_multiplyComplex, imaginary part

Percentage Accurate: 99.1% → 99.5%

Time: 1.8s

Alternatives: 5

Speedup: 1.0×

• 23.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 99.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma y.re x.im (* x.re 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_im, (x_46_re * 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_im, Float64(x_46_re * y_46_im))
end

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x.re \cdot y.im + x.im \cdot y.re \]

2. Taylor expanded in x.re around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-def 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma x.re y.im (* 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(x_46_re, y_46_im, (y_46_re * x_46_im));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 75.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -8500000000000:\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{elif}\;x.re \cdot y.im \leq -6 \cdot 10^{-20} \lor \neg \left(x.re \cdot y.im \leq -5 \cdot 10^{-61}\right) \land x.re \cdot y.im \leq 3.1 \cdot 10^{-47}:\\ \;\;\;\;y.re \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= (* x.re y.im) -8500000000000.0)
   (* x.re y.im)
   (if (or (<= (* x.re y.im) -6e-20)
           (and (not (<= (* x.re y.im) -5e-61)) (<= (* x.re y.im) 3.1e-47)))
     (* y.re x.im)
     (* x.re 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_im) <= -8500000000000.0) {
		tmp = x_46_re * y_46_im;
	} else if (((x_46_re * y_46_im) <= -6e-20) || (!((x_46_re * y_46_im) <= -5e-61) && ((x_46_re * y_46_im) <= 3.1e-47))) {
		tmp = y_46_re * x_46_im;
	} else {
		tmp = x_46_re * y_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) <= (-8500000000000.0d0)) then
        tmp = x_46re * y_46im
    else if (((x_46re * y_46im) <= (-6d-20)) .or. (.not. ((x_46re * y_46im) <= (-5d-61))) .and. ((x_46re * y_46im) <= 3.1d-47)) then
        tmp = y_46re * x_46im
    else
        tmp = x_46re * y_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) <= -8500000000000.0) {
		tmp = x_46_re * y_46_im;
	} else if (((x_46_re * y_46_im) <= -6e-20) || (!((x_46_re * y_46_im) <= -5e-61) && ((x_46_re * y_46_im) <= 3.1e-47))) {
		tmp = y_46_re * x_46_im;
	} else {
		tmp = x_46_re * y_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) <= -8500000000000.0:
		tmp = x_46_re * y_46_im
	elif ((x_46_re * y_46_im) <= -6e-20) or (not ((x_46_re * y_46_im) <= -5e-61) and ((x_46_re * y_46_im) <= 3.1e-47)):
		tmp = y_46_re * x_46_im
	else:
		tmp = x_46_re * 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_im) <= -8500000000000.0)
		tmp = Float64(x_46_re * y_46_im);
	elseif ((Float64(x_46_re * y_46_im) <= -6e-20) || (!(Float64(x_46_re * y_46_im) <= -5e-61) && (Float64(x_46_re * y_46_im) <= 3.1e-47)))
		tmp = Float64(y_46_re * x_46_im);
	else
		tmp = Float64(x_46_re * y_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) <= -8500000000000.0)
		tmp = x_46_re * y_46_im;
	elseif (((x_46_re * y_46_im) <= -6e-20) || (~(((x_46_re * y_46_im) <= -5e-61)) && ((x_46_re * y_46_im) <= 3.1e-47)))
		tmp = y_46_re * x_46_im;
	else
		tmp = x_46_re * y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], -8500000000000.0], N[(x$46$re * y$46$im), $MachinePrecision], If[Or[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], -6e-20], And[N[Not[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], -5e-61]], $MachinePrecision], LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], 3.1e-47]]], N[(y$46$re * x$46$im), $MachinePrecision], N[(x$46$re * y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.im) < -8.5e12 or -6.00000000000000057e-20 < (*.f64 x.re y.im) < -4.9999999999999999e-61 or 3.0999999999999998e-47 < (*.f64 x.re y.im)

   1. Initial program 99.3%

      \[x.re \cdot y.im + x.im \cdot y.re \]

   2. Taylor expanded in x.re around inf 77.2%

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

   if -8.5e12 < (*.f64 x.re y.im) < -6.00000000000000057e-20 or -4.9999999999999999e-61 < (*.f64 x.re y.im) < 3.0999999999999998e-47

   1. Initial program 100.0%

      \[x.re \cdot y.im + x.im \cdot y.re \]

   2. Taylor expanded in x.re around 0 79.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -8500000000000:\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{elif}\;x.re \cdot y.im \leq -6 \cdot 10^{-20} \lor \neg \left(x.re \cdot y.im \leq -5 \cdot 10^{-61}\right) \land x.re \cdot y.im \leq 3.1 \cdot 10^{-47}:\\ \;\;\;\;y.re \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.im\\ \end{array} \]

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (+ (* x.re y.im) (* y.re 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_im) + (y_46_re * 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_46im) + (y_46re * 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_im) + (y_46_re * x_46_im);
}

Python

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x.re \cdot y.im + x.im \cdot y.re \]

2. Final simplification 99.6%

   \[\leadsto x.re \cdot y.im + y.re \cdot x.im \]

Alternative 5: 51.0% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* x.re 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_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_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_im;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x.re \cdot y.im + x.im \cdot y.re \]

2. Taylor expanded in x.re around inf 54.7%

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

3. Final simplification 54.7%

   \[\leadsto x.re \cdot y.im \]

Reproduce

herbie shell --seed 2023194 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))