math.cube on complex, real part

Percentage Accurate: 82.3% → 99.8%

Time: 3.5s

Alternatives: 8

Speedup: 0.9×

• 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: 82.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := x.im \cdot \left|x.re\right|\\ t_1 := \left|x.re\right| - x.im\\ t_2 := x.im + \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left|x.re\right| \cdot \left|x.re\right| - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + t\_0\right) \cdot x.im \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_2 \cdot \left|x.re\right|, \left(-2 \cdot t\_0\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(x.im + x.im, -x.im, t\_1 \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* x.im (fabs x.re)))
       (t_1 (- (fabs x.re) x.im))
       (t_2 (+ x.im (fabs x.re))))
  (*
   (copysign 1.0 x.re)
   (if (<=
        (-
         (* (- (* (fabs x.re) (fabs x.re)) (* x.im x.im)) (fabs x.re))
         (* (+ (* (fabs x.re) x.im) t_0) x.im))
        -1000000.0)
     (fma t_1 (* t_2 (fabs x.re)) (* (* -2.0 t_0) x.im))
     (* (fabs x.re) (fma (+ x.im x.im) (- x.im) (* t_1 t_2)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * fabs(x_46_re);
	double t_1 = fabs(x_46_re) - x_46_im;
	double t_2 = x_46_im + fabs(x_46_re);
	double tmp;
	if (((((fabs(x_46_re) * fabs(x_46_re)) - (x_46_im * x_46_im)) * fabs(x_46_re)) - (((fabs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= -1000000.0) {
		tmp = fma(t_1, (t_2 * fabs(x_46_re)), ((-2.0 * t_0) * x_46_im));
	} else {
		tmp = fabs(x_46_re) * fma((x_46_im + x_46_im), -x_46_im, (t_1 * t_2));
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_im * abs(x_46_re))
	t_1 = Float64(abs(x_46_re) - x_46_im)
	t_2 = Float64(x_46_im + abs(x_46_re))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(abs(x_46_re) * abs(x_46_re)) - Float64(x_46_im * x_46_im)) * abs(x_46_re)) - Float64(Float64(Float64(abs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= -1000000.0)
		tmp = fma(t_1, Float64(t_2 * abs(x_46_re)), Float64(Float64(-2.0 * t_0) * x_46_im));
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(x_46_im + x_46_im), Float64(-x_46_im), Float64(t_1 * t_2)));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$re], $MachinePrecision] - x$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(x$46$im + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * x$46$im), $MachinePrecision] + t$95$0), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -1000000.0], N[(t$95$1 * N[(t$95$2 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(x$46$im + x$46$im), $MachinePrecision] * (-x$46$im) + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1e6

   1. Initial program 82.3%

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

   3. Applied rewrites 91.3%

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

   if -1e6 < (-.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 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. add-flip-rev N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

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

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

   3. Applied rewrites 94.0%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := x.im \cdot \left|x.re\right|\\ t_1 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(t\_1 - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + t\_0\right) \cdot x.im \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \left|x.re\right|, \left(t\_0 \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(x.im + x.im, -x.im, \left(\left|x.re\right| - x.im\right) \cdot \left(x.im + \left|x.re\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* x.im (fabs x.re))) (t_1 (* (fabs x.re) (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))
        -4e-8)
     (fma t_1 (fabs x.re) (* (* t_0 x.im) -3.0))
     (*
      (fabs x.re)
      (fma
       (+ x.im x.im)
       (- x.im)
       (* (- (fabs x.re) x.im) (+ x.im (fabs x.re)))))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * fabs(x_46_re);
	double t_1 = fabs(x_46_re) * 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)) <= -4e-8) {
		tmp = fma(t_1, fabs(x_46_re), ((t_0 * x_46_im) * -3.0));
	} else {
		tmp = fabs(x_46_re) * fma((x_46_im + x_46_im), -x_46_im, ((fabs(x_46_re) - x_46_im) * (x_46_im + fabs(x_46_re))));
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(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)) <= -4e-8)
		tmp = fma(t_1, abs(x_46_re), Float64(Float64(t_0 * x_46_im) * -3.0));
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(x_46_im + x_46_im), Float64(-x_46_im), Float64(Float64(abs(x_46_re) - x_46_im) * Float64(x_46_im + abs(x_46_re)))));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(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], -4e-8], N[(t$95$1 * N[Abs[x$46$re], $MachinePrecision] + N[(N[(t$95$0 * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(x$46$im + x$46$im), $MachinePrecision] * (-x$46$im) + N[(N[(N[Abs[x$46$re], $MachinePrecision] - x$46$im), $MachinePrecision] * N[(x$46$im + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -4.0000000000000001e-8

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.2%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-neg.f64 N/A

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

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

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 88.0%

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

   5. Applied rewrites 88.0%

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

   if -4.0000000000000001e-8 < (-.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 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. add-flip-rev N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

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

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

   3. Applied rewrites 94.0%

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

Alternative 3: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := x.im \cdot \left|x.re\right|\\ t_1 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(t\_1 - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + t\_0\right) \cdot x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \left|x.re\right|, \left(t\_0 \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left|x.re\right|\right)}^{3}\\ \end{array} \end{array} \]

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))
        -1e-310)
     (fma t_1 (fabs x.re) (* (* t_0 x.im) -3.0))
     (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)) <= -1e-310) {
		tmp = fma(t_1, fabs(x_46_re), ((t_0 * x_46_im) * -3.0));
	} 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)) <= -1e-310)
		tmp = fma(t_1, abs(x_46_re), Float64(Float64(t_0 * x_46_im) * -3.0));
	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], -1e-310], N[(t$95$1 * N[Abs[x$46$re], $MachinePrecision] + N[(N[(t$95$0 * x$46$im), $MachinePrecision] * -3.0), $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)) < -9.9999999999999694e-311

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.2%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-neg.f64 N/A

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

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

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 88.0%

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

   5. Applied rewrites 88.0%

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

   if -9.9999999999999694e-311 < (-.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 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. add-flip-rev N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

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

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 58.8%

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

   6. Applied rewrites 58.8%

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

Alternative 4: 96.1% accurate, 0.9× speedup

Math

\[\mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 1.25 \cdot 10^{-91}:\\ \;\;\;\;\left(\left(x.im \cdot \left|x.re\right|\right) \cdot -3\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(-3 \cdot x.im, x.im, \left|x.re\right| \cdot \left|x.re\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.re)
 (if (<= (fabs x.re) 1.25e-91)
   (* (* (* x.im (fabs x.re)) -3.0) x.im)
   (*
    (fabs x.re)
    (fma (* -3.0 x.im) x.im (* (fabs x.re) (fabs x.re)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_re) <= 1.25e-91) {
		tmp = ((x_46_im * fabs(x_46_re)) * -3.0) * x_46_im;
	} else {
		tmp = fabs(x_46_re) * fma((-3.0 * x_46_im), x_46_im, (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) <= 1.25e-91)
		tmp = Float64(Float64(Float64(x_46_im * abs(x_46_re)) * -3.0) * x_46_im);
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(-3.0 * x_46_im), x_46_im, Float64(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], 1.25e-91], N[(N[(N[(x$46$im * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$im + N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.25e-91

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 50.2%

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

   4. Applied rewrites 50.2%

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 55.9%

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 55.9%

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

   6. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 55.9%

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

   8. Applied rewrites 55.9%

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

   if 1.25e-91 < x.re

   1. Initial program 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.2%

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

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

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

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqr-neg-rev N/A

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

      14. pow2 N/A

          \[\leadsto x.re \cdot \left(x.re \cdot x.re - \color{blue}{{x.im}^{2}} \cdot 3\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto x.re \cdot \left(x.re \cdot x.re - \color{blue}{{x.im}^{2}} \cdot 3\right) \]

      16. lower-*.f64 87.4%

          \[\leadsto x.re \cdot \left(x.re \cdot x.re - \color{blue}{{x.im}^{2} \cdot 3}\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto x.re \cdot \left(x.re \cdot x.re - \color{blue}{{x.im}^{2}} \cdot 3\right) \]

      18. pow2 N/A

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

      19. lower-*.f64 87.4%

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

   5. Applied rewrites 87.4%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-out N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. remove-double-neg N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 90.7%

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

   7. Applied rewrites 90.7%

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

Alternative 5: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := x.im \cdot \left|x.re\right|\\ t_1 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(t\_1 - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + t\_0\right) \cdot x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_0 \cdot -3\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left|x.re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* x.im (fabs x.re))) (t_1 (* (fabs x.re) (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))
        -1e-310)
     (* (* t_0 -3.0) x.im)
     (* t_1 (fabs x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * fabs(x_46_re);
	double t_1 = fabs(x_46_re) * 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)) <= -1e-310) {
		tmp = (t_0 * -3.0) * x_46_im;
	} else {
		tmp = t_1 * fabs(x_46_re);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * Math.abs(x_46_re);
	double t_1 = Math.abs(x_46_re) * Math.abs(x_46_re);
	double tmp;
	if ((((t_1 - (x_46_im * x_46_im)) * Math.abs(x_46_re)) - (((Math.abs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= -1e-310) {
		tmp = (t_0 * -3.0) * x_46_im;
	} else {
		tmp = t_1 * Math.abs(x_46_re);
	}
	return Math.copySign(1.0, x_46_re) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_im * math.fabs(x_46_re)
	t_1 = math.fabs(x_46_re) * math.fabs(x_46_re)
	tmp = 0
	if (((t_1 - (x_46_im * x_46_im)) * math.fabs(x_46_re)) - (((math.fabs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= -1e-310:
		tmp = (t_0 * -3.0) * x_46_im
	else:
		tmp = t_1 * math.fabs(x_46_re)
	return math.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)) <= -1e-310)
		tmp = Float64(Float64(t_0 * -3.0) * x_46_im);
	else
		tmp = Float64(t_1 * abs(x_46_re));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_im * abs(x_46_re);
	t_1 = abs(x_46_re) * abs(x_46_re);
	tmp = 0.0;
	if ((((t_1 - (x_46_im * x_46_im)) * abs(x_46_re)) - (((abs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= -1e-310)
		tmp = (t_0 * -3.0) * x_46_im;
	else
		tmp = t_1 * abs(x_46_re);
	end
	tmp_2 = (sign(x_46_re) * abs(1.0)) * 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], -1e-310], N[(N[(t$95$0 * -3.0), $MachinePrecision] * x$46$im), $MachinePrecision], N[(t$95$1 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.9999999999999694e-311

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 50.2%

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

   4. Applied rewrites 50.2%

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 55.9%

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 55.9%

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

   6. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 55.9%

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

   8. Applied rewrites 55.9%

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

   if -9.9999999999999694e-311 < (-.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 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.2%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-neg.f64 N/A

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

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

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 88.0%

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 58.8%

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

   8. Applied rewrites 58.8%

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

      1. lift-pow.f64 N/A

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

      2. pow3 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. lift-*.f64 58.7%

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

   10. Applied rewrites 58.7%

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

Alternative 6: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(t\_0 - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + x.im \cdot \left|x.re\right|\right) \cdot x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-3 \cdot \left|x.re\right|\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left|x.re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.re) (fabs x.re))))
  (*
   (copysign 1.0 x.re)
   (if (<=
        (-
         (* (- t_0 (* x.im x.im)) (fabs x.re))
         (* (+ (* (fabs x.re) x.im) (* x.im (fabs x.re))) x.im))
        -1e-310)
     (* (* (* -3.0 (fabs x.re)) x.im) x.im)
     (* t_0 (fabs x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) * fabs(x_46_re);
	double tmp;
	if ((((t_0 - (x_46_im * x_46_im)) * fabs(x_46_re)) - (((fabs(x_46_re) * x_46_im) + (x_46_im * fabs(x_46_re))) * x_46_im)) <= -1e-310) {
		tmp = ((-3.0 * fabs(x_46_re)) * x_46_im) * x_46_im;
	} else {
		tmp = t_0 * fabs(x_46_re);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = Math.abs(x_46_re) * Math.abs(x_46_re);
	double tmp;
	if ((((t_0 - (x_46_im * x_46_im)) * Math.abs(x_46_re)) - (((Math.abs(x_46_re) * x_46_im) + (x_46_im * Math.abs(x_46_re))) * x_46_im)) <= -1e-310) {
		tmp = ((-3.0 * Math.abs(x_46_re)) * x_46_im) * x_46_im;
	} else {
		tmp = t_0 * Math.abs(x_46_re);
	}
	return Math.copySign(1.0, x_46_re) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = math.fabs(x_46_re) * math.fabs(x_46_re)
	tmp = 0
	if (((t_0 - (x_46_im * x_46_im)) * math.fabs(x_46_re)) - (((math.fabs(x_46_re) * x_46_im) + (x_46_im * math.fabs(x_46_re))) * x_46_im)) <= -1e-310:
		tmp = ((-3.0 * math.fabs(x_46_re)) * x_46_im) * x_46_im
	else:
		tmp = t_0 * math.fabs(x_46_re)
	return math.copysign(1.0, x_46_re) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_re) * abs(x_46_re))
	tmp = 0.0
	if (Float64(Float64(Float64(t_0 - Float64(x_46_im * x_46_im)) * abs(x_46_re)) - Float64(Float64(Float64(abs(x_46_re) * x_46_im) + Float64(x_46_im * abs(x_46_re))) * x_46_im)) <= -1e-310)
		tmp = Float64(Float64(Float64(-3.0 * abs(x_46_re)) * x_46_im) * x_46_im);
	else
		tmp = Float64(t_0 * abs(x_46_re));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = abs(x_46_re) * abs(x_46_re);
	tmp = 0.0;
	if ((((t_0 - (x_46_im * x_46_im)) * abs(x_46_re)) - (((abs(x_46_re) * x_46_im) + (x_46_im * abs(x_46_re))) * x_46_im)) <= -1e-310)
		tmp = ((-3.0 * abs(x_46_re)) * x_46_im) * x_46_im;
	else
		tmp = t_0 * abs(x_46_re);
	end
	tmp_2 = (sign(x_46_re) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(t$95$0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * x$46$im), $MachinePrecision] + N[(x$46$im * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -1e-310], N[(N[(N[(-3.0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.9999999999999694e-311

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 50.2%

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

   4. Applied rewrites 50.2%

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 55.9%

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 55.9%

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

   6. Applied rewrites 55.9%

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

   if -9.9999999999999694e-311 < (-.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 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.2%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-neg.f64 N/A

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

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

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 88.0%

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 58.8%

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

   8. Applied rewrites 58.8%

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

      1. lift-pow.f64 N/A

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

      2. pow3 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. lift-*.f64 58.7%

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

   10. Applied rewrites 58.7%

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

Alternative 7: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := x.im \cdot \left|x.re\right|\\ t_1 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(t\_1 - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + t\_0\right) \cdot x.im \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(-3 \cdot x.im\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left|x.re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* x.im (fabs x.re))) (t_1 (* (fabs x.re) (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))
        -1e-310)
     (* (* -3.0 x.im) t_0)
     (* t_1 (fabs x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * fabs(x_46_re);
	double t_1 = fabs(x_46_re) * 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)) <= -1e-310) {
		tmp = (-3.0 * x_46_im) * t_0;
	} else {
		tmp = t_1 * fabs(x_46_re);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * Math.abs(x_46_re);
	double t_1 = Math.abs(x_46_re) * Math.abs(x_46_re);
	double tmp;
	if ((((t_1 - (x_46_im * x_46_im)) * Math.abs(x_46_re)) - (((Math.abs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= -1e-310) {
		tmp = (-3.0 * x_46_im) * t_0;
	} else {
		tmp = t_1 * Math.abs(x_46_re);
	}
	return Math.copySign(1.0, x_46_re) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_im * math.fabs(x_46_re)
	t_1 = math.fabs(x_46_re) * math.fabs(x_46_re)
	tmp = 0
	if (((t_1 - (x_46_im * x_46_im)) * math.fabs(x_46_re)) - (((math.fabs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= -1e-310:
		tmp = (-3.0 * x_46_im) * t_0
	else:
		tmp = t_1 * math.fabs(x_46_re)
	return math.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)) <= -1e-310)
		tmp = Float64(Float64(-3.0 * x_46_im) * t_0);
	else
		tmp = Float64(t_1 * abs(x_46_re));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_im * abs(x_46_re);
	t_1 = abs(x_46_re) * abs(x_46_re);
	tmp = 0.0;
	if ((((t_1 - (x_46_im * x_46_im)) * abs(x_46_re)) - (((abs(x_46_re) * x_46_im) + t_0) * x_46_im)) <= -1e-310)
		tmp = (-3.0 * x_46_im) * t_0;
	else
		tmp = t_1 * abs(x_46_re);
	end
	tmp_2 = (sign(x_46_re) * abs(1.0)) * 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], -1e-310], N[(N[(-3.0 * x$46$im), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$1 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.9999999999999694e-311

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 50.2%

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

   4. Applied rewrites 50.2%

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 55.9%

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 55.9%

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

   6. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 55.9%

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

   8. Applied rewrites 55.9%

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

   if -9.9999999999999694e-311 < (-.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 82.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.2%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-neg.f64 N/A

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

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

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 88.0%

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-pow.f64 58.8%

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

   8. Applied rewrites 58.8%

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

      1. lift-pow.f64 N/A

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

      2. pow3 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. lift-*.f64 58.7%

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

   10. Applied rewrites 58.7%

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

Alternative 8: 58.7% accurate, 3.8× 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 82.3%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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 82.2%

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

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

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-neg.f64 N/A

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

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

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

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

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

   16. lower-fma.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 88.0%

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

5. Applied rewrites 88.0%

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

6. Taylor expanded in x.re around inf

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

   1. lower-pow.f64 58.8%

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

8. Applied rewrites 58.8%

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

   1. lift-pow.f64 N/A

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

   2. pow3 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. lift-*.f64 58.7%

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

10. Applied rewrites 58.7%

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

Developer Target 1: 87.1% 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 2025212 
(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)))