math.cube on complex, imaginary part

Percentage Accurate: 82.4% → 96.3%

Time: 2.7s

Alternatives: 9

Speedup: 1.7×

• 56.2% 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: 82.4% 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: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := x.re - \left|x.im\right|\\ t_1 := \left|x.im\right| + x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1 \cdot \left|x.im\right|, \left(\left(x.re + x.re\right) \cdot \left|x.im\right|\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(t\_0, t\_1, \left(x.re + x.re\right) \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (- x.re (fabs x.im))) (t_1 (+ (fabs x.im) x.re)))
   (*
    (copysign 1.0 x.im)
    (if (<= (fabs x.im) 1e-58)
      (fma t_0 (* t_1 (fabs x.im)) (* (* (+ x.re x.re) (fabs x.im)) x.re))
      (* (fabs x.im) (fma t_0 t_1 (* (+ x.re x.re) x.re)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re - fabs(x_46_im);
	double t_1 = fabs(x_46_im) + x_46_re;
	double tmp;
	if (fabs(x_46_im) <= 1e-58) {
		tmp = fma(t_0, (t_1 * fabs(x_46_im)), (((x_46_re + x_46_re) * fabs(x_46_im)) * x_46_re));
	} else {
		tmp = fabs(x_46_im) * fma(t_0, t_1, ((x_46_re + x_46_re) * x_46_re));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re - abs(x_46_im))
	t_1 = Float64(abs(x_46_im) + x_46_re)
	tmp = 0.0
	if (abs(x_46_im) <= 1e-58)
		tmp = fma(t_0, Float64(t_1 * abs(x_46_im)), Float64(Float64(Float64(x_46_re + x_46_re) * abs(x_46_im)) * x_46_re));
	else
		tmp = Float64(abs(x_46_im) * fma(t_0, t_1, Float64(Float64(x_46_re + x_46_re) * x_46_re)));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 1e-58], N[(t$95$0 * N[(t$95$1 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re + x$46$re), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[(t$95$0 * t$95$1 + N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1e-58

   1. Initial program 82.4%

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

   3. Applied rewrites 91.6%

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

   if 1e-58 < x.im

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 90.5%

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

Alternative 2: 96.2% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 10^{-58}:\\ \;\;\;\;3 \cdot \left(\left(\left|x.im\right| \cdot x.re\right) \cdot x.re\right) - \left(\left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(x.re - \left|x.im\right|, \left|x.im\right| + x.re, \left(x.re + x.re\right) \cdot x.re\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (*
  (copysign 1.0 x.im)
  (if (<= (fabs x.im) 1e-58)
    (-
     (* 3.0 (* (* (fabs x.im) x.re) x.re))
     (* (* (fabs x.im) (fabs x.im)) (fabs x.im)))
    (*
     (fabs x.im)
     (fma (- x.re (fabs x.im)) (+ (fabs x.im) x.re) (* (+ x.re x.re) x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 1e-58) {
		tmp = (3.0 * ((fabs(x_46_im) * x_46_re) * x_46_re)) - ((fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im));
	} else {
		tmp = fabs(x_46_im) * fma((x_46_re - fabs(x_46_im)), (fabs(x_46_im) + x_46_re), ((x_46_re + x_46_re) * x_46_re));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 1e-58)
		tmp = Float64(Float64(3.0 * Float64(Float64(abs(x_46_im) * x_46_re) * x_46_re)) - Float64(Float64(abs(x_46_im) * abs(x_46_im)) * abs(x_46_im)));
	else
		tmp = Float64(abs(x_46_im) * fma(Float64(x_46_re - abs(x_46_im)), Float64(abs(x_46_im) + x_46_re), Float64(Float64(x_46_re + x_46_re) * x_46_re)));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 1e-58], N[(N[(3.0 * N[(N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(x$46$re - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision] + N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1e-58

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 85.9%

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

   if 1e-58 < x.im

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 90.5%

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

Alternative 3: 96.2% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 10^{-58}:\\ \;\;\;\;3 \cdot \left(\left(\left|x.im\right| \cdot x.re\right) \cdot x.re\right) - \left(\left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(-\left|x.im\right|, \left|x.im\right|, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (*
  (copysign 1.0 x.im)
  (if (<= (fabs x.im) 1e-58)
    (-
     (* 3.0 (* (* (fabs x.im) x.re) x.re))
     (* (* (fabs x.im) (fabs x.im)) (fabs x.im)))
    (* (fabs x.im) (fma (- (fabs x.im)) (fabs x.im) (* (* x.re x.re) 3.0))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 1e-58) {
		tmp = (3.0 * ((fabs(x_46_im) * x_46_re) * x_46_re)) - ((fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im));
	} else {
		tmp = fabs(x_46_im) * fma(-fabs(x_46_im), fabs(x_46_im), ((x_46_re * x_46_re) * 3.0));
	}
	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) <= 1e-58)
		tmp = Float64(Float64(3.0 * Float64(Float64(abs(x_46_im) * x_46_re) * x_46_re)) - Float64(Float64(abs(x_46_im) * abs(x_46_im)) * abs(x_46_im)));
	else
		tmp = Float64(abs(x_46_im) * fma(Float64(-abs(x_46_im)), abs(x_46_im), Float64(Float64(x_46_re * x_46_re) * 3.0)));
	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], 1e-58], N[(N[(3.0 * N[(N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision] + N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1e-58

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 85.9%

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

   if 1e-58 < x.im

   1. Initial program 82.4%

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

   3. Applied rewrites 91.6%

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

   5. Applied rewrites 90.4%

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

Alternative 4: 90.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

3. Applied rewrites 91.6%

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

5. Applied rewrites 90.4%

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

Alternative 5: 90.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\left(3 \cdot \left(\left|x.re\right| \cdot \left|x.re\right|\right) - x.im \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(3 \cdot \left|x.re\right|\right) \cdot \left|x.re\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= (fabs x.re) 8e+153)
   (* (- (* 3.0 (* (fabs x.re) (fabs x.re))) (* x.im x.im)) x.im)
   (* x.im (* (* 3.0 (fabs x.re)) (fabs x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_re) <= 8e+153) {
		tmp = ((3.0 * (fabs(x_46_re) * fabs(x_46_re))) - (x_46_im * x_46_im)) * x_46_im;
	} else {
		tmp = x_46_im * ((3.0 * fabs(x_46_re)) * 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) <= 8d+153) then
        tmp = ((3.0d0 * (abs(x_46re) * abs(x_46re))) - (x_46im * x_46im)) * x_46im
    else
        tmp = x_46im * ((3.0d0 * abs(x_46re)) * 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) <= 8e+153) {
		tmp = ((3.0 * (Math.abs(x_46_re) * Math.abs(x_46_re))) - (x_46_im * x_46_im)) * x_46_im;
	} else {
		tmp = x_46_im * ((3.0 * Math.abs(x_46_re)) * Math.abs(x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_re) <= 8e+153:
		tmp = ((3.0 * (math.fabs(x_46_re) * math.fabs(x_46_re))) - (x_46_im * x_46_im)) * x_46_im
	else:
		tmp = x_46_im * ((3.0 * math.fabs(x_46_re)) * math.fabs(x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_re) <= 8e+153)
		tmp = Float64(Float64(Float64(3.0 * Float64(abs(x_46_re) * abs(x_46_re))) - Float64(x_46_im * x_46_im)) * x_46_im);
	else
		tmp = Float64(x_46_im * Float64(Float64(3.0 * abs(x_46_re)) * 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) <= 8e+153)
		tmp = ((3.0 * (abs(x_46_re) * abs(x_46_re))) - (x_46_im * x_46_im)) * x_46_im;
	else
		tmp = x_46_im * ((3.0 * abs(x_46_re)) * 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], 8e+153], N[(N[(N[(3.0 * N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(x$46$im * N[(N[(3.0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 8e153

   1. Initial program 82.4%

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

   3. Applied rewrites 91.6%

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

   5. Applied rewrites 87.3%

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

   if 8e153 < x.re

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 90.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift--.f64 N/A

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

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

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. difference-of-squares-rev N/A

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

      11. lift-*.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. associate-+r- N/A

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

      14. lift-*.f64 N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-neg-rev N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 49.9%

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

   8. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 49.9%

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 49.9%

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

   10. Applied rewrites 49.9%

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

Alternative 6: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ t_1 := \left(x.re \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + \left|x.im\right| \cdot x.re\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-301}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left|x.im\right| \cdot \mathsf{fma}\left(x.re + x.re, x.re, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im)))
        (t_1
         (+
          (* (- (* x.re x.re) (* (fabs x.im) (fabs x.im))) (fabs x.im))
          (* (+ (* x.re (fabs x.im)) (* (fabs x.im) x.re)) x.re))))
   (*
    (copysign 1.0 x.im)
    (if (<= t_1 -2e-301)
      t_0
      (if (<= t_1 INFINITY)
        (* (fabs x.im) (fma (+ x.re x.re) x.re (* x.re x.re)))
        t_0)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im);
	double t_1 = (((x_46_re * x_46_re) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + (fabs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-301) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fabs(x_46_im) * fma((x_46_re + x_46_re), x_46_re, (x_46_re * x_46_re));
	} else {
		tmp = t_0;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(Float64(-abs(x_46_im)) * abs(x_46_im)) * abs(x_46_im))
	t_1 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + Float64(abs(x_46_im) * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_1 <= -2e-301)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(abs(x_46_im) * fma(Float64(x_46_re + x_46_re), x_46_re, Float64(x_46_re * x_46_re)));
	else
		tmp = t_0;
	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[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re * x$46$re), $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[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] * 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[t$95$1, -2e-301], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$re + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 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)) < -2.00000000000000013e-301 or +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 82.4%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 58.2%

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

   4. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 58.1%

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

   6. Applied rewrites 58.1%

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

   if -2.00000000000000013e-301 < (+.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 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 90.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift--.f64 N/A

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

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

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. difference-of-squares-rev N/A

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

      11. lift-*.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. associate-+r- N/A

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

      14. lift-*.f64 N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-neg-rev N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 49.9%

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

   8. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 50.0%

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 50.0%

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

   10. Applied rewrites 50.0%

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

Alternative 7: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ t_1 := \left(x.re \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + \left|x.im\right| \cdot x.re\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-301}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left|x.im\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im)))
        (t_1
         (+
          (* (- (* x.re x.re) (* (fabs x.im) (fabs x.im))) (fabs x.im))
          (* (+ (* x.re (fabs x.im)) (* (fabs x.im) x.re)) x.re))))
   (*
    (copysign 1.0 x.im)
    (if (<= t_1 -2e-301)
      t_0
      (if (<= t_1 INFINITY) (* (* (* x.re x.re) 3.0) (fabs x.im)) t_0)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im);
	double t_1 = (((x_46_re * x_46_re) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + (fabs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-301) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x_46_re * x_46_re) * 3.0) * fabs(x_46_im);
	} else {
		tmp = t_0;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (-Math.abs(x_46_im) * Math.abs(x_46_im)) * Math.abs(x_46_im);
	double t_1 = (((x_46_re * x_46_re) - (Math.abs(x_46_im) * Math.abs(x_46_im))) * Math.abs(x_46_im)) + (((x_46_re * Math.abs(x_46_im)) + (Math.abs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-301) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_re * x_46_re) * 3.0) * Math.abs(x_46_im);
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (-math.fabs(x_46_im) * math.fabs(x_46_im)) * math.fabs(x_46_im)
	t_1 = (((x_46_re * x_46_re) - (math.fabs(x_46_im) * math.fabs(x_46_im))) * math.fabs(x_46_im)) + (((x_46_re * math.fabs(x_46_im)) + (math.fabs(x_46_im) * x_46_re)) * x_46_re)
	tmp = 0
	if t_1 <= -2e-301:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = ((x_46_re * x_46_re) * 3.0) * math.fabs(x_46_im)
	else:
		tmp = t_0
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(Float64(-abs(x_46_im)) * abs(x_46_im)) * abs(x_46_im))
	t_1 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + Float64(abs(x_46_im) * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_1 <= -2e-301)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x_46_re * x_46_re) * 3.0) * abs(x_46_im));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (-abs(x_46_im) * abs(x_46_im)) * abs(x_46_im);
	t_1 = (((x_46_re * x_46_re) - (abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + (((x_46_re * abs(x_46_im)) + (abs(x_46_im) * x_46_re)) * x_46_re);
	tmp = 0.0;
	if (t_1 <= -2e-301)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = ((x_46_re * x_46_re) * 3.0) * abs(x_46_im);
	else
		tmp = t_0;
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re * x$46$re), $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[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] * 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[t$95$1, -2e-301], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 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)) < -2.00000000000000013e-301 or +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 82.4%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 58.2%

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

   4. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 58.1%

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

   6. Applied rewrites 58.1%

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

   if -2.00000000000000013e-301 < (+.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 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 90.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift--.f64 N/A

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

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

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. difference-of-squares-rev N/A

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

      11. lift-*.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. associate-+r- N/A

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

      14. lift-*.f64 N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-neg-rev N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 49.9%

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

   8. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 49.9%

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 49.9%

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

   10. Applied rewrites 49.9%

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

Alternative 8: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ t_1 := \left(x.re \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + \left|x.im\right| \cdot x.re\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-301}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left|x.im\right| \cdot \left(\left(3 \cdot x.re\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im)))
        (t_1
         (+
          (* (- (* x.re x.re) (* (fabs x.im) (fabs x.im))) (fabs x.im))
          (* (+ (* x.re (fabs x.im)) (* (fabs x.im) x.re)) x.re))))
   (*
    (copysign 1.0 x.im)
    (if (<= t_1 -2e-301)
      t_0
      (if (<= t_1 INFINITY) (* (fabs x.im) (* (* 3.0 x.re) x.re)) t_0)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im);
	double t_1 = (((x_46_re * x_46_re) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + (fabs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-301) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fabs(x_46_im) * ((3.0 * x_46_re) * x_46_re);
	} else {
		tmp = t_0;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (-Math.abs(x_46_im) * Math.abs(x_46_im)) * Math.abs(x_46_im);
	double t_1 = (((x_46_re * x_46_re) - (Math.abs(x_46_im) * Math.abs(x_46_im))) * Math.abs(x_46_im)) + (((x_46_re * Math.abs(x_46_im)) + (Math.abs(x_46_im) * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-301) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.abs(x_46_im) * ((3.0 * x_46_re) * x_46_re);
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (-math.fabs(x_46_im) * math.fabs(x_46_im)) * math.fabs(x_46_im)
	t_1 = (((x_46_re * x_46_re) - (math.fabs(x_46_im) * math.fabs(x_46_im))) * math.fabs(x_46_im)) + (((x_46_re * math.fabs(x_46_im)) + (math.fabs(x_46_im) * x_46_re)) * x_46_re)
	tmp = 0
	if t_1 <= -2e-301:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = math.fabs(x_46_im) * ((3.0 * x_46_re) * x_46_re)
	else:
		tmp = t_0
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(Float64(-abs(x_46_im)) * abs(x_46_im)) * abs(x_46_im))
	t_1 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + Float64(abs(x_46_im) * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_1 <= -2e-301)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(abs(x_46_im) * Float64(Float64(3.0 * x_46_re) * x_46_re));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (-abs(x_46_im) * abs(x_46_im)) * abs(x_46_im);
	t_1 = (((x_46_re * x_46_re) - (abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + (((x_46_re * abs(x_46_im)) + (abs(x_46_im) * x_46_re)) * x_46_re);
	tmp = 0.0;
	if (t_1 <= -2e-301)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = abs(x_46_im) * ((3.0 * x_46_re) * x_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re * x$46$re), $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[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] * 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[t$95$1, -2e-301], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 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)) < -2.00000000000000013e-301 or +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 82.4%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 58.2%

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

   4. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 58.1%

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

   6. Applied rewrites 58.1%

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

   if -2.00000000000000013e-301 < (+.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 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 90.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift--.f64 N/A

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

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

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. difference-of-squares-rev N/A

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

      11. lift-*.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. associate-+r- N/A

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

      14. lift-*.f64 N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-neg-rev N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 49.9%

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

   8. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 49.9%

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 49.9%

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

   10. Applied rewrites 49.9%

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

Alternative 9: 58.1% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 58.2%

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

4. Applied rewrites 58.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow3 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 58.1%

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

6. Applied rewrites 58.1%

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

Developer Target 1: 91.6% 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 2025183 
(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)))