_multiplyComplex, real part

Percentage Accurate: 99.2% → 99.6%

Time: 3.9s

Alternatives: 7

Speedup: 1.0×

• 23.8% 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

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma (- y.im) x.im (* y.re x.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_im, (y_46_re * x_46_re));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -2 \cdot 10^{+55} \lor \neg \left(x.re \cdot y.re \leq 5 \cdot 10^{+112}\right):\\ \;\;\;\;\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.re y.re) -2e+55) (not (<= (* x.re y.re) 5e+112)))
   (fma y.im x.im (* 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) {
	double tmp;
	if (((x_46_re * y_46_re) <= -2e+55) || !((x_46_re * y_46_re) <= 5e+112)) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re));
	} else {
		tmp = -x_46_im * y_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) <= -2e+55) || !(Float64(x_46_re * y_46_re) <= 5e+112))
		tmp = fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re));
	else
		tmp = Float64(Float64(-x_46_im) * y_46_im);
	end
	return 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], -2e+55], N[Not[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 5e+112]], $MachinePrecision]], N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[((-x$46$im) * y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.re) < -2.00000000000000002e55 or 5e112 < (*.f64 x.re y.re)

   1. Initial program 94.7%

      \[x.re \cdot y.re - x.im \cdot y.im \]
   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 95.8

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

   4. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y.re, x.re, \left(-y.im\right) \cdot x.im\right)} \]
   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 98.9

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

   6. Applied rewrites 87.8%

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

   if -2.00000000000000002e55 < (*.f64 x.re y.re) < 5e112

   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 78.0

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

   5. Applied rewrites 78.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -2 \cdot 10^{+55} \lor \neg \left(x.re \cdot y.re \leq 5 \cdot 10^{+112}\right):\\ \;\;\;\;\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -2 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)\\ \mathbf{elif}\;x.re \cdot y.re \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\left(-x.im\right) \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= (* x.re y.re) -2e+55)
   (fma y.im x.im (* y.re x.re))
   (if (<= (* x.re y.re) 5e+112)
     (* (- x.im) y.im)
     (fma y.re x.re (* y.im 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) <= -2e+55) {
		tmp = fma(y_46_im, x_46_im, (y_46_re * x_46_re));
	} else if ((x_46_re * y_46_re) <= 5e+112) {
		tmp = -x_46_im * y_46_im;
	} else {
		tmp = fma(y_46_re, x_46_re, (y_46_im * 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) <= -2e+55)
		tmp = fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re));
	elseif (Float64(x_46_re * y_46_re) <= 5e+112)
		tmp = Float64(Float64(-x_46_im) * y_46_im);
	else
		tmp = fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im));
	end
	return 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], -2e+55], N[(y$46$im * x$46$im + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 5e+112], N[((-x$46$im) * y$46$im), $MachinePrecision], N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x.re y.re) < -2.00000000000000002e55

   1. Initial program 94.8%

      \[x.re \cdot y.re - x.im \cdot y.im \]
   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 94.8

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

   4. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y.re, x.re, \left(-y.im\right) \cdot x.im\right)} \]
   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 87.3%

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

   if -2.00000000000000002e55 < (*.f64 x.re y.re) < 5e112

   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 78.0

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

   5. Applied rewrites 78.0%

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

   if 5e112 < (*.f64 x.re y.re)

   1. Initial program 94.6%

      \[x.re \cdot y.re - x.im \cdot y.im \]
   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 97.3

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

   4. Applied rewrites 97.3%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. sub0-neg N/A

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

      6. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(y.re, x.re, \frac{\mathsf{neg}\left(\color{blue}{{y.im}^{\left(\frac{3}{2}\right)} \cdot {y.im}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(y.im \cdot y.im + 0 \cdot y.im\right)} \cdot x.im\right) \]

      7. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(y.re, x.re, \frac{\mathsf{neg}\left(\color{blue}{{\left(y.im \cdot y.im\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(y.im \cdot y.im + 0 \cdot y.im\right)} \cdot x.im\right) \]

      8. sqr-neg N/A

         \[\leadsto \mathsf{fma}\left(y.re, x.re, \frac{\mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(y.im\right)\right) \cdot \left(\mathsf{neg}\left(y.im\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(y.im \cdot y.im + 0 \cdot y.im\right)} \cdot x.im\right) \]

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. pow-prod-down N/A

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

      12. sqr-pow N/A

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

      13. lift-neg.f64 N/A

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

      14. cube-neg N/A

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

      15. sub0-neg N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. flip3-- N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg 88.4

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

   6. Applied rewrites 88.4%

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

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re - x.im \cdot y.im\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.re y.re) (* x.im y.im))))
   (if (<= t_0 INFINITY) t_0 (fma y.re x.re (* y.im x.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) - (x_46_im * y_46_im);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(y_46_re, x_46_re, (y_46_im * x_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * y_46_re) - Float64(x_46_im * y_46_im))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) < +inf.0

   1. Initial program 100.0%

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

   if +inf.0 < (-.f64 (*.f64 x.re y.re) (*.f64 x.im y.im))

   1. Initial program 0.0%

      \[x.re \cdot y.re - x.im \cdot y.im \]
   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 20.0

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

   4. Applied rewrites 20.0%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. flip3-- N/A

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

      4. metadata-eval N/A

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

      5. sub0-neg N/A

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

      6. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(y.re, x.re, \frac{\mathsf{neg}\left(\color{blue}{{y.im}^{\left(\frac{3}{2}\right)} \cdot {y.im}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(y.im \cdot y.im + 0 \cdot y.im\right)} \cdot x.im\right) \]

      7. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(y.re, x.re, \frac{\mathsf{neg}\left(\color{blue}{{\left(y.im \cdot y.im\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(y.im \cdot y.im + 0 \cdot y.im\right)} \cdot x.im\right) \]

      8. sqr-neg N/A

         \[\leadsto \mathsf{fma}\left(y.re, x.re, \frac{\mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(y.im\right)\right) \cdot \left(\mathsf{neg}\left(y.im\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(y.im \cdot y.im + 0 \cdot y.im\right)} \cdot x.im\right) \]

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. pow-prod-down N/A

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

      12. sqr-pow N/A

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

      13. lift-neg.f64 N/A

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

      14. cube-neg N/A

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

      15. sub0-neg N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. flip3-- N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg 80.0

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

   6. Applied rewrites 80.0%

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

Alternative 5: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.re x.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 fma(y_46_re, x_46_re, (-y_46_im * 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(Float64(-y_46_im) * 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) * x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.0%

   \[x.re \cdot y.re - x.im \cdot y.im \]
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 98.4

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

4. Applied rewrites 98.4%

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

Alternative 6: 52.3% 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 98.0%

   \[x.re \cdot y.re - x.im \cdot y.im \]
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 54.8

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

5. Applied rewrites 54.8%

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

Alternative 7: 3.7% accurate, 2.3× speedup

Math

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

FPCore

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

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_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_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 y_46_im * x_46_im;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.0%

   \[x.re \cdot y.re - x.im \cdot y.im \]
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 54.8

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

5. Applied rewrites 54.8%

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

7. Applied rewrites 4.7%

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

Reproduce

herbie shell --seed 2024308 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))