math.cube on complex, real part

Percentage Accurate: 83.1% → 99.8%

Time: 3.3s

Alternatives: 6

Speedup: 0.9×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Initial Program: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| - x.im\\ t_1 := x.im + \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1 \cdot \left|x.re\right|, \left(-2 \cdot \left(x.im \cdot \left|x.re\right|\right)\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot x.im, x.im, t\_1 \cdot t\_0\right) \cdot \left|x.re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (- (fabs x.re) x.im)) (t_1 (+ x.im (fabs x.re))))
   (*
    (copysign 1.0 x.re)
    (if (<= (fabs x.re) 5e+22)
      (fma t_0 (* t_1 (fabs x.re)) (* (* -2.0 (* x.im (fabs x.re))) x.im))
      (* (fma (* -2.0 x.im) x.im (* t_1 t_0)) (fabs x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) - x_46_im;
	double t_1 = x_46_im + fabs(x_46_re);
	double tmp;
	if (fabs(x_46_re) <= 5e+22) {
		tmp = fma(t_0, (t_1 * fabs(x_46_re)), ((-2.0 * (x_46_im * fabs(x_46_re))) * x_46_im));
	} else {
		tmp = fma((-2.0 * x_46_im), x_46_im, (t_1 * t_0)) * fabs(x_46_re);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_re) - x_46_im)
	t_1 = Float64(x_46_im + abs(x_46_re))
	tmp = 0.0
	if (abs(x_46_re) <= 5e+22)
		tmp = fma(t_0, Float64(t_1 * abs(x_46_re)), Float64(Float64(-2.0 * Float64(x_46_im * abs(x_46_re))) * x_46_im));
	else
		tmp = Float64(fma(Float64(-2.0 * x_46_im), x_46_im, Float64(t_1 * t_0)) * abs(x_46_re));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] - x$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 5e+22], N[(t$95$0 * N[(t$95$1 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(x$46$im * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * x$46$im), $MachinePrecision] * x$46$im + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.9999999999999996e22

   1. Initial program 83.1%

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

   3. Applied rewrites 91.7%

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

   if 4.9999999999999996e22 < x.re

   1. Initial program 83.1%

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

   3. Applied rewrites 91.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

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

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

      12. count-2 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-in N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. lift--.f64 N/A

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

   5. Applied rewrites 94.0%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := x.im \cdot \left|x.re\right|\\ t_1 := \left|x.re\right| - x.im\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left|x.re\right| \cdot \left|x.re\right| - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + t\_0\right) \cdot x.im \leq -2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_0, \left(-2 \cdot t\_0\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot x.im, x.im, \left(x.im + \left|x.re\right|\right) \cdot t\_1\right) \cdot \left|x.re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (fabs x.re))) (t_1 (- (fabs x.re) x.im)))
   (*
    (copysign 1.0 x.re)
    (if (<=
         (-
          (* (- (* (fabs x.re) (fabs x.re)) (* x.im x.im)) (fabs x.re))
          (* (+ (* (fabs x.re) x.im) t_0) x.im))
         -2e+90)
      (fma t_1 t_0 (* (* -2.0 t_0) x.im))
      (* (fma (* -2.0 x.im) x.im (* (+ x.im (fabs x.re)) t_1)) (fabs x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * fabs(x_46_re);
	double t_1 = fabs(x_46_re) - x_46_im;
	double tmp;
	if (((((fabs(x_46_re) * fabs(x_46_re)) - (x_46_im * x_46_im)) * fabs(x_46_re)) - (((fabs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= -2e+90) {
		tmp = fma(t_1, t_0, ((-2.0 * t_0) * x_46_im));
	} else {
		tmp = fma((-2.0 * x_46_im), x_46_im, ((x_46_im + fabs(x_46_re)) * t_1)) * fabs(x_46_re);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_im * abs(x_46_re))
	t_1 = Float64(abs(x_46_re) - x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(abs(x_46_re) * abs(x_46_re)) - Float64(x_46_im * x_46_im)) * abs(x_46_re)) - Float64(Float64(Float64(abs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= -2e+90)
		tmp = fma(t_1, t_0, Float64(Float64(-2.0 * t_0) * x_46_im));
	else
		tmp = Float64(fma(Float64(-2.0 * x_46_im), x_46_im, Float64(Float64(x_46_im + abs(x_46_re)) * t_1)) * abs(x_46_re));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$re], $MachinePrecision] - x$46$im), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * x$46$im), $MachinePrecision] + t$95$0), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e+90], N[(t$95$1 * t$95$0 + N[(N[(-2.0 * t$95$0), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * x$46$im), $MachinePrecision] * x$46$im + N[(N[(x$46$im + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1.99999999999999993e90

   1. Initial program 83.1%

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

   3. Applied rewrites 91.7%

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

   4. Taylor expanded in x.re around 0

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

   6. Applied rewrites 60.0%

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

   if -1.99999999999999993e90 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 83.1%

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

   3. Applied rewrites 91.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

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

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

      12. count-2 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-in N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. lift--.f64 N/A

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

   5. Applied rewrites 94.0%

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

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(t\_0 - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + x.im \cdot \left|x.re\right|\right) \cdot x.im \leq -2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left|x.re\right|, 3 \cdot \left(\left(\left(-\left|x.re\right|\right) \cdot x.im\right) \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot x.im, x.im, \left(x.im + \left|x.re\right|\right) \cdot \left(\left|x.re\right| - x.im\right)\right) \cdot \left|x.re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (fabs x.re) (fabs x.re))))
   (*
    (copysign 1.0 x.re)
    (if (<=
         (-
          (* (- t_0 (* x.im x.im)) (fabs x.re))
          (* (+ (* (fabs x.re) x.im) (* x.im (fabs x.re))) x.im))
         -2e+90)
      (fma t_0 (fabs x.re) (* 3.0 (* (* (- (fabs x.re)) x.im) x.im)))
      (*
       (fma (* -2.0 x.im) x.im (* (+ x.im (fabs x.re)) (- (fabs x.re) x.im)))
       (fabs x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) * fabs(x_46_re);
	double tmp;
	if ((((t_0 - (x_46_im * x_46_im)) * fabs(x_46_re)) - (((fabs(x_46_re) * x_46_im) + (x_46_im * fabs(x_46_re))) * x_46_im)) <= -2e+90) {
		tmp = fma(t_0, fabs(x_46_re), (3.0 * ((-fabs(x_46_re) * x_46_im) * x_46_im)));
	} else {
		tmp = fma((-2.0 * x_46_im), x_46_im, ((x_46_im + fabs(x_46_re)) * (fabs(x_46_re) - x_46_im))) * fabs(x_46_re);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_re) * abs(x_46_re))
	tmp = 0.0
	if (Float64(Float64(Float64(t_0 - Float64(x_46_im * x_46_im)) * abs(x_46_re)) - Float64(Float64(Float64(abs(x_46_re) * x_46_im) + Float64(x_46_im * abs(x_46_re))) * x_46_im)) <= -2e+90)
		tmp = fma(t_0, abs(x_46_re), Float64(3.0 * Float64(Float64(Float64(-abs(x_46_re)) * x_46_im) * x_46_im)));
	else
		tmp = Float64(fma(Float64(-2.0 * x_46_im), x_46_im, Float64(Float64(x_46_im + abs(x_46_re)) * Float64(abs(x_46_re) - x_46_im))) * abs(x_46_re));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(t$95$0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * x$46$im), $MachinePrecision] + N[(x$46$im * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e+90], N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision] + N[(3.0 * N[(N[((-N[Abs[x$46$re], $MachinePrecision]) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * x$46$im), $MachinePrecision] * x$46$im + N[(N[(x$46$im + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1.99999999999999993e90

   1. Initial program 83.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. associate-+l+ N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 83.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot x.re, x.re, 3 \cdot \left(\left(\color{blue}{\left(-1 \cdot x.re\right)} \cdot x.im\right) \cdot x.im\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot x.re, x.re, 3 \cdot \left(\left(\color{blue}{\left(-1 \cdot x.re\right)} \cdot x.im\right) \cdot x.im\right)\right) \]

      15. lower-*.f64 88.8

          \[\leadsto \mathsf{fma}\left(x.re \cdot x.re, x.re, 3 \cdot \left(\color{blue}{\left(\left(-1 \cdot x.re\right) \cdot x.im\right)} \cdot x.im\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot x.re, x.re, 3 \cdot \left(\left(\color{blue}{\left(-1 \cdot x.re\right)} \cdot x.im\right) \cdot x.im\right)\right) \]

      17. mul-1-neg N/A

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

      18. lower-neg.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if -1.99999999999999993e90 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 83.1%

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

   3. Applied rewrites 91.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

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

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

      12. count-2 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-in N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. lift--.f64 N/A

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

   5. Applied rewrites 94.0%

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

Alternative 4: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 6 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left|x.re\right|, \left(x.im \cdot \left|x.re\right|\right) \cdot \left(-3 \cdot x.im\right)\right)\\ \mathbf{elif}\;\left|x.re\right| \leq 10^{+224}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(\left|x.re\right|, \left|x.re\right|, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0 \cdot t\_0} \cdot \left|x.re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (fabs x.re) (fabs x.re))))
   (*
    (copysign 1.0 x.re)
    (if (<= (fabs x.re) 6e+22)
      (fma t_0 (fabs x.re) (* (* x.im (fabs x.re)) (* -3.0 x.im)))
      (if (<= (fabs x.re) 1e+224)
        (* (fabs x.re) (fma (fabs x.re) (fabs x.re) (* -3.0 (* x.im x.im))))
        (* (sqrt (* t_0 t_0)) (fabs x.re)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) * fabs(x_46_re);
	double tmp;
	if (fabs(x_46_re) <= 6e+22) {
		tmp = fma(t_0, fabs(x_46_re), ((x_46_im * fabs(x_46_re)) * (-3.0 * x_46_im)));
	} else if (fabs(x_46_re) <= 1e+224) {
		tmp = fabs(x_46_re) * fma(fabs(x_46_re), fabs(x_46_re), (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = sqrt((t_0 * t_0)) * fabs(x_46_re);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_re) * abs(x_46_re))
	tmp = 0.0
	if (abs(x_46_re) <= 6e+22)
		tmp = fma(t_0, abs(x_46_re), Float64(Float64(x_46_im * abs(x_46_re)) * Float64(-3.0 * x_46_im)));
	elseif (abs(x_46_re) <= 1e+224)
		tmp = Float64(abs(x_46_re) * fma(abs(x_46_re), abs(x_46_re), Float64(-3.0 * Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(sqrt(Float64(t_0 * t_0)) * abs(x_46_re));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 6e+22], N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision] + N[(N[(x$46$im * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[(-3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 1e+224], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.re < 6e22

   1. Initial program 83.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. associate-+l+ N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 83.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 88.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 88.8

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

   5. Applied rewrites 88.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 88.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 88.8

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

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

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 88.8

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

   7. Applied rewrites 88.8%

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

   if 6e22 < x.re < 9.9999999999999997e223

   1. Initial program 83.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. associate-+l+ N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 83.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 88.3

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

   5. Applied rewrites 88.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

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

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

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

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

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

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

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

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

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

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 91.2

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

   7. Applied rewrites 91.2%

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

   if 9.9999999999999997e223 < x.re

   1. Initial program 83.1%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 59.3

         \[\leadsto {x.re}^{\color{blue}{3}} \]

   4. Applied rewrites 59.3%

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

      1. lift-pow.f64 N/A

         \[\leadsto {x.re}^{\color{blue}{3}} \]

      2. unpow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 59.3

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

   6. Applied rewrites 59.3%

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

      1. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{x.re \cdot x.re} \cdot \sqrt{x.re \cdot x.re}\right) \cdot x.re \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re \]

      4. lower-unsound-*.f32 N/A

         \[\leadsto \sqrt{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re \]

      6. lower-unsound-*.f64 55.4

         \[\leadsto \sqrt{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re \]

   8. Applied rewrites 55.4%

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

Alternative 5: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 10^{+224}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(\left|x.re\right|, \left|x.re\right|, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0 \cdot t\_0} \cdot \left|x.re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (fabs x.re) (fabs x.re))))
   (*
    (copysign 1.0 x.re)
    (if (<= (fabs x.re) 1e+224)
      (* (fabs x.re) (fma (fabs x.re) (fabs x.re) (* -3.0 (* x.im x.im))))
      (* (sqrt (* t_0 t_0)) (fabs x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) * fabs(x_46_re);
	double tmp;
	if (fabs(x_46_re) <= 1e+224) {
		tmp = fabs(x_46_re) * fma(fabs(x_46_re), fabs(x_46_re), (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = sqrt((t_0 * t_0)) * fabs(x_46_re);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_re) * abs(x_46_re))
	tmp = 0.0
	if (abs(x_46_re) <= 1e+224)
		tmp = Float64(abs(x_46_re) * fma(abs(x_46_re), abs(x_46_re), Float64(-3.0 * Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(sqrt(Float64(t_0 * t_0)) * abs(x_46_re));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 1e+224], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 9.9999999999999997e223

   1. Initial program 83.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. associate-+l+ N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 83.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 88.3

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

   5. Applied rewrites 88.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

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

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

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

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

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

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

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

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

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

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 91.2

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

   7. Applied rewrites 91.2%

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

   if 9.9999999999999997e223 < x.re

   1. Initial program 83.1%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 59.3

         \[\leadsto {x.re}^{\color{blue}{3}} \]

   4. Applied rewrites 59.3%

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

      1. lift-pow.f64 N/A

         \[\leadsto {x.re}^{\color{blue}{3}} \]

      2. unpow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 59.3

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

   6. Applied rewrites 59.3%

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

      1. rem-square-sqrt N/A

         \[\leadsto \left(\sqrt{x.re \cdot x.re} \cdot \sqrt{x.re \cdot x.re}\right) \cdot x.re \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re \]

      4. lower-unsound-*.f32 N/A

         \[\leadsto \sqrt{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re \]

      6. lower-unsound-*.f64 55.4

         \[\leadsto \sqrt{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)} \cdot x.re \]

   8. Applied rewrites 55.4%

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

Alternative 6: 59.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(x_46_re * 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_re;
end

Wolfram

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

Derivation

1. Initial program 83.1%

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

2. Taylor expanded in x.re around inf

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

   1. lower-pow.f64 59.3

      \[\leadsto {x.re}^{\color{blue}{3}} \]

4. Applied rewrites 59.3%

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

   1. lift-pow.f64 N/A

      \[\leadsto {x.re}^{\color{blue}{3}} \]

   2. unpow3 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-*.f64 59.3

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

6. Applied rewrites 59.3%

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

Developer Target 1: 87.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * 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_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025170 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform c (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

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