math.cube on complex, real part

Percentage Accurate: 82.5% → 99.9%

Time: 3.6s

Alternatives: 7

Speedup: 1.8×

• 55.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 82.5% 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: 99.9% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 4 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \mathsf{fma}\left(-3 \cdot x.im, \left|x.re\right|, 0\right), {\left(\left|x.re\right|\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot x.im, x.im, \left(x.im + \left|x.re\right|\right) \cdot \left(\left|x.re\right| - x.im\right)\right) \cdot \left|x.re\right|\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (*
  (copysign 1.0 x.re)
  (if (<= (fabs x.re) 4e+90)
    (fma x.im (fma (* -3.0 x.im) (fabs x.re) 0.0) (pow (fabs x.re) 3.0))
    (*
     (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 tmp;
	if (fabs(x_46_re) <= 4e+90) {
		tmp = fma(x_46_im, fma((-3.0 * x_46_im), fabs(x_46_re), 0.0), pow(fabs(x_46_re), 3.0));
	} 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)
	tmp = 0.0
	if (abs(x_46_re) <= 4e+90)
		tmp = fma(x_46_im, fma(Float64(-3.0 * x_46_im), abs(x_46_re), 0.0), (abs(x_46_re) ^ 3.0));
	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_] := 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], 4e+90], N[(x$46$im * N[(N[(-3.0 * x$46$im), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + 0.0), $MachinePrecision] + N[Power[N[Abs[x$46$re], $MachinePrecision], 3.0], $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 x.re < 3.99999999999999987e90

   1. Initial program 82.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. distribute-lft-neg-in N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 91.0%

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

   4. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 88.1%

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

   6. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, \mathsf{fma}\left(-3, x.im \cdot x.re, x.re \cdot \left(x.re + -1 \cdot x.re\right)\right), {x.re}^{3}\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 88.1%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(-3 \cdot x.im, x.re, \left(0 \cdot x.re\right) \cdot x.re\right), {x.re}^{3}\right) \]

      12. mul0-lft N/A

          \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(-3 \cdot x.im, x.re, 0 \cdot x.re\right), {x.re}^{3}\right) \]

      13. mul0-lft 88.1%

          \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(-3 \cdot x.im, x.re, 0\right), {x.re}^{3}\right) \]

   8. Applied rewrites 88.1%

      \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(-3 \cdot x.im, \color{blue}{x.re}, 0\right), {x.re}^{3}\right) \]

   if 3.99999999999999987e90 < x.re

   1. Initial program 82.5%

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

   3. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, \left(-2 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\right)} \]
   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 94.1%

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.im \cdot 3\right) \cdot \left|x.re\right|, -x.im, \left(\left|x.re\right| \cdot \left|x.re\right|\right) \cdot \left|x.re\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot x.im, x.im, \left(x.im + \left|x.re\right|\right) \cdot \left(\left|x.re\right| - x.im\right)\right) \cdot \left|x.re\right|\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (*
  (copysign 1.0 x.re)
  (if (<= (fabs x.re) 5e-60)
    (fma
     (* (* x.im 3.0) (fabs x.re))
     (- x.im)
     (* (* (fabs x.re) (fabs x.re)) (fabs x.re)))
    (*
     (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 tmp;
	if (fabs(x_46_re) <= 5e-60) {
		tmp = fma(((x_46_im * 3.0) * fabs(x_46_re)), -x_46_im, ((fabs(x_46_re) * fabs(x_46_re)) * fabs(x_46_re)));
	} 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)
	tmp = 0.0
	if (abs(x_46_re) <= 5e-60)
		tmp = fma(Float64(Float64(x_46_im * 3.0) * abs(x_46_re)), Float64(-x_46_im), Float64(Float64(abs(x_46_re) * abs(x_46_re)) * abs(x_46_re)));
	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_] := 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-60], N[(N[(N[(x$46$im * 3.0), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * (-x$46$im) + N[(N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $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 x.re < 5.0000000000000001e-60

   1. Initial program 82.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, x.re, 3 \cdot \left(\left(\left(-x.im\right) \cdot x.im\right) \cdot x.re\right)\right)} \]
   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 88.1%

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

      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 88.1%

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

   5. Applied rewrites 88.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

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

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

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

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

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

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

      12. lift-neg.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 88.0%

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

   7. Applied rewrites 88.0%

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

   if 5.0000000000000001e-60 < x.re

   1. Initial program 82.5%

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

   3. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, \left(-2 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\right)} \]
   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 94.1%

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

Alternative 3: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.im \cdot 3\right) \cdot \left|x.re\right|, -x.im, t\_0 \cdot \left|x.re\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(-3 \cdot x.im, x.im, t\_0\right)\\ \end{array} \end{array} \]

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-41)
      (fma (* (* x.im 3.0) (fabs x.re)) (- x.im) (* t_0 (fabs x.re)))
      (* (fabs x.re) (fma (* -3.0 x.im) x.im t_0))))))

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-41) {
		tmp = fma(((x_46_im * 3.0) * fabs(x_46_re)), -x_46_im, (t_0 * fabs(x_46_re)));
	} else {
		tmp = fabs(x_46_re) * fma((-3.0 * x_46_im), x_46_im, t_0);
	}
	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-41)
		tmp = fma(Float64(Float64(x_46_im * 3.0) * abs(x_46_re)), Float64(-x_46_im), Float64(t_0 * abs(x_46_re)));
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(-3.0 * x_46_im), x_46_im, t_0));
	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-41], N[(N[(N[(x$46$im * 3.0), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * (-x$46$im) + N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$im + t$95$0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.00000000000000001e-41

   1. Initial program 82.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, x.re, 3 \cdot \left(\left(\left(-x.im\right) \cdot x.im\right) \cdot x.re\right)\right)} \]
   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 88.1%

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

      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 88.1%

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

   5. Applied rewrites 88.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

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

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

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

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

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

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

      12. lift-neg.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 88.0%

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

   7. Applied rewrites 88.0%

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

   if 1.00000000000000001e-41 < x.re

   1. Initial program 82.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.5%

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

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

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

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

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

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

      11. sub-flip-reverse N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 87.7%

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

   5. Applied rewrites 87.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqr-neg-rev N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

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

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

      22. sqr-neg-rev N/A

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

      23. sqr-neg-rev N/A

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

      24. lift-*.f64 N/A

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

   7. Applied rewrites 90.8%

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

Alternative 4: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := x.im \cdot \left|x.re\right|\\ t_1 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(t\_1 - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + t\_0\right) \cdot x.im \leq 5 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \left|x.re\right|, t\_0 \cdot \left(-3 \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left|x.re\right|\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (fabs x.re))) (t_1 (* (fabs x.re) (fabs x.re))))
   (*
    (copysign 1.0 x.re)
    (if (<=
         (-
          (* (- t_1 (* x.im x.im)) (fabs x.re))
          (* (+ (* (fabs x.re) x.im) t_0) x.im))
         5e-262)
      (fma t_1 (fabs x.re) (* t_0 (* -3.0 x.im)))
      (pow (fabs x.re) 3.0)))))

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) * fabs(x_46_re);
	double tmp;
	if ((((t_1 - (x_46_im * x_46_im)) * fabs(x_46_re)) - (((fabs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= 5e-262) {
		tmp = fma(t_1, fabs(x_46_re), (t_0 * (-3.0 * x_46_im)));
	} else {
		tmp = pow(fabs(x_46_re), 3.0);
	}
	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) * abs(x_46_re))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 - 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)) <= 5e-262)
		tmp = fma(t_1, abs(x_46_re), Float64(t_0 * Float64(-3.0 * x_46_im)));
	else
		tmp = abs(x_46_re) ^ 3.0;
	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] * 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$1 - 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], 5e-262], N[(t$95$1 * N[Abs[x$46$re], $MachinePrecision] + N[(t$95$0 * N[(-3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Abs[x$46$re], $MachinePrecision], 3.0], $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)) < 4.99999999999999992e-262

   1. Initial program 82.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, x.re, 3 \cdot \left(\left(\left(-x.im\right) \cdot x.im\right) \cdot x.re\right)\right)} \]
   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 88.1%

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

      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 88.1%

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

   5. Applied rewrites 88.1%

      \[\leadsto \mathsf{fma}\left(x.re \cdot x.re, x.re, \color{blue}{\left(3 \cdot \left(-x.im\right)\right) \cdot \left(x.re \cdot x.im\right)}\right) \]
   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 88.1%

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

      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 88.1%

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

      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 88.1%

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

   7. Applied rewrites 88.1%

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

   if 4.99999999999999992e-262 < (-.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 82.5%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 58.9%

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

   4. Applied rewrites 58.9%

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

Alternative 5: 92.5% accurate, 0.9× speedup

Math

\[\mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 2 \cdot 10^{+221}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(\left|x.re\right|, \left|x.re\right|, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|x.re\right| \cdot \left|x.re\right|\right) \cdot \left|x.re\right|\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (*
  (copysign 1.0 x.re)
  (if (<= (fabs x.re) 2e+221)
    (* (fabs x.re) (fma (fabs x.re) (fabs x.re) (* -3.0 (* x.im x.im))))
    (* (* (fabs x.re) (fabs x.re)) (fabs x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_re) <= 2e+221) {
		tmp = fabs(x_46_re) * fma(fabs(x_46_re), fabs(x_46_re), (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = (fabs(x_46_re) * fabs(x_46_re)) * fabs(x_46_re);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_re) <= 2e+221)
		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(Float64(abs(x_46_re) * abs(x_46_re)) * abs(x_46_re));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := 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], 2e+221], 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[(N[Abs[x$46$re], $MachinePrecision] * 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 < 2.0000000000000001e221

   1. Initial program 82.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.5%

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

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

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

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

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

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

      11. sub-flip-reverse N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 87.7%

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

   5. Applied rewrites 87.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. add-flip N/A

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

      10. sub-flip N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-neg.f64 N/A

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

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

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

      22. lift-*.f64 N/A

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

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

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

   7. Applied rewrites 91.0%

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

   if 2.0000000000000001e221 < x.re

   1. Initial program 82.5%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 58.9%

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

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   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. lower-*.f64 N/A

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

      4. lower-*.f64 58.8%

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

   6. Applied rewrites 58.8%

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

Alternative 6: 90.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.5%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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 82.5%

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

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-neg.f64 N/A

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

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

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

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

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

   11. sub-flip-reverse N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 87.7%

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

5. Applied rewrites 87.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

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

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

   5. lift-*.f64 N/A

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

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

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

   7. lift-neg.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. sqr-neg-rev N/A

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

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

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

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

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

   17. associate-*l* N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

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

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

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

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

   22. sqr-neg-rev N/A

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

   23. sqr-neg-rev N/A

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

   24. lift-*.f64 N/A

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

7. Applied rewrites 90.8%

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

Alternative 7: 58.8% 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 82.5%

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

2. Taylor expanded in x.re around inf

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

   1. lower-pow.f64 58.9%

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

4. Applied rewrites 58.9%

   \[\leadsto \color{blue}{{x.re}^{3}} \]
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. lower-*.f64 N/A

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

   4. lower-*.f64 58.8%

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

6. Applied rewrites 58.8%

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

Developer Target 1: 87.4% 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 2025183 
(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)))