math.cube on complex, imaginary part

Percentage Accurate: 82.3% → 99.8%

Time: 4.0s

Alternatives: 10

Speedup: 1.7×

• 55.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (+
 (* (- (* x.re x.re) (* x.im x.im)) x.im)
 (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

Wolfram

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

Initial Program: 82.3% accurate, 1.0× speedup

Math

\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (+
 (* (- (* x.re x.re) (* x.im x.im)) x.im)
 (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re + x.re, \left|x.im\right|, \left|x.im\right| \cdot x.re\right), \left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - \left|x.im\right|\right) \cdot \left(\left|x.im\right| + x.re\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 50000000.0)
   (fma
    x.re
    (fma (+ x.re x.re) (fabs x.im) (* (fabs x.im) x.re))
    (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im)))
   (*
    (fabs x.im)
    (fma
     (+ x.re x.re)
     x.re
     (* (- x.re (fabs x.im)) (+ (fabs x.im) x.re)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 50000000.0) {
		tmp = fma(x_46_re, fma((x_46_re + x_46_re), fabs(x_46_im), (fabs(x_46_im) * x_46_re)), ((-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im)));
	} else {
		tmp = fabs(x_46_im) * fma((x_46_re + x_46_re), x_46_re, ((x_46_re - fabs(x_46_im)) * (fabs(x_46_im) + x_46_re)));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 50000000.0)
		tmp = fma(x_46_re, fma(Float64(x_46_re + x_46_re), abs(x_46_im), Float64(abs(x_46_im) * x_46_re)), Float64(Float64(Float64(-abs(x_46_im)) * abs(x_46_im)) * abs(x_46_im)));
	else
		tmp = Float64(abs(x_46_im) * fma(Float64(x_46_re + x_46_re), x_46_re, Float64(Float64(x_46_re - abs(x_46_im)) * Float64(abs(x_46_im) + x_46_re))));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 50000000.0], N[(x$46$re * N[(N[(x$46$re + x$46$re), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$re + N[(N[(x$46$re - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5e7

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. distribute-rgt-in N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. associate-+r+ N/A

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

   3. Applied rewrites 88.3%

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

   if 5e7 < x.im

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

      \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := x.re - \left|x.im\right|\\ t_1 := \left|x.im\right| + x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1 \cdot \left|x.im\right|, \left(\left(x.re + x.re\right) \cdot \left|x.im\right|\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(x.re + x.re, x.re, t\_0 \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (- x.re (fabs x.im))) (t_1 (+ (fabs x.im) x.re)))
  (*
   (copysign 1.0 x.im)
   (if (<= (fabs x.im) 50000000.0)
     (fma
      t_0
      (* t_1 (fabs x.im))
      (* (* (+ x.re x.re) (fabs x.im)) x.re))
     (* (fabs x.im) (fma (+ x.re x.re) x.re (* t_0 t_1)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re - fabs(x_46_im);
	double t_1 = fabs(x_46_im) + x_46_re;
	double tmp;
	if (fabs(x_46_im) <= 50000000.0) {
		tmp = fma(t_0, (t_1 * fabs(x_46_im)), (((x_46_re + x_46_re) * fabs(x_46_im)) * x_46_re));
	} else {
		tmp = fabs(x_46_im) * fma((x_46_re + x_46_re), x_46_re, (t_0 * t_1));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re - abs(x_46_im))
	t_1 = Float64(abs(x_46_im) + x_46_re)
	tmp = 0.0
	if (abs(x_46_im) <= 50000000.0)
		tmp = fma(t_0, Float64(t_1 * abs(x_46_im)), Float64(Float64(Float64(x_46_re + x_46_re) * abs(x_46_im)) * x_46_re));
	else
		tmp = Float64(abs(x_46_im) * fma(Float64(x_46_re + x_46_re), x_46_re, Float64(t_0 * t_1)));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 50000000.0], N[(t$95$0 * N[(t$95$1 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re + x$46$re), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$re + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < 5e7

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   3. Applied rewrites 91.9%

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

   if 5e7 < x.im

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

      \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 50000000:\\ \;\;\;\;\left(\left(3 \cdot \left|x.im\right|\right) \cdot x.re\right) \cdot x.re - \left(\left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - \left|x.im\right|\right) \cdot \left(\left|x.im\right| + x.re\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 50000000.0)
   (-
    (* (* (* 3.0 (fabs x.im)) x.re) x.re)
    (* (* (fabs x.im) (fabs x.im)) (fabs x.im)))
   (*
    (fabs x.im)
    (fma
     (+ x.re x.re)
     x.re
     (* (- x.re (fabs x.im)) (+ (fabs x.im) x.re)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 50000000.0) {
		tmp = (((3.0 * fabs(x_46_im)) * x_46_re) * x_46_re) - ((fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im));
	} else {
		tmp = fabs(x_46_im) * fma((x_46_re + x_46_re), x_46_re, ((x_46_re - fabs(x_46_im)) * (fabs(x_46_im) + x_46_re)));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 50000000.0)
		tmp = Float64(Float64(Float64(Float64(3.0 * abs(x_46_im)) * x_46_re) * x_46_re) - Float64(Float64(abs(x_46_im) * abs(x_46_im)) * abs(x_46_im)));
	else
		tmp = Float64(abs(x_46_im) * fma(Float64(x_46_re + x_46_re), x_46_re, Float64(Float64(x_46_re - abs(x_46_im)) * Float64(abs(x_46_im) + x_46_re))));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 50000000.0], N[(N[(N[(N[(3.0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$re + N[(N[(x$46$re - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5e7

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. distribute-rgt-in N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. associate-+r+ N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. lower--.f64 N/A

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

   3. Applied rewrites 86.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right)} - \left(x.im \cdot x.im\right) \cdot x.im \]

      2. lift-*.f64 N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot x.re\right)} - \left(x.im \cdot x.im\right) \cdot x.im \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re} - \left(x.im \cdot x.im\right) \cdot x.im \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re} - \left(x.im \cdot x.im\right) \cdot x.im \]

      5. lift-*.f64 N/A

         \[\leadsto \left(3 \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot x.im\right) \cdot x.re\right)} \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(\color{blue}{\left(2 + 1\right)} \cdot x.im\right) \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      8. distribute-lft1-in N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot x.im + x.im\right)} \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      9. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(2, x.im, x.im\right)} \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      10. lower-*.f64 86.1%

          \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2, x.im, x.im\right) \cdot x.re\right)} \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      11. lift-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(2 \cdot x.im + x.im\right)} \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      12. distribute-lft1-in N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 + 1\right) \cdot x.im\right)} \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\color{blue}{3} \cdot x.im\right) \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      14. lower-*.f64 86.1%

          \[\leadsto \left(\color{blue}{\left(3 \cdot x.im\right)} \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re} - \left(x.im \cdot x.im\right) \cdot x.im \]

   if 5e7 < x.im

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

      \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.8% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\left(\left(3 \cdot \left|x.im\right|\right) \cdot \left|x.re\right|\right) \cdot \left|x.re\right| - \left(\left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(\left|x.re\right| + \left|x.re\right|, \left|x.re\right|, \left(\left|x.re\right| - \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 5e+95)
   (-
    (* (* (* 3.0 (fabs x.im)) (fabs x.re)) (fabs x.re))
    (* (* (fabs x.im) (fabs x.im)) (fabs x.im)))
   (*
    (fabs x.im)
    (fma
     (+ (fabs x.re) (fabs x.re))
     (fabs x.re)
     (* (- (fabs x.re) (fabs x.im)) (fabs x.im)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 5e+95) {
		tmp = (((3.0 * fabs(x_46_im)) * fabs(x_46_re)) * fabs(x_46_re)) - ((fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im));
	} else {
		tmp = fabs(x_46_im) * fma((fabs(x_46_re) + fabs(x_46_re)), fabs(x_46_re), ((fabs(x_46_re) - fabs(x_46_im)) * fabs(x_46_im)));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 5e+95)
		tmp = Float64(Float64(Float64(Float64(3.0 * abs(x_46_im)) * abs(x_46_re)) * abs(x_46_re)) - Float64(Float64(abs(x_46_im) * abs(x_46_im)) * abs(x_46_im)));
	else
		tmp = Float64(abs(x_46_im) * fma(Float64(abs(x_46_re) + abs(x_46_re)), abs(x_46_re), Float64(Float64(abs(x_46_re) - abs(x_46_im)) * abs(x_46_im))));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 5e+95], N[(N[(N[(N[(3.0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(N[Abs[x$46$re], $MachinePrecision] + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + N[(N[(N[Abs[x$46$re], $MachinePrecision] - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000002e95

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. distribute-rgt-in N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. associate-+r+ N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. lower--.f64 N/A

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

   3. Applied rewrites 86.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right)} - \left(x.im \cdot x.im\right) \cdot x.im \]

      2. lift-*.f64 N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot x.re\right)} - \left(x.im \cdot x.im\right) \cdot x.im \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re} - \left(x.im \cdot x.im\right) \cdot x.im \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re} - \left(x.im \cdot x.im\right) \cdot x.im \]

      5. lift-*.f64 N/A

         \[\leadsto \left(3 \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(3 \cdot x.im\right) \cdot x.re\right)} \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(\color{blue}{\left(2 + 1\right)} \cdot x.im\right) \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      8. distribute-lft1-in N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot x.im + x.im\right)} \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      9. lift-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(2, x.im, x.im\right)} \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      10. lower-*.f64 86.1%

          \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2, x.im, x.im\right) \cdot x.re\right)} \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      11. lift-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(2 \cdot x.im + x.im\right)} \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      12. distribute-lft1-in N/A

          \[\leadsto \left(\color{blue}{\left(\left(2 + 1\right) \cdot x.im\right)} \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\color{blue}{3} \cdot x.im\right) \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

      14. lower-*.f64 86.1%

          \[\leadsto \left(\color{blue}{\left(3 \cdot x.im\right)} \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot x.im \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re} - \left(x.im \cdot x.im\right) \cdot x.im \]

   if 5.0000000000000002e95 < x.im

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in x.re around 0

      \[\leadsto x.im \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - x.im\right) \cdot \color{blue}{x.im}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 75.7%

      \[\leadsto x.im \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - x.im\right) \cdot \color{blue}{x.im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.8% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 5 \cdot 10^{+95}:\\ \;\;\;\;3 \cdot \left(\left(\left|x.im\right| \cdot \left|x.re\right|\right) \cdot \left|x.re\right|\right) - \left(\left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(\left|x.re\right| + \left|x.re\right|, \left|x.re\right|, \left(\left|x.re\right| - \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 5e+95)
   (-
    (* 3.0 (* (* (fabs x.im) (fabs x.re)) (fabs x.re)))
    (* (* (fabs x.im) (fabs x.im)) (fabs x.im)))
   (*
    (fabs x.im)
    (fma
     (+ (fabs x.re) (fabs x.re))
     (fabs x.re)
     (* (- (fabs x.re) (fabs x.im)) (fabs x.im)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 5e+95) {
		tmp = (3.0 * ((fabs(x_46_im) * fabs(x_46_re)) * fabs(x_46_re))) - ((fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im));
	} else {
		tmp = fabs(x_46_im) * fma((fabs(x_46_re) + fabs(x_46_re)), fabs(x_46_re), ((fabs(x_46_re) - fabs(x_46_im)) * fabs(x_46_im)));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 5e+95)
		tmp = Float64(Float64(3.0 * Float64(Float64(abs(x_46_im) * abs(x_46_re)) * abs(x_46_re))) - Float64(Float64(abs(x_46_im) * abs(x_46_im)) * abs(x_46_im)));
	else
		tmp = Float64(abs(x_46_im) * fma(Float64(abs(x_46_re) + abs(x_46_re)), abs(x_46_re), Float64(Float64(abs(x_46_re) - abs(x_46_im)) * abs(x_46_im))));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 5e+95], N[(N[(3.0 * N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(N[Abs[x$46$re], $MachinePrecision] + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + N[(N[(N[Abs[x$46$re], $MachinePrecision] - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000002e95

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. distribute-rgt-in N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. associate-+r+ N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. lower--.f64 N/A

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

   3. Applied rewrites 86.1%

      \[\leadsto \color{blue}{3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) - \left(x.im \cdot x.im\right) \cdot x.im} \]

   if 5.0000000000000002e95 < x.im

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in x.re around 0

      \[\leadsto x.im \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - x.im\right) \cdot \color{blue}{x.im}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 75.7%

      \[\leadsto x.im \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - x.im\right) \cdot \color{blue}{x.im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.5% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(3 \cdot \left|x.re\right|, \left|x.re\right|, \left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(\left|x.re\right| + \left|x.re\right|, \left|x.re\right|, \left(\left|x.re\right| - \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 5e+95)
   (*
    (fabs x.im)
    (fma
     (* 3.0 (fabs x.re))
     (fabs x.re)
     (* (- (fabs x.im)) (fabs x.im))))
   (*
    (fabs x.im)
    (fma
     (+ (fabs x.re) (fabs x.re))
     (fabs x.re)
     (* (- (fabs x.re) (fabs x.im)) (fabs x.im)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 5e+95) {
		tmp = fabs(x_46_im) * fma((3.0 * fabs(x_46_re)), fabs(x_46_re), (-fabs(x_46_im) * fabs(x_46_im)));
	} else {
		tmp = fabs(x_46_im) * fma((fabs(x_46_re) + fabs(x_46_re)), fabs(x_46_re), ((fabs(x_46_re) - fabs(x_46_im)) * fabs(x_46_im)));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 5e+95)
		tmp = Float64(abs(x_46_im) * fma(Float64(3.0 * abs(x_46_re)), abs(x_46_re), Float64(Float64(-abs(x_46_im)) * abs(x_46_im))));
	else
		tmp = Float64(abs(x_46_im) * fma(Float64(abs(x_46_re) + abs(x_46_re)), abs(x_46_re), Float64(Float64(abs(x_46_re) - abs(x_46_im)) * abs(x_46_im))));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 5e+95], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(3.0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(N[Abs[x$46$re], $MachinePrecision] + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + N[(N[(N[Abs[x$46$re], $MachinePrecision] - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000002e95

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

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

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(-x.im, x.im, \left(x.re \cdot x.re\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto x.im \cdot \color{blue}{\left(\left(-x.im\right) \cdot x.im + \left(x.re \cdot x.re\right) \cdot 3\right)} \]

      2. +-commutative N/A

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3 + \left(-x.im\right) \cdot x.im\right)} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3 - \left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)} \]

      4. sub-flip N/A

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 3} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot 3 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto x.im \cdot \left(\color{blue}{x.re \cdot \left(x.re \cdot 3\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re \cdot 3\right) \cdot x.re} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)\right)\right) \]

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

         \[\leadsto x.im \cdot \left(\left(x.re \cdot 3\right) \cdot x.re + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(-x.im\right) \cdot x.im\right)\right)}\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto x.im \cdot \left(\left(x.re \cdot 3\right) \cdot x.re + \color{blue}{\left(-x.im\right) \cdot x.im}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto x.im \cdot \color{blue}{\mathsf{fma}\left(x.re \cdot 3, x.re, \left(-x.im\right) \cdot x.im\right)} \]

      12. *-commutative N/A

          \[\leadsto x.im \cdot \mathsf{fma}\left(\color{blue}{3 \cdot x.re}, x.re, \left(-x.im\right) \cdot x.im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto x.im \cdot \mathsf{fma}\left(\color{blue}{3 \cdot x.re}, x.re, \left(-x.im\right) \cdot x.im\right) \]

      14. lower-*.f64 90.2%

          \[\leadsto x.im \cdot \mathsf{fma}\left(3 \cdot x.re, x.re, \color{blue}{\left(-x.im\right) \cdot x.im}\right) \]

   7. Applied rewrites 90.2%

      \[\leadsto x.im \cdot \color{blue}{\mathsf{fma}\left(3 \cdot x.re, x.re, \left(-x.im\right) \cdot x.im\right)} \]

   if 5.0000000000000002e95 < x.im

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in x.re around 0

      \[\leadsto x.im \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - x.im\right) \cdot \color{blue}{x.im}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 75.7%

      \[\leadsto x.im \cdot \mathsf{fma}\left(x.re + x.re, x.re, \left(x.re - x.im\right) \cdot \color{blue}{x.im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := -\left|x.im\right|\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 4 \cdot 10^{+166}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(t\_0, \left|x.im\right|, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (- (fabs x.im))))
  (*
   (copysign 1.0 x.im)
   (if (<= (fabs x.im) 4e+166)
     (* (fabs x.im) (fma t_0 (fabs x.im) (* (* x.re x.re) 3.0)))
     (* (* t_0 (fabs x.im)) (fabs x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = -fabs(x_46_im);
	double tmp;
	if (fabs(x_46_im) <= 4e+166) {
		tmp = fabs(x_46_im) * fma(t_0, fabs(x_46_im), ((x_46_re * x_46_re) * 3.0));
	} else {
		tmp = (t_0 * fabs(x_46_im)) * fabs(x_46_im);
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(-abs(x_46_im))
	tmp = 0.0
	if (abs(x_46_im) <= 4e+166)
		tmp = Float64(abs(x_46_im) * fma(t_0, abs(x_46_im), Float64(Float64(x_46_re * x_46_re) * 3.0)));
	else
		tmp = Float64(Float64(t_0 * abs(x_46_im)) * abs(x_46_im));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = (-N[Abs[x$46$im], $MachinePrecision])}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 4e+166], N[(N[Abs[x$46$im], $MachinePrecision] * N[(t$95$0 * N[Abs[x$46$im], $MachinePrecision] + N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.9999999999999998e166

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

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

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(-x.im, x.im, \left(x.re \cdot x.re\right) \cdot 3\right)} \]

   if 3.9999999999999998e166 < x.im

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. lower-pow.f64 59.0%

         \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. lift-pow.f64 N/A

         \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

      3. pow3 N/A

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

      4. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(x.im \cdot x.im\right) \cdot x.im\right) \]

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 58.9%

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

   8. Applied rewrites 58.9%

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

Alternative 8: 90.2% accurate, 1.7× speedup

Math

\[x.im \cdot \mathsf{fma}\left(3 \cdot x.re, x.re, \left(-x.im\right) \cdot x.im\right) \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (* x.im (fma (* 3.0 x.re) x.re (* (- x.im) x.im))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * fma((3.0 * x_46_re), x_46_re, (-x_46_im * x_46_im));
}

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_im * fma(Float64(3.0 * x_46_re), x_46_re, Float64(Float64(-x_46_im) * x_46_im)))
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$im * N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$re + N[((-x$46$im) * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. sub-flip N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. distribute-rgt-out N/A

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

   13. associate-*l* N/A

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

   14. distribute-rgt-neg-in N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. remove-double-neg N/A

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

   17. lift-*.f64 N/A

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

3. Applied rewrites 93.5%

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

5. Applied rewrites 90.8%

   \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(-x.im, x.im, \left(x.re \cdot x.re\right) \cdot 3\right)} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto x.im \cdot \color{blue}{\left(\left(-x.im\right) \cdot x.im + \left(x.re \cdot x.re\right) \cdot 3\right)} \]

   2. +-commutative N/A

      \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3 + \left(-x.im\right) \cdot x.im\right)} \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3 - \left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)} \]

   4. sub-flip N/A

      \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 3} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot 3 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto x.im \cdot \left(\color{blue}{x.re \cdot \left(x.re \cdot 3\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re \cdot 3\right) \cdot x.re} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-x.im\right)\right)\right) \cdot x.im\right)\right)\right) \]

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

      \[\leadsto x.im \cdot \left(\left(x.re \cdot 3\right) \cdot x.re + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(-x.im\right) \cdot x.im\right)\right)}\right)\right)\right) \]

   10. remove-double-neg N/A

       \[\leadsto x.im \cdot \left(\left(x.re \cdot 3\right) \cdot x.re + \color{blue}{\left(-x.im\right) \cdot x.im}\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto x.im \cdot \color{blue}{\mathsf{fma}\left(x.re \cdot 3, x.re, \left(-x.im\right) \cdot x.im\right)} \]

   12. *-commutative N/A

       \[\leadsto x.im \cdot \mathsf{fma}\left(\color{blue}{3 \cdot x.re}, x.re, \left(-x.im\right) \cdot x.im\right) \]

   13. lower-*.f64 N/A

       \[\leadsto x.im \cdot \mathsf{fma}\left(\color{blue}{3 \cdot x.re}, x.re, \left(-x.im\right) \cdot x.im\right) \]

   14. lower-*.f64 90.2%

       \[\leadsto x.im \cdot \mathsf{fma}\left(3 \cdot x.re, x.re, \color{blue}{\left(-x.im\right) \cdot x.im}\right) \]

7. Applied rewrites 90.2%

   \[\leadsto x.im \cdot \color{blue}{\mathsf{fma}\left(3 \cdot x.re, x.re, \left(-x.im\right) \cdot x.im\right)} \]
8. Add Preprocessing

Alternative 9: 89.5% accurate, 0.9× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 7.5 \cdot 10^{+137}:\\ \;\;\;\;\left|x.im\right| \cdot \left(\left(3 \cdot x.re\right) \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 7.5e+137)
   (*
    (fabs x.im)
    (- (* (* 3.0 x.re) x.re) (* (fabs x.im) (fabs x.im))))
   (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 7.5e+137) {
		tmp = fabs(x_46_im) * (((3.0 * x_46_re) * x_46_re) - (fabs(x_46_im) * fabs(x_46_im)));
	} else {
		tmp = (-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im);
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (Math.abs(x_46_im) <= 7.5e+137) {
		tmp = Math.abs(x_46_im) * (((3.0 * x_46_re) * x_46_re) - (Math.abs(x_46_im) * Math.abs(x_46_im)));
	} else {
		tmp = (-Math.abs(x_46_im) * Math.abs(x_46_im)) * Math.abs(x_46_im);
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_im) <= 7.5e+137:
		tmp = math.fabs(x_46_im) * (((3.0 * x_46_re) * x_46_re) - (math.fabs(x_46_im) * math.fabs(x_46_im)))
	else:
		tmp = (-math.fabs(x_46_im) * math.fabs(x_46_im)) * math.fabs(x_46_im)
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 7.5e+137)
		tmp = Float64(abs(x_46_im) * Float64(Float64(Float64(3.0 * x_46_re) * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))));
	else
		tmp = Float64(Float64(Float64(-abs(x_46_im)) * abs(x_46_im)) * abs(x_46_im));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (abs(x_46_im) <= 7.5e+137)
		tmp = abs(x_46_im) * (((3.0 * x_46_re) * x_46_re) - (abs(x_46_im) * abs(x_46_im)));
	else
		tmp = (-abs(x_46_im) * abs(x_46_im)) * abs(x_46_im);
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 7.5e+137], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.5000000000000002e137

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

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

   5. Applied rewrites 87.4%

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

      1. lift-*.f64 N/A

         \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 3} - x.im \cdot x.im\right) \]

      2. lift-*.f64 N/A

         \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot 3 - x.im \cdot x.im\right) \]

      3. associate-*l* N/A

         \[\leadsto x.im \cdot \left(\color{blue}{x.re \cdot \left(x.re \cdot 3\right)} - x.im \cdot x.im\right) \]

      4. *-commutative N/A

         \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re \cdot 3\right) \cdot x.re} - x.im \cdot x.im\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re \cdot 3\right) \cdot x.re} - x.im \cdot x.im\right) \]

      6. *-commutative N/A

         \[\leadsto x.im \cdot \left(\color{blue}{\left(3 \cdot x.re\right)} \cdot x.re - x.im \cdot x.im\right) \]

      7. lower-*.f64 87.4%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(3 \cdot x.re\right)} \cdot x.re - x.im \cdot x.im\right) \]

   7. Applied rewrites 87.4%

      \[\leadsto x.im \cdot \left(\color{blue}{\left(3 \cdot x.re\right) \cdot x.re} - x.im \cdot x.im\right) \]

   if 7.5000000000000002e137 < x.im

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. remove-double-neg N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. associate-*l* N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. remove-double-neg N/A

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

      17. lift-*.f64 N/A

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

   3. Applied rewrites 93.5%

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

   4. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. lower-pow.f64 59.0%

         \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

      2. lift-pow.f64 N/A

         \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

      3. pow3 N/A

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

      4. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(x.im \cdot x.im\right) \cdot x.im\right) \]

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 58.9%

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

   8. Applied rewrites 58.9%

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

Alternative 10: 58.9% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (* (* (- x.im) x.im) x.im))

C

double code(double x_46_re, double x_46_im) {
	return (-x_46_im * x_46_im) * x_46_im;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (-x_46im * x_46im) * x_46im
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (-x_46_im * x_46_im) * x_46_im;
}

Python

def code(x_46_re, x_46_im):
	return (-x_46_im * x_46_im) * x_46_im

Julia

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

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (-x_46_im * x_46_im) * x_46_im;
end

Wolfram

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

Derivation

1. Initial program 82.3%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. sub-flip N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. distribute-rgt-out N/A

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

   13. associate-*l* N/A

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

   14. distribute-rgt-neg-in N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. remove-double-neg N/A

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

   17. lift-*.f64 N/A

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

3. Applied rewrites 93.5%

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

4. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

   2. lower-pow.f64 59.0%

      \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

6. Applied rewrites 59.0%

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

   2. lift-pow.f64 N/A

      \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

   3. pow3 N/A

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

   4. lift-*.f64 N/A

      \[\leadsto -1 \cdot \left(\left(x.im \cdot x.im\right) \cdot x.im\right) \]

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-neg.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 58.9%

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

8. Applied rewrites 58.9%

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

Developer Target 1: 91.8% accurate, 1.1× speedup

Math

\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (+
 (* (* x.re x.im) (* 2.0 x.re))
 (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025217 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform c (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))