowering-*.f64 99.2

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

4. Applied egg-rr 99.2%

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

Alternative 2: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \cdot x.re \leq -1 \cdot 10^{+92}:\\ \;\;\;\;y.re \cdot x.re\\ \mathbf{elif}\;y.re \cdot x.re \leq 2 \cdot 10^{+31}:\\ \;\;\;\;0 - x.im \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= (* y.re x.re) -1e+92)
   (* y.re x.re)
   (if (<= (* y.re x.re) 2e+31) (- 0.0 (* 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 ((y_46_re * x_46_re) <= -1e+92) {
		tmp = y_46_re * x_46_re;
	} else if ((y_46_re * x_46_re) <= 2e+31) {
		tmp = 0.0 - (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 ((y_46re * x_46re) <= (-1d+92)) then
        tmp = y_46re * x_46re
    else if ((y_46re * x_46re) <= 2d+31) then
        tmp = 0.0d0 - (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 ((y_46_re * x_46_re) <= -1e+92) {
		tmp = y_46_re * x_46_re;
	} else if ((y_46_re * x_46_re) <= 2e+31) {
		tmp = 0.0 - (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 (y_46_re * x_46_re) <= -1e+92:
		tmp = y_46_re * x_46_re
	elif (y_46_re * x_46_re) <= 2e+31:
		tmp = 0.0 - (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(y_46_re * x_46_re) <= -1e+92)
		tmp = Float64(y_46_re * x_46_re);
	elseif (Float64(y_46_re * x_46_re) <= 2e+31)
		tmp = Float64(0.0 - 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 ((y_46_re * x_46_re) <= -1e+92)
		tmp = y_46_re * x_46_re;
	elseif ((y_46_re * x_46_re) <= 2e+31)
		tmp = 0.0 - (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[LessEqual[N[(y$46$re * x$46$re), $MachinePrecision], -1e+92], N[(y$46$re * x$46$re), $MachinePrecision], If[LessEqual[N[(y$46$re * x$46$re), $MachinePrecision], 2e+31], N[(0.0 - N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re * x$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.re) < -1e92 or 1.9999999999999999e31 < (*.f64 x.re y.re)

   1. Initial program 97.8%

      \[x.re \cdot y.re - x.im \cdot y.im \]
   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. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 82.8

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

   5. Simplified 82.8%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 82.8

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

   7. Applied egg-rr 82.8%

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

   if -1e92 < (*.f64 x.re y.re) < 1.9999999999999999e31

   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

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

      2. neg-sub0 N/A

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

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

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

      4. +-rgt-identity N/A

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

      5. accelerator-lowering-fma.f64 78.2

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

   5. Simplified 78.2%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 78.2

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

   7. Applied egg-rr 78.2%

      \[\leadsto 0 - \color{blue}{y.im \cdot x.im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \cdot x.re \leq -1 \cdot 10^{+92}:\\ \;\;\;\;y.re \cdot x.re\\ \mathbf{elif}\;y.re \cdot x.re \leq 2 \cdot 10^{+31}:\\ \;\;\;\;0 - x.im \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* 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 (y_46_re * x_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 = (y_46re * x_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 (y_46_re * x_46_re) - (x_46_im * y_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (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 Float64(Float64(y_46_re * x_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 = (y_46_re * x_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[(y$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Final simplification 99.2%

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

Alternative 4: 51.3% 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 99.2%

   \[x.re \cdot y.re - x.im \cdot y.im \]
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. +-rgt-identity N/A

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

   2. accelerator-lowering-fma.f64 47.6

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

5. Simplified 47.6%

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

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 47.6

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

7. Applied egg-rr 47.6%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))