_multiplyComplex, imaginary part

Percentage Accurate: 99.2% → 99.7%

Time: 2.7s

Alternatives: 5

Speedup: 1.0×

• 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: 99.2% 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: 99.7% 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 98.8%

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

   1. +-commutative 98.8%

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

   2. *-commutative 98.8%

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

   3. fma-def 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 99.5% 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 98.8%

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

   1. fma-def 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 76.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -5.6 \cdot 10^{+170} \lor \neg \left(x.re \cdot y.im \leq -2.1 \cdot 10^{+165}\right) \land \left(x.re \cdot y.im \leq -9 \cdot 10^{-18} \lor \neg \left(x.re \cdot y.im \leq 2.85 \cdot 10^{-47}\right)\right):\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.re y.im) -5.6e+170)
         (and (not (<= (* x.re y.im) -2.1e+165))
              (or (<= (* x.re y.im) -9e-18)
                  (not (<= (* x.re y.im) 2.85e-47)))))
   (* 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) {
	double tmp;
	if (((x_46_re * y_46_im) <= -5.6e+170) || (!((x_46_re * y_46_im) <= -2.1e+165) && (((x_46_re * y_46_im) <= -9e-18) || !((x_46_re * y_46_im) <= 2.85e-47)))) {
		tmp = x_46_re * y_46_im;
	} 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) <= (-5.6d+170)) .or. (.not. ((x_46re * y_46im) <= (-2.1d+165))) .and. ((x_46re * y_46im) <= (-9d-18)) .or. (.not. ((x_46re * y_46im) <= 2.85d-47))) then
        tmp = x_46re * y_46im
    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) <= -5.6e+170) || (!((x_46_re * y_46_im) <= -2.1e+165) && (((x_46_re * y_46_im) <= -9e-18) || !((x_46_re * y_46_im) <= 2.85e-47)))) {
		tmp = x_46_re * y_46_im;
	} 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) <= -5.6e+170) or (not ((x_46_re * y_46_im) <= -2.1e+165) and (((x_46_re * y_46_im) <= -9e-18) or not ((x_46_re * y_46_im) <= 2.85e-47))):
		tmp = x_46_re * y_46_im
	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) <= -5.6e+170) || (!(Float64(x_46_re * y_46_im) <= -2.1e+165) && ((Float64(x_46_re * y_46_im) <= -9e-18) || !(Float64(x_46_re * y_46_im) <= 2.85e-47))))
		tmp = Float64(x_46_re * y_46_im);
	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) <= -5.6e+170) || (~(((x_46_re * y_46_im) <= -2.1e+165)) && (((x_46_re * y_46_im) <= -9e-18) || ~(((x_46_re * y_46_im) <= 2.85e-47)))))
		tmp = x_46_re * y_46_im;
	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], -5.6e+170], And[N[Not[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], -2.1e+165]], $MachinePrecision], Or[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], -9e-18], N[Not[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], 2.85e-47]], $MachinePrecision]]]], N[(x$46$re * y$46$im), $MachinePrecision], N[(y$46$re * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.im) < -5.6000000000000003e170 or -2.1000000000000001e165 < (*.f64 x.re y.im) < -8.99999999999999987e-18 or 2.85000000000000023e-47 < (*.f64 x.re y.im)

   1. Initial program 97.9%

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

   3. Taylor expanded in x.re around inf 74.5%

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

   if -5.6000000000000003e170 < (*.f64 x.re y.im) < -2.1000000000000001e165 or -8.99999999999999987e-18 < (*.f64 x.re y.im) < 2.85000000000000023e-47

   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 83.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -5.6 \cdot 10^{+170} \lor \neg \left(x.re \cdot y.im \leq -2.1 \cdot 10^{+165}\right) \land \left(x.re \cdot y.im \leq -9 \cdot 10^{-18} \lor \neg \left(x.re \cdot y.im \leq 2.85 \cdot 10^{-47}\right)\right):\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% 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 98.8%

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

3. Final simplification 98.8%

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

Alternative 5: 52.2% accurate, 2.3× speedup

Math

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

FPCore

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in x.re around 0 52.1%

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

4. Final simplification 52.1%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))