_multiplyComplex, real part

Percentage Accurate: 99.0% → 99.6%

Time: 2.0s

Alternatives: 4

Speedup: 1.0×

• 76.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)
use fmin_fmax_functions
    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: 99.0% accurate, 1.0× speedup

Math

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

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)
use fmin_fmax_functions
    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, 1.0× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. add-flip N/A

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

   4. sub-flip N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. distribute-lft-neg-out N/A

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

   10. remove-double-neg N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. distribute-lft-neg-in N/A

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

   16. lower-*.f64 N/A

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

   17. lower-neg.f64 99.6%

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

3. Applied rewrites 99.6%

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

Alternative 2: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(-y.im\right) \cdot x.im\\ \mathbf{if}\;x.im \cdot y.im \leq -1 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x.im \cdot y.im \leq 4 \cdot 10^{+55}:\\ \;\;\;\;x.re \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* (- y.im) x.im)))
  (if (<= (* x.im y.im) -1e-78)
    t_0
    (if (<= (* x.im y.im) 4e+55) (* x.re y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -y_46_im * x_46_im;
	double tmp;
	if ((x_46_im * y_46_im) <= -1e-78) {
		tmp = t_0;
	} else if ((x_46_im * y_46_im) <= 4e+55) {
		tmp = x_46_re * y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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 = -y_46im * x_46im
    if ((x_46im * y_46im) <= (-1d-78)) then
        tmp = t_0
    else if ((x_46im * y_46im) <= 4d+55) then
        tmp = x_46re * y_46re
    else
        tmp = t_0
    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 = -y_46_im * x_46_im;
	double tmp;
	if ((x_46_im * y_46_im) <= -1e-78) {
		tmp = t_0;
	} else if ((x_46_im * y_46_im) <= 4e+55) {
		tmp = x_46_re * y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -y_46_im * x_46_im
	tmp = 0
	if (x_46_im * y_46_im) <= -1e-78:
		tmp = t_0
	elif (x_46_im * y_46_im) <= 4e+55:
		tmp = x_46_re * y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-y_46_im) * x_46_im)
	tmp = 0.0
	if (Float64(x_46_im * y_46_im) <= -1e-78)
		tmp = t_0;
	elseif (Float64(x_46_im * y_46_im) <= 4e+55)
		tmp = Float64(x_46_re * y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -y_46_im * x_46_im;
	tmp = 0.0;
	if ((x_46_im * y_46_im) <= -1e-78)
		tmp = t_0;
	elseif ((x_46_im * y_46_im) <= 4e+55)
		tmp = x_46_re * y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-y$46$im) * x$46$im), $MachinePrecision]}, If[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], -1e-78], t$95$0, If[LessEqual[N[(x$46$im * y$46$im), $MachinePrecision], 4e+55], N[(x$46$re * y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x.im y.im) < -1e-78 or 4e55 < (*.f64 x.im y.im)

   1. Initial program 99.0%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 52.7%

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

   4. Applied rewrites 52.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot y.im\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot y.im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(y.im \cdot x.im\right) \]

      5. distribute-lft-neg-out N/A

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

      6. lift-neg.f64 N/A

         \[\leadsto \left(-y.im\right) \cdot x.im \]

      7. lift-*.f64 52.7%

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

   6. Applied rewrites 52.7%

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

   if -1e-78 < (*.f64 x.im y.im) < 4e55

   1. Initial program 99.0%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 52.7%

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

   4. Applied rewrites 52.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot y.im\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot y.im\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(y.im \cdot x.im\right) \]

      5. distribute-lft-neg-out N/A

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

      6. lift-neg.f64 N/A

         \[\leadsto \left(-y.im\right) \cdot x.im \]

      7. lift-*.f64 52.7%

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

   6. Applied rewrites 52.7%

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

   7. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{x.re \cdot y.re} \]
   8. Step-by-step derivation

      1. lower-*.f64 51.3%

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

   9. Applied rewrites 51.3%

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

Alternative 3: 51.3% accurate, 2.4× speedup

Math

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

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)
use fmin_fmax_functions
    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 99.0%

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

2. Taylor expanded in x.re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 52.7%

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

4. Applied rewrites 52.7%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(x.im \cdot y.im\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(x.im \cdot y.im\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(y.im \cdot x.im\right) \]

   5. distribute-lft-neg-out N/A

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

   6. lift-neg.f64 N/A

      \[\leadsto \left(-y.im\right) \cdot x.im \]

   7. lift-*.f64 52.7%

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

6. Applied rewrites 52.7%

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

7. Taylor expanded in x.re around inf

   \[\leadsto \color{blue}{x.re \cdot y.re} \]
8. Step-by-step derivation

   1. lower-*.f64 51.3%

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

9. Applied rewrites 51.3%

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

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))