_multiplyComplex, imaginary part

Percentage Accurate: 99.1% → 99.6%

Time: 3.6s

Alternatives: 3

Speedup: 1.2×

• 23.5% 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

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.im x.re (* y.re 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_im, x_46_re, (y_46_re * x_46_im));
}

Julia

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$im * x$46$re + N[(y$46$re * x$46$im), $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. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-pow1 N/A

      \[\leadsto \color{blue}{\sqrt{{\left(x.re \cdot y.im\right)}^{2}}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \cdot y.re \]

   8. pow2 N/A

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

   9. rem-sqrt-square-rev N/A

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

   10. remove-double-neg N/A

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

   11. lift-*.f64 N/A

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

   12. unpow1 N/A

       \[\leadsto \left|x.re \cdot y.im\right| + \color{blue}{{\left(x.im \cdot y.re\right)}^{1}} \]

   13. metadata-eval N/A

       \[\leadsto \left|x.re \cdot y.im\right| + {\left(x.im \cdot y.re\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \]

   14. sqrt-pow1 N/A

       \[\leadsto \left|x.re \cdot y.im\right| + \color{blue}{\sqrt{{\left(x.im \cdot y.re\right)}^{2}}} \]

   15. pow2 N/A

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

   16. rem-sqrt-square-rev N/A

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

   17. rem-sqrt-square-rev N/A

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

   18. pow2 N/A

       \[\leadsto \left|x.re \cdot y.im\right| + \sqrt{\color{blue}{{\left(x.im \cdot y.re\right)}^{2}}} \]

   19. sqrt-pow1 N/A

       \[\leadsto \left|x.re \cdot y.im\right| + \color{blue}{{\left(x.im \cdot y.re\right)}^{\left(\frac{2}{2}\right)}} \]

   20. metadata-eval N/A

       \[\leadsto \left|x.re \cdot y.im\right| + {\left(x.im \cdot y.re\right)}^{\color{blue}{1}} \]

   21. unpow1 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -0.02 \lor \neg \left(x.re \cdot y.im \leq 10^{-35}\right):\\ \;\;\;\;y.im \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.re y.im) -0.02) (not (<= (* x.re y.im) 1e-35)))
   (* y.im x.re)
   (* y.re 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_im) <= -0.02) || !((x_46_re * y_46_im) <= 1e-35)) {
		tmp = y_46_im * x_46_re;
	} else {
		tmp = y_46_re * 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_46im) <= (-0.02d0)) .or. (.not. ((x_46re * y_46im) <= 1d-35))) then
        tmp = y_46im * x_46re
    else
        tmp = y_46re * 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_im) <= -0.02) || !((x_46_re * y_46_im) <= 1e-35)) {
		tmp = y_46_im * x_46_re;
	} else {
		tmp = y_46_re * 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_im) <= -0.02) or not ((x_46_re * y_46_im) <= 1e-35):
		tmp = y_46_im * x_46_re
	else:
		tmp = y_46_re * 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_im) <= -0.02) || !(Float64(x_46_re * y_46_im) <= 1e-35))
		tmp = Float64(y_46_im * x_46_re);
	else
		tmp = Float64(y_46_re * 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_im) <= -0.02) || ~(((x_46_re * y_46_im) <= 1e-35)))
		tmp = y_46_im * x_46_re;
	else
		tmp = y_46_re * 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$46$im), $MachinePrecision], -0.02], N[Not[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], 1e-35]], $MachinePrecision]], N[(y$46$im * x$46$re), $MachinePrecision], N[(y$46$re * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.im) < -0.0200000000000000004 or 1.00000000000000001e-35 < (*.f64 x.re y.im)

   1. Initial program 98.5%

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

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

      2. lower-*.f64 24.3

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

   5. Applied rewrites 24.3%

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

   7. Applied rewrites 15.7%

      \[\leadsto \left|y.re \cdot x.im\right| \]

   8. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.9

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

   10. Applied rewrites 77.9%

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

   if -0.0200000000000000004 < (*.f64 x.re y.im) < 1.00000000000000001e-35

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -0.02 \lor \neg \left(x.re \cdot y.im \leq 10^{-35}\right):\\ \;\;\;\;y.im \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.9% 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 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 51.9

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

5. Applied rewrites 51.9%

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

7. Applied rewrites 29.2%

   \[\leadsto \left|y.re \cdot x.im\right| \]

8. Taylor expanded in x.re around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 51.9

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

10. Applied rewrites 51.9%

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

Reproduce

herbie shell --seed 2024343 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))