math.cube on complex, imaginary part

Percentage Accurate: 82.3% → 99.8%

Time: 3.5s

Alternatives: 8

Speedup: 1.2×

• 55.6% 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, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| + \left|x.re\right|\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\left|x.re\right|, \mathsf{fma}\left(t\_0, \left|x.im\right|, \left|x.im\right| \cdot \left|x.re\right|\right), \left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(t\_0, \left|x.re\right|, \left(\left|x.re\right| - \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (+ (fabs x.re) (fabs x.re))))
  (*
   (copysign 1.0 x.im)
   (if (<= (fabs x.im) 2.5e+72)
     (fma
      (fabs x.re)
      (fma t_0 (fabs x.im) (* (fabs x.im) (fabs x.re)))
      (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im)))
     (*
      (fabs x.im)
      (fma
       t_0
       (fabs x.re)
       (* (- (fabs x.re) (fabs x.im)) (fabs x.im))))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) + fabs(x_46_re);
	double tmp;
	if (fabs(x_46_im) <= 2.5e+72) {
		tmp = fma(fabs(x_46_re), fma(t_0, fabs(x_46_im), (fabs(x_46_im) * fabs(x_46_re))), ((-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im)));
	} else {
		tmp = fabs(x_46_im) * fma(t_0, 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)
	t_0 = Float64(abs(x_46_re) + abs(x_46_re))
	tmp = 0.0
	if (abs(x_46_im) <= 2.5e+72)
		tmp = fma(abs(x_46_re), fma(t_0, abs(x_46_im), Float64(abs(x_46_im) * abs(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(t_0, 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_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] + N[Abs[x$46$re], $MachinePrecision]), $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], 2.5e+72], N[(N[Abs[x$46$re], $MachinePrecision] * N[(t$95$0 * N[Abs[x$46$im], $MachinePrecision] + N[(N[Abs[x$46$im], $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[(t$95$0 * 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 < 2.5e72

   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 87.7%

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

   if 2.5e72 < 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 94.0%

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

   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 75.1%

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(3, \left(\left|x.im\right| \cdot \left|x.re\right|\right) \cdot \left|x.re\right|, \left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(\left|x.re\right| + \left|x.re\right|, \left|x.re\right|, \left(\left|x.re\right| - \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 2.5e+72)
   (fma
    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) <= 2.5e+72) {
		tmp = fma(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) <= 2.5e+72)
		tmp = fma(3.0, Float64(Float64(abs(x_46_im) * abs(x_46_re)) * abs(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(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], 2.5e+72], 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] + 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 < 2.5e72

   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 85.6%

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

   if 2.5e72 < 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 94.0%

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

   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 75.1%

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

Alternative 3: 99.5% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 2.5 \cdot 10^{+72}:\\ \;\;\;\;3 \cdot \left(\left(\left|x.im\right| \cdot \left|x.re\right|\right) \cdot \left|x.re\right|\right) - \left(\left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(\left|x.re\right| + \left|x.re\right|, \left|x.re\right|, \left(\left|x.re\right| - \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 2.5e+72)
   (-
    (* 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) <= 2.5e+72) {
		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) <= 2.5e+72)
		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], 2.5e+72], 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 < 2.5e72

   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 85.6%

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

   if 2.5e72 < 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 94.0%

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

   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 75.1%

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

Alternative 4: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := 3 \cdot \left|x.re\right|\\ t_1 := \left(\left|x.re\right| \cdot \left|x.re\right| - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right| + \left(\left|x.re\right| \cdot \left|x.im\right| + \left|x.im\right| \cdot \left|x.re\right|\right) \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{+259}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(t\_0, \left|x.re\right|, \left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(t\_0 \cdot \left|x.im\right|\right) \cdot \left|x.re\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(\left|x.re\right| + \left|x.re\right|, \left|x.re\right|, \left(\left|x.re\right| - \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* 3.0 (fabs x.re)))
       (t_1
        (+
         (*
          (- (* (fabs x.re) (fabs x.re)) (* (fabs x.im) (fabs x.im)))
          (fabs x.im))
         (*
          (+ (* (fabs x.re) (fabs x.im)) (* (fabs x.im) (fabs x.re)))
          (fabs x.re)))))
  (*
   (copysign 1.0 x.im)
   (if (<= t_1 1e+259)
     (*
      (fabs x.im)
      (fma t_0 (fabs x.re) (* (- (fabs x.im)) (fabs x.im))))
     (if (<= t_1 INFINITY)
       (* (* t_0 (fabs x.im)) (fabs x.re))
       (*
        (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 t_0 = 3.0 * fabs(x_46_re);
	double t_1 = (((fabs(x_46_re) * fabs(x_46_re)) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im)) + (((fabs(x_46_re) * fabs(x_46_im)) + (fabs(x_46_im) * fabs(x_46_re))) * fabs(x_46_re));
	double tmp;
	if (t_1 <= 1e+259) {
		tmp = fabs(x_46_im) * fma(t_0, fabs(x_46_re), (-fabs(x_46_im) * fabs(x_46_im)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * fabs(x_46_im)) * fabs(x_46_re);
	} 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)
	t_0 = Float64(3.0 * abs(x_46_re))
	t_1 = Float64(Float64(Float64(Float64(abs(x_46_re) * abs(x_46_re)) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + Float64(Float64(Float64(abs(x_46_re) * abs(x_46_im)) + Float64(abs(x_46_im) * abs(x_46_re))) * abs(x_46_re)))
	tmp = 0.0
	if (t_1 <= 1e+259)
		tmp = Float64(abs(x_46_im) * fma(t_0, abs(x_46_re), Float64(Float64(-abs(x_46_im)) * abs(x_46_im))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * abs(x_46_im)) * abs(x_46_re));
	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_] := Block[{t$95$0 = N[(3.0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, 1e+259], N[(N[Abs[x$46$im], $MachinePrecision] * N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision] + N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $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 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 9.9999999999999993e258

   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 94.0%

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

   5. Applied rewrites 87.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft1-in N/A

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

      12. *-commutative N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-lft1-in N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 90.9%

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

   7. Applied rewrites 90.9%

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

   if 9.9999999999999993e258 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   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 87.7%

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

   4. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.4%

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

   6. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt1-in N/A

         \[\leadsto x.re \cdot \left(\left(\left(2 + 1\right) \cdot x.re\right) \cdot x.im\right) \]

      8. metadata-eval N/A

         \[\leadsto x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 56.1%

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

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      15. associate-*l* N/A

          \[\leadsto \left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.re \]

      16. *-commutative N/A

          \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

      17. lift-*.f64 N/A

          \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

      18. lower-*.f64 56.1%

          \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

   8. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

      2. lift-*.f64 N/A

         \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

      3. *-commutative N/A

         \[\leadsto \left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\left(2 + 1\right) \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      6. distribute-lft1-in N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

          \[\leadsto \left(\left(\left(2 + 1\right) \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      12. lower-*.f64 56.1%

          \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

   10. Applied rewrites 56.1%

       \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   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 94.0%

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

   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 75.1%

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

Alternative 5: 96.5% accurate, 0.9× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 4.6 \cdot 10^{-92}:\\ \;\;\;\;\left(\left(x.re \cdot \left|x.im\right|\right) \cdot x.re\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(3 \cdot x.re, x.re, \left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 4.6e-92)
   (* (* (* x.re (fabs x.im)) x.re) 3.0)
   (*
    (fabs x.im)
    (fma (* 3.0 x.re) 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) <= 4.6e-92) {
		tmp = ((x_46_re * fabs(x_46_im)) * x_46_re) * 3.0;
	} else {
		tmp = fabs(x_46_im) * fma((3.0 * x_46_re), 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) <= 4.6e-92)
		tmp = Float64(Float64(Float64(x_46_re * abs(x_46_im)) * x_46_re) * 3.0);
	else
		tmp = Float64(abs(x_46_im) * fma(Float64(3.0 * x_46_re), x_46_re, Float64(Float64(-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], 4.6e-92], N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$re + N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.6000000000000003e-92

   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 87.7%

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

   4. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.4%

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

   6. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt1-in N/A

         \[\leadsto x.re \cdot \left(\left(\left(2 + 1\right) \cdot x.re\right) \cdot x.im\right) \]

      8. metadata-eval N/A

         \[\leadsto x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 56.1%

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

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      15. associate-*l* N/A

          \[\leadsto \left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.re \]

      16. *-commutative N/A

          \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

      17. lift-*.f64 N/A

          \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

      18. lower-*.f64 56.1%

          \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

   8. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

         \[\leadsto \left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3 \]

      7. *-commutative N/A

         \[\leadsto \left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3 \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3 \]

      11. lift-*.f64 50.4%

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

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3 \]

      13. *-commutative N/A

          \[\leadsto \left(x.im \cdot \left(x.re \cdot x.re\right)\right) \cdot 3 \]

      14. lift-*.f64 N/A

          \[\leadsto \left(x.im \cdot \left(x.re \cdot x.re\right)\right) \cdot 3 \]

      15. associate-*l* N/A

          \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3 \]

      16. lift-*.f64 N/A

          \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3 \]

      17. lower-*.f64 56.1%

          \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3 \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3 \]

      19. *-commutative N/A

          \[\leadsto \left(\left(x.re \cdot x.im\right) \cdot x.re\right) \cdot 3 \]

      20. lower-*.f64 56.1%

          \[\leadsto \left(\left(x.re \cdot x.im\right) \cdot x.re\right) \cdot 3 \]

   10. Applied rewrites 56.1%

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

   if 4.6000000000000003e-92 < 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 94.0%

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

   5. Applied rewrites 87.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft1-in N/A

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

      12. *-commutative N/A

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-lft1-in N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 90.9%

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

   7. Applied rewrites 90.9%

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

Alternative 6: 95.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 2 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \left(\left(\left|x.re\right| \cdot \left|x.re\right|\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot \left|x.re\right|\right) \cdot x.im\right) \cdot \left|x.re\right|\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (if (<= (fabs x.re) 2e+154)
  (* x.im (- (* (* (fabs x.re) (fabs x.re)) 3.0) (* x.im x.im)))
  (* (* (* 3.0 (fabs x.re)) x.im) (fabs x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_re) <= 2e+154) {
		tmp = x_46_im * (((fabs(x_46_re) * fabs(x_46_re)) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = ((3.0 * fabs(x_46_re)) * x_46_im) * fabs(x_46_re);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(x_46re) <= 2d+154) then
        tmp = x_46im * (((abs(x_46re) * abs(x_46re)) * 3.0d0) - (x_46im * x_46im))
    else
        tmp = ((3.0d0 * abs(x_46re)) * x_46im) * abs(x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (Math.abs(x_46_re) <= 2e+154) {
		tmp = x_46_im * (((Math.abs(x_46_re) * Math.abs(x_46_re)) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = ((3.0 * Math.abs(x_46_re)) * x_46_im) * Math.abs(x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_re) <= 2e+154:
		tmp = x_46_im * (((math.fabs(x_46_re) * math.fabs(x_46_re)) * 3.0) - (x_46_im * x_46_im))
	else:
		tmp = ((3.0 * math.fabs(x_46_re)) * x_46_im) * math.fabs(x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_re) <= 2e+154)
		tmp = Float64(x_46_im * Float64(Float64(Float64(abs(x_46_re) * abs(x_46_re)) * 3.0) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(Float64(Float64(3.0 * abs(x_46_re)) * x_46_im) * abs(x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (abs(x_46_re) <= 2e+154)
		tmp = x_46_im * (((abs(x_46_re) * abs(x_46_re)) * 3.0) - (x_46_im * x_46_im));
	else
		tmp = ((3.0 * abs(x_46_re)) * x_46_im) * abs(x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 2e+154], N[(x$46$im * N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.0000000000000001e154

   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 94.0%

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

   5. Applied rewrites 87.6%

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

   if 2.0000000000000001e154 < x.re

   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 87.7%

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

   4. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.4%

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

   6. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt1-in N/A

         \[\leadsto x.re \cdot \left(\left(\left(2 + 1\right) \cdot x.re\right) \cdot x.im\right) \]

      8. metadata-eval N/A

         \[\leadsto x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 56.1%

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

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      15. associate-*l* N/A

          \[\leadsto \left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.re \]

      16. *-commutative N/A

          \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

      17. lift-*.f64 N/A

          \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

      18. lower-*.f64 56.1%

          \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

   8. Applied rewrites 56.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

      2. lift-*.f64 N/A

         \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

      3. *-commutative N/A

         \[\leadsto \left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.re \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\left(2 + 1\right) \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      6. distribute-lft1-in N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

          \[\leadsto \left(\left(\left(2 + 1\right) \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

      12. lower-*.f64 56.1%

          \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

   10. Applied rewrites 56.1%

       \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 56.1% accurate, 2.7× speedup

Math

\[\left(\left(x.re \cdot x.im\right) \cdot x.re\right) \cdot 3 \]

FPCore

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

C

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

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) * x_46re) * 3.0d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * x_46_re) * 3.0;
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * x_46_re) * 3.0

Julia

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

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * x_46_re) * 3.0;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] * 3.0), $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. 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 87.7%

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

4. Taylor expanded in x.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 50.4%

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

6. Applied rewrites 50.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt1-in N/A

      \[\leadsto x.re \cdot \left(\left(\left(2 + 1\right) \cdot x.re\right) \cdot x.im\right) \]

   8. metadata-eval N/A

      \[\leadsto x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) \]

   9. lift-*.f64 N/A

      \[\leadsto x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) \]

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 56.1%

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

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

   14. lift-*.f64 N/A

       \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

   15. associate-*l* N/A

       \[\leadsto \left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.re \]

   16. *-commutative N/A

       \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

   17. lift-*.f64 N/A

       \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

   18. lower-*.f64 56.1%

       \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

8. Applied rewrites 56.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

      \[\leadsto \left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3 \]

   7. *-commutative N/A

      \[\leadsto \left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 \]

   8. associate-*l* N/A

      \[\leadsto \left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3 \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3 \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3 \]

   11. lift-*.f64 50.4%

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

   12. lift-*.f64 N/A

       \[\leadsto \left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3 \]

   13. *-commutative N/A

       \[\leadsto \left(x.im \cdot \left(x.re \cdot x.re\right)\right) \cdot 3 \]

   14. lift-*.f64 N/A

       \[\leadsto \left(x.im \cdot \left(x.re \cdot x.re\right)\right) \cdot 3 \]

   15. associate-*l* N/A

       \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3 \]

   16. lift-*.f64 N/A

       \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3 \]

   17. lower-*.f64 56.1%

       \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3 \]

   18. lift-*.f64 N/A

       \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3 \]

   19. *-commutative N/A

       \[\leadsto \left(\left(x.re \cdot x.im\right) \cdot x.re\right) \cdot 3 \]

   20. lower-*.f64 56.1%

       \[\leadsto \left(\left(x.re \cdot x.im\right) \cdot x.re\right) \cdot 3 \]

10. Applied rewrites 56.1%

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

Alternative 8: 56.1% accurate, 2.7× speedup

Math

\[\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return (3.0 * (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 = (3.0d0 * (x_46im * x_46re)) * x_46re
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (3.0 * (x_46_im * x_46_re)) * x_46_re;
}

Python

def code(x_46_re, x_46_im):
	return (3.0 * (x_46_im * x_46_re)) * x_46_re

Julia

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

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (3.0 * (x_46_im * x_46_re)) * x_46_re;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $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. 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 87.7%

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

4. Taylor expanded in x.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 50.4%

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

6. Applied rewrites 50.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt1-in N/A

      \[\leadsto x.re \cdot \left(\left(\left(2 + 1\right) \cdot x.re\right) \cdot x.im\right) \]

   8. metadata-eval N/A

      \[\leadsto x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) \]

   9. lift-*.f64 N/A

      \[\leadsto x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right) \]

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 56.1%

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

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

   14. lift-*.f64 N/A

       \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re \]

   15. associate-*l* N/A

       \[\leadsto \left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.re \]

   16. *-commutative N/A

       \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

   17. lift-*.f64 N/A

       \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

   18. lower-*.f64 56.1%

       \[\leadsto \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re \]

8. Applied rewrites 56.1%

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

Developer Target 1: 91.1% 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 2025213 
(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)))