_multiplyComplex, real part

Percentage Accurate: 99.3% → 99.3%

Time: 3.4s

Alternatives: 4

Speedup: 1.0×

• 23.8% 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: 99.3% 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: 99.3% 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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 73.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \cdot y.im \leq -4 \cdot 10^{+54} \lor \neg \left(x.im \cdot y.im \leq -2 \cdot 10^{-86}\right) \land \left(x.im \cdot y.im \leq -2 \cdot 10^{-182} \lor \neg \left(x.im \cdot y.im \leq 10^{+43}\right)\right):\\ \;\;\;\;y.im \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= (* x.im y.im) -4e+54)
         (and (not (<= (* x.im y.im) -2e-86))
              (or (<= (* x.im y.im) -2e-182) (not (<= (* x.im y.im) 1e+43)))))
   (* y.im (- x.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_im * y_46_im) <= -4e+54) || (!((x_46_im * y_46_im) <= -2e-86) && (((x_46_im * y_46_im) <= -2e-182) || !((x_46_im * y_46_im) <= 1e+43)))) {
		tmp = y_46_im * -x_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_46im * y_46im) <= (-4d+54)) .or. (.not. ((x_46im * y_46im) <= (-2d-86))) .and. ((x_46im * y_46im) <= (-2d-182)) .or. (.not. ((x_46im * y_46im) <= 1d+43))) then
        tmp = y_46im * -x_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_im * y_46_im) <= -4e+54) || (!((x_46_im * y_46_im) <= -2e-86) && (((x_46_im * y_46_im) <= -2e-182) || !((x_46_im * y_46_im) <= 1e+43)))) {
		tmp = y_46_im * -x_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_im * y_46_im) <= -4e+54) or (not ((x_46_im * y_46_im) <= -2e-86) and (((x_46_im * y_46_im) <= -2e-182) or not ((x_46_im * y_46_im) <= 1e+43))):
		tmp = y_46_im * -x_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_im * y_46_im) <= -4e+54) || (!(Float64(x_46_im * y_46_im) <= -2e-86) && ((Float64(x_46_im * y_46_im) <= -2e-182) || !(Float64(x_46_im * y_46_im) <= 1e+43))))
		tmp = Float64(y_46_im * Float64(-x_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_im * y_46_im) <= -4e+54) || (~(((x_46_im * y_46_im) <= -2e-86)) && (((x_46_im * y_46_im) <= -2e-182) || ~(((x_46_im * y_46_im) <= 1e+43)))))
		tmp = y_46_im * -x_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[Or[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], -4e+54], And[N[Not[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], -2e-86]], $MachinePrecision], Or[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], -2e-182], N[Not[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], 1e+43]], $MachinePrecision]]]], N[(y$46$im * (-x$46$im)), $MachinePrecision], N[(x$46$re * y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.im y.im) < -4.0000000000000003e54 or -2.00000000000000017e-86 < (*.f64 x.im y.im) < -2.0000000000000001e-182 or 1.00000000000000001e43 < (*.f64 x.im y.im)

   1. Initial program 99.2%

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

   3. Taylor expanded in x.re around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. distribute-rgt-neg-out 86.8%

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

   5. Simplified 86.8%

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

   if -4.0000000000000003e54 < (*.f64 x.im y.im) < -2.00000000000000017e-86 or -2.0000000000000001e-182 < (*.f64 x.im y.im) < 1.00000000000000001e43

   1. Initial program 100.0%

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

   3. Taylor expanded in x.re around inf 78.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.im \leq -4 \cdot 10^{+54} \lor \neg \left(x.im \cdot y.im \leq -2 \cdot 10^{-86}\right) \land \left(x.im \cdot y.im \leq -2 \cdot 10^{-182} \lor \neg \left(x.im \cdot y.im \leq 10^{+43}\right)\right):\\ \;\;\;\;y.im \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot y.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.0% 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. Add Preprocessing

3. Taylor expanded in x.re around inf 48.2%

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

Alternative 4: 3.9% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* 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_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_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_im * y_46_im;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x.re around 0 53.8%

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

   1. mul-1-neg 53.8%

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

   2. distribute-rgt-neg-out 53.8%

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

5. Simplified 53.8%

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

   1. expm1-log1p-u 35.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(-y.im\right)\right)\right)} \]

   2. expm1-undefine 24.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(-y.im\right)\right)} - 1} \]

   3. *-commutative 24.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-y.im\right) \cdot x.im}\right)} - 1 \]

   4. add-sqr-sqrt 10.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-y.im} \cdot \sqrt{-y.im}\right)} \cdot x.im\right)} - 1 \]

   5. sqrt-unprod 9.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-y.im\right) \cdot \left(-y.im\right)}} \cdot x.im\right)} - 1 \]

   6. sqr-neg 9.0%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{y.im \cdot y.im}} \cdot x.im\right)} - 1 \]

   7. sqrt-unprod 1.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{y.im} \cdot \sqrt{y.im}\right)} \cdot x.im\right)} - 1 \]

   8. add-sqr-sqrt 2.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{y.im} \cdot x.im\right)} - 1 \]

7. Applied egg-rr 2.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y.im \cdot x.im\right)} - 1} \]
8. Step-by-step derivation

   1. log1p-undefine 2.3%

      \[\leadsto e^{\color{blue}{\log \left(1 + y.im \cdot x.im\right)}} - 1 \]

   2. rem-exp-log 2.5%

      \[\leadsto \color{blue}{\left(1 + y.im \cdot x.im\right)} - 1 \]

   3. +-commutative 2.5%

      \[\leadsto \color{blue}{\left(y.im \cdot x.im + 1\right)} - 1 \]

   4. associate--l+ 2.5%

      \[\leadsto \color{blue}{y.im \cdot x.im + \left(1 - 1\right)} \]

   5. metadata-eval 2.5%

      \[\leadsto y.im \cdot x.im + \color{blue}{0} \]

   6. mul0-lft 2.5%

      \[\leadsto y.im \cdot x.im + \color{blue}{0 \cdot x.im} \]

   7. distribute-rgt-in 2.5%

      \[\leadsto \color{blue}{x.im \cdot \left(y.im + 0\right)} \]

   8. +-rgt-identity 2.5%

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

9. Simplified 2.5%

   \[\leadsto \color{blue}{x.im \cdot y.im} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))