_multiplyComplex, imaginary part

Percentage Accurate: 99.2% → 99.5%

Time: 3.0s

Alternatives: 4

Speedup: 1.0×

• 23.3% 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: 99.5% 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 98.0%

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

   1. fma-define 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 73.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -9.5 \cdot 10^{-145} \lor \neg \left(x.re \cdot y.im \leq 0.065\right) \land \left(x.re \cdot y.im \leq 1.35 \cdot 10^{+85} \lor \neg \left(x.re \cdot y.im \leq 1.35 \cdot 10^{+148}\right)\right):\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.re y.im) -9.5e-145)
         (and (not (<= (* x.re y.im) 0.065))
              (or (<= (* x.re y.im) 1.35e+85)
                  (not (<= (* x.re y.im) 1.35e+148)))))
   (* 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) {
	double tmp;
	if (((x_46_re * y_46_im) <= -9.5e-145) || (!((x_46_re * y_46_im) <= 0.065) && (((x_46_re * y_46_im) <= 1.35e+85) || !((x_46_re * y_46_im) <= 1.35e+148)))) {
		tmp = x_46_re * y_46_im;
	} else {
		tmp = x_46_im * 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) :: tmp
    if (((x_46re * y_46im) <= (-9.5d-145)) .or. (.not. ((x_46re * y_46im) <= 0.065d0)) .and. ((x_46re * y_46im) <= 1.35d+85) .or. (.not. ((x_46re * y_46im) <= 1.35d+148))) then
        tmp = x_46re * y_46im
    else
        tmp = x_46im * 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 tmp;
	if (((x_46_re * y_46_im) <= -9.5e-145) || (!((x_46_re * y_46_im) <= 0.065) && (((x_46_re * y_46_im) <= 1.35e+85) || !((x_46_re * y_46_im) <= 1.35e+148)))) {
		tmp = x_46_re * y_46_im;
	} 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):
	tmp = 0
	if ((x_46_re * y_46_im) <= -9.5e-145) or (not ((x_46_re * y_46_im) <= 0.065) and (((x_46_re * y_46_im) <= 1.35e+85) or not ((x_46_re * y_46_im) <= 1.35e+148))):
		tmp = x_46_re * y_46_im
	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)
	tmp = 0.0
	if ((Float64(x_46_re * y_46_im) <= -9.5e-145) || (!(Float64(x_46_re * y_46_im) <= 0.065) && ((Float64(x_46_re * y_46_im) <= 1.35e+85) || !(Float64(x_46_re * y_46_im) <= 1.35e+148))))
		tmp = Float64(x_46_re * y_46_im);
	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)
	tmp = 0.0;
	if (((x_46_re * y_46_im) <= -9.5e-145) || (~(((x_46_re * y_46_im) <= 0.065)) && (((x_46_re * y_46_im) <= 1.35e+85) || ~(((x_46_re * y_46_im) <= 1.35e+148)))))
		tmp = x_46_re * y_46_im;
	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_] := If[Or[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], -9.5e-145], And[N[Not[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], 0.065]], $MachinePrecision], Or[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], 1.35e+85], N[Not[LessEqual[N[(x$46$re * y$46$im), $MachinePrecision], 1.35e+148]], $MachinePrecision]]]], N[(x$46$re * y$46$im), $MachinePrecision], N[(x$46$im * y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.im) < -9.49999999999999981e-145 or 0.065000000000000002 < (*.f64 x.re y.im) < 1.34999999999999992e85 or 1.35000000000000009e148 < (*.f64 x.re y.im)

   1. Initial program 96.6%

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

   3. Taylor expanded in x.re around inf 77.7%

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

   if -9.49999999999999981e-145 < (*.f64 x.re y.im) < 0.065000000000000002 or 1.34999999999999992e85 < (*.f64 x.re y.im) < 1.35000000000000009e148

   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 87.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.im \leq -9.5 \cdot 10^{-145} \lor \neg \left(x.re \cdot y.im \leq 0.065\right) \land \left(x.re \cdot y.im \leq 1.35 \cdot 10^{+85} \lor \neg \left(x.re \cdot y.im \leq 1.35 \cdot 10^{+148}\right)\right):\\ \;\;\;\;x.re \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (+ (* 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) {
	return (x_46_im * y_46_re) + (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_46im * y_46re) + (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_im * y_46_re) + (x_46_re * y_46_im);
}

Python

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Final simplification 98.0%

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

Alternative 4: 52.0% 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 98.0%

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

3. Taylor expanded in x.re around 0 51.0%

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

4. Final simplification 51.0%

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

Reproduce

herbie shell --seed 2024077 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))