_multiplyComplex, imaginary part

Percentage Accurate: 99.3% → 99.2%

Time: 3.6s

Alternatives: 5

Speedup: 0.4×

• 24.1% 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: 99.3% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re + x.re \cdot y.im\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(x.im + y.im \cdot \frac{x.re}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.im y.re) (* x.re y.im))))
   (if (<= t_0 2e+273) t_0 (* y.re (+ x.im (* y.im (/ x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) + (x_46_re * y_46_im);
	double tmp;
	if (t_0 <= 2e+273) {
		tmp = t_0;
	} else {
		tmp = y_46_re * (x_46_im + (y_46_im * (x_46_re / y_46_re)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im * y_46re) + (x_46re * y_46im)
    if (t_0 <= 2d+273) then
        tmp = t_0
    else
        tmp = y_46re * (x_46im + (y_46im * (x_46re / y_46re)))
    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 t_0 = (x_46_im * y_46_re) + (x_46_re * y_46_im);
	double tmp;
	if (t_0 <= 2e+273) {
		tmp = t_0;
	} else {
		tmp = y_46_re * (x_46_im + (y_46_im * (x_46_re / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * y_46_re) + (x_46_re * y_46_im)
	tmp = 0
	if t_0 <= 2e+273:
		tmp = t_0
	else:
		tmp = y_46_re * (x_46_im + (y_46_im * (x_46_re / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) + Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (t_0 <= 2e+273)
		tmp = t_0;
	else
		tmp = Float64(y_46_re * Float64(x_46_im + Float64(y_46_im * Float64(x_46_re / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * y_46_re) + (x_46_re * y_46_im);
	tmp = 0.0;
	if (t_0 <= 2e+273)
		tmp = t_0;
	else
		tmp = y_46_re * (x_46_im + (y_46_im * (x_46_re / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] + N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+273], t$95$0, N[(y$46$re * N[(x$46$im + N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x.re y.im) (*.f64 x.im y.re)) < 1.99999999999999989e273

   1. Initial program 100.0%

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

   if 1.99999999999999989e273 < (+.f64 (*.f64 x.re y.im) (*.f64 x.im y.re))

   1. Initial program 87.8%

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

   3. Taylor expanded in y.re around inf 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-/l* 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.re + x.re \cdot y.im \leq 2 \cdot 10^{+273}:\\ \;\;\;\;x.im \cdot y.re + x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(x.im + y.im \cdot \frac{x.re}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma 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 fma(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 fma(x_46_re, y_46_im, Float64(x_46_im * y_46_re))
end

Wolfram

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

Derivation

1. Initial program 97.6%

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

   1. fma-define 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. Add Preprocessing

5. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re + x.re \cdot y.im\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) + (x_46_re * y_46_im);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = x_46_im * y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * y_46_re) + (x_46_re * y_46_im)
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = x_46_im * y_46_re
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * y_46_re) + (x_46_re * y_46_im);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = x_46_im * y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x.re around 0 83.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.re + x.re \cdot y.im \leq \infty:\\ \;\;\;\;x.im \cdot y.re + x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \cdot y.re \leq -1.45 \cdot 10^{-112} \lor \neg \left(x.im \cdot y.re \leq 7 \cdot 10^{-20}\right):\\ \;\;\;\;x.im \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.im y.re) -1.45e-112) (not (<= (* x.im y.re) 7e-20)))
   (* x.im y.re)
   (* 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_im * y_46_re) <= -1.45e-112) || !((x_46_im * y_46_re) <= 7e-20)) {
		tmp = x_46_im * y_46_re;
	} 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_46im * y_46re) <= (-1.45d-112)) .or. (.not. ((x_46im * y_46re) <= 7d-20))) then
        tmp = x_46im * y_46re
    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_im * y_46_re) <= -1.45e-112) || !((x_46_im * y_46_re) <= 7e-20)) {
		tmp = x_46_im * y_46_re;
	} 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_im * y_46_re) <= -1.45e-112) or not ((x_46_im * y_46_re) <= 7e-20):
		tmp = x_46_im * y_46_re
	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_im * y_46_re) <= -1.45e-112) || !(Float64(x_46_im * y_46_re) <= 7e-20))
		tmp = Float64(x_46_im * y_46_re);
	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_im * y_46_re) <= -1.45e-112) || ~(((x_46_im * y_46_re) <= 7e-20)))
		tmp = x_46_im * y_46_re;
	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[Or[LessEqual[N[(x$46$im * y$46$re), $MachinePrecision], -1.45e-112], N[Not[LessEqual[N[(x$46$im * y$46$re), $MachinePrecision], 7e-20]], $MachinePrecision]], N[(x$46$im * y$46$re), $MachinePrecision], N[(x$46$re * y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.im y.re) < -1.44999999999999996e-112 or 7.00000000000000007e-20 < (*.f64 x.im y.re)

   1. Initial program 95.8%

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

   3. Taylor expanded in x.re around 0 76.1%

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

   if -1.44999999999999996e-112 < (*.f64 x.im y.re) < 7.00000000000000007e-20

   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 inf 84.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.re \leq -1.45 \cdot 10^{-112} \lor \neg \left(x.im \cdot y.re \leq 7 \cdot 10^{-20}\right):\\ \;\;\;\;x.im \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.2% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* 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_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_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_im * y_46_re;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in x.re around 0 51.4%

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

4. Final simplification 51.4%

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

Reproduce

herbie shell --seed 2024079 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))