_multiplyComplex, real part

Percentage Accurate: 99.3% → 99.7%

Time: 9.5s

Alternatives: 5

Speedup: 1.0×

• 23.6% 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: 99.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma (- 0.0 y.im) x.im (* x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return fma((0.0 - y_46_im), x_46_im, (x_46_re * y_46_re));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return fma(Float64(0.0 - y_46_im), x_46_im, Float64(x_46_re * y_46_re))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(0.0 - y$46$im), $MachinePrecision] * x$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(0 - y.im\right), x.im, \left(x.re \cdot y.re\right)\right) \]

   7. --lowering--.f64 N/A

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

   8. *-lowering-*.f64 99.6%

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

4. Applied egg-rr 99.6%

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

   1. sub0-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(y.im\right)\right), x.im, \mathsf{*.f64}\left(x.re, y.re\right)\right) \]

   2. neg-lowering-neg.f64 99.6%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{neg.f64}\left(y.im\right), x.im, \mathsf{*.f64}\left(x.re, y.re\right)\right) \]

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.re x.re (* y.im (- 0.0 x.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, (y_46_im * (0.0 - x_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(y_46_im * Float64(0.0 - x_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[(y$46$im * N[(0.0 - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(y.re, x.re, \left(0 - x.im \cdot y.im\right)\right) \]

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

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

   6. *-lowering-*.f64 99.2%

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

4. Applied egg-rr 99.2%

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

   1. sub0-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(y.re, x.re, \left(\mathsf{neg}\left(x.im \cdot y.im\right)\right)\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(y.re, x.re, \mathsf{neg.f64}\left(\left(x.im \cdot y.im\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(y.re, x.re, \mathsf{neg.f64}\left(\left(y.im \cdot x.im\right)\right)\right) \]

   4. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{fma.f64}\left(y.re, x.re, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y.im, x.im\right)\right)\right) \]

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 3: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -5 \cdot 10^{-11} \lor \neg \left(x.re \cdot y.re \leq 2 \cdot 10^{+54}\right):\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(0 - x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.re y.re) -5e-11) (not (<= (* x.re y.re) 2e+54)))
   (* x.re y.re)
   (* y.im (- 0.0 x.im))))

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) <= -5e-11) || !((x_46_re * y_46_re) <= 2e+54)) {
		tmp = x_46_re * y_46_re;
	} else {
		tmp = y_46_im * (0.0 - x_46_im);
	}
	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) <= (-5d-11)) .or. (.not. ((x_46re * y_46re) <= 2d+54))) then
        tmp = x_46re * y_46re
    else
        tmp = y_46im * (0.0d0 - x_46im)
    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) <= -5e-11) || !((x_46_re * y_46_re) <= 2e+54)) {
		tmp = x_46_re * y_46_re;
	} else {
		tmp = y_46_im * (0.0 - x_46_im);
	}
	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) <= -5e-11) or not ((x_46_re * y_46_re) <= 2e+54):
		tmp = x_46_re * y_46_re
	else:
		tmp = y_46_im * (0.0 - x_46_im)
	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) <= -5e-11) || !(Float64(x_46_re * y_46_re) <= 2e+54))
		tmp = Float64(x_46_re * y_46_re);
	else
		tmp = Float64(y_46_im * Float64(0.0 - x_46_im));
	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) <= -5e-11) || ~(((x_46_re * y_46_re) <= 2e+54)))
		tmp = x_46_re * y_46_re;
	else
		tmp = y_46_im * (0.0 - x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], -5e-11], N[Not[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 2e+54]], $MachinePrecision]], N[(x$46$re * y$46$re), $MachinePrecision], N[(y$46$im * N[(0.0 - x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.re) < -5.00000000000000018e-11 or 2.0000000000000002e54 < (*.f64 x.re y.re)

   1. Initial program 97.6%

      \[x.re \cdot y.re - x.im \cdot y.im \]
   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 84.8%

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

   5. Simplified 84.8%

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

   if -5.00000000000000018e-11 < (*.f64 x.re y.re) < 2.0000000000000002e54

   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 80.8%

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

   5. Simplified 80.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(x.im \cdot y.im\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(y.im \cdot x.im\right)\right) \]

      4. *-lowering-*.f64 80.8%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y.im, x.im\right)\right) \]

   7. Applied egg-rr 80.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -5 \cdot 10^{-11} \lor \neg \left(x.re \cdot y.re \leq 2 \cdot 10^{+54}\right):\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(0 - x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* x.re y.re) (* y.im x.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) - (y_46_im * x_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) - (y_46im * x_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) - (y_46_im * x_46_im);
}

Python

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

Derivation

1. Initial program 98.8%

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

3. Final simplification 98.8%

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

Alternative 5: 52.2% 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 98.8%

   \[x.re \cdot y.re - x.im \cdot y.im \]
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. *-lowering-*.f64 54.6%

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

5. Simplified 54.6%

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

Reproduce

herbie shell --seed 2024130 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))