_multiplyComplex, real part

Percentage Accurate: 99.2% → 99.6%

Time: 2.0s

Alternatives: 4

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* x.re y.re) (* x.im 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_re) - (x_46_im * 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_46re) - (x_46im * 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_re) - (x_46_im * y_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (x_46_re * y_46_re) - (x_46_im * y_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_re) - Float64(x_46_im * 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_re) - (x_46_im * y_46_im);
end

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* x.re y.re) (* x.im 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_re) - (x_46_im * 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_46re) - (x_46im * 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_re) - (x_46_im * y_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (x_46_re * y_46_re) - (x_46_im * y_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_re) - Float64(x_46_im * 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_re) - (x_46_im * y_46_im);
end

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. fma-neg 100.0%

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

   2. distribute-rgt-neg-in 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -5.8 \cdot 10^{-90}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq 0.49 \lor \neg \left(x.re \cdot y.re \leq 5.6 \cdot 10^{+63}\right) \land x.re \cdot y.re \leq 1.9 \cdot 10^{+95}:\\ \;\;\;\;-x.im \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= (* x.re y.re) -5.8e-90)
   (* x.re y.re)
   (if (or (<= (* x.re y.re) 0.49)
           (and (not (<= (* x.re y.re) 5.6e+63)) (<= (* x.re y.re) 1.9e+95)))
     (- (* 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 tmp;
	if ((x_46_re * y_46_re) <= -5.8e-90) {
		tmp = x_46_re * y_46_re;
	} else if (((x_46_re * y_46_re) <= 0.49) || (!((x_46_re * y_46_re) <= 5.6e+63) && ((x_46_re * y_46_re) <= 1.9e+95))) {
		tmp = -(x_46_im * y_46_im);
	} else {
		tmp = 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) :: tmp
    if ((x_46re * y_46re) <= (-5.8d-90)) then
        tmp = x_46re * y_46re
    else if (((x_46re * y_46re) <= 0.49d0) .or. (.not. ((x_46re * y_46re) <= 5.6d+63)) .and. ((x_46re * y_46re) <= 1.9d+95)) then
        tmp = -(x_46im * y_46im)
    else
        tmp = 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 tmp;
	if ((x_46_re * y_46_re) <= -5.8e-90) {
		tmp = x_46_re * y_46_re;
	} else if (((x_46_re * y_46_re) <= 0.49) || (!((x_46_re * y_46_re) <= 5.6e+63) && ((x_46_re * y_46_re) <= 1.9e+95))) {
		tmp = -(x_46_im * y_46_im);
	} else {
		tmp = x_46_re * 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_re) <= -5.8e-90:
		tmp = x_46_re * y_46_re
	elif ((x_46_re * y_46_re) <= 0.49) or (not ((x_46_re * y_46_re) <= 5.6e+63) and ((x_46_re * y_46_re) <= 1.9e+95)):
		tmp = -(x_46_im * y_46_im)
	else:
		tmp = x_46_re * 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_re) <= -5.8e-90)
		tmp = Float64(x_46_re * y_46_re);
	elseif ((Float64(x_46_re * y_46_re) <= 0.49) || (!(Float64(x_46_re * y_46_re) <= 5.6e+63) && (Float64(x_46_re * y_46_re) <= 1.9e+95)))
		tmp = Float64(-Float64(x_46_im * y_46_im));
	else
		tmp = 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)
	tmp = 0.0;
	if ((x_46_re * y_46_re) <= -5.8e-90)
		tmp = x_46_re * y_46_re;
	elseif (((x_46_re * y_46_re) <= 0.49) || (~(((x_46_re * y_46_re) <= 5.6e+63)) && ((x_46_re * y_46_re) <= 1.9e+95)))
		tmp = -(x_46_im * y_46_im);
	else
		tmp = 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_] := If[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], -5.8e-90], N[(x$46$re * y$46$re), $MachinePrecision], If[Or[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 0.49], And[N[Not[LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 5.6e+63]], $MachinePrecision], LessEqual[N[(x$46$re * y$46$re), $MachinePrecision], 1.9e+95]]], (-N[(x$46$im * y$46$im), $MachinePrecision]), N[(x$46$re * y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.re y.re) < -5.79999999999999967e-90 or 0.48999999999999999 < (*.f64 x.re y.re) < 5.59999999999999974e63 or 1.9e95 < (*.f64 x.re y.re)

   1. Initial program 99.3%

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

   2. Taylor expanded in x.re around inf 81.7%

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

   if -5.79999999999999967e-90 < (*.f64 x.re y.re) < 0.48999999999999999 or 5.59999999999999974e63 < (*.f64 x.re y.re) < 1.9e95

   1. Initial program 100.0%

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

   2. Taylor expanded in x.re around 0 85.2%

      \[\leadsto \color{blue}{-1 \cdot \left(y.im \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 85.2%

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

      2. distribute-rgt-neg-in 85.2%

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

   4. Simplified 85.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot y.re \leq -5.8 \cdot 10^{-90}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{elif}\;x.re \cdot y.re \leq 0.49 \lor \neg \left(x.re \cdot y.re \leq 5.6 \cdot 10^{+63}\right) \land x.re \cdot y.re \leq 1.9 \cdot 10^{+95}:\\ \;\;\;\;-x.im \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* x.re y.re) (* x.im 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_re) - (x_46_im * 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_46re) - (x_46im * 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_re) - (x_46_im * y_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (x_46_re * y_46_re) - (x_46_im * y_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_re) - Float64(x_46_im * 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_re) - (x_46_im * y_46_im);
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Final simplification 99.6%

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

Alternative 4: 52.8% accurate, 2.3× speedup

Math

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

FPCore

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x.re around inf 55.8%

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

3. Final simplification 55.8%

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

Reproduce

herbie shell --seed 2023171 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))