math.cube on complex, real part

Percentage Accurate: 82.4% → 99.8%

Time: 3.7s

Alternatives: 9

Speedup: 1.8×

• 55.3% 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: 82.4% 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

\[\begin{array}{l} t_0 := -1 \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \mathsf{fma}\left(x.im, \mathsf{fma}\left(-2, \left|x.re\right|, t\_0\right), \left|x.re\right| \cdot \left(\left|x.re\right| + t\_0\right)\right), {\left(\left|x.re\right|\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(x.im + x.im, -x.im, \left(\left|x.re\right| - x.im\right) \cdot \left(x.im + \left|x.re\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* -1.0 (fabs x.re))))
  (*
   (copysign 1.0 x.re)
   (if (<= (fabs x.re) 1e+99)
     (fma
      x.im
      (fma
       x.im
       (fma -2.0 (fabs x.re) t_0)
       (* (fabs x.re) (+ (fabs x.re) t_0)))
      (pow (fabs x.re) 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 = -1.0 * fabs(x_46_re);
	double tmp;
	if (fabs(x_46_re) <= 1e+99) {
		tmp = fma(x_46_im, fma(x_46_im, fma(-2.0, fabs(x_46_re), t_0), (fabs(x_46_re) * (fabs(x_46_re) + t_0))), pow(fabs(x_46_re), 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(-1.0 * abs(x_46_re))
	tmp = 0.0
	if (abs(x_46_re) <= 1e+99)
		tmp = fma(x_46_im, fma(x_46_im, fma(-2.0, abs(x_46_re), t_0), Float64(abs(x_46_re) * Float64(abs(x_46_re) + t_0))), (abs(x_46_re) ^ 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[(-1.0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 1e+99], N[(x$46$im * N[(x$46$im * N[(-2.0 * N[Abs[x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision] + N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Abs[x$46$re], $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 x.re < 9.9999999999999997e98

   1. Initial program 82.4%

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

   3. Applied rewrites 91.0%

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

   4. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{x.im \cdot \left(-2 \cdot x.re + -1 \cdot x.re\right) + x.re \cdot \left(x.re + -1 \cdot x.re\right)}, {x.re}^{3}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(x.im, \color{blue}{-2 \cdot x.re + -1 \cdot x.re}, x.re \cdot \left(x.re + -1 \cdot x.re\right)\right), {x.re}^{3}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(x.im, \mathsf{fma}\left(-2, \color{blue}{x.re}, -1 \cdot x.re\right), x.re \cdot \left(x.re + -1 \cdot x.re\right)\right), {x.re}^{3}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(x.im, \mathsf{fma}\left(-2, x.re, -1 \cdot x.re\right), x.re \cdot \left(x.re + -1 \cdot x.re\right)\right), {x.re}^{3}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(x.im, \mathsf{fma}\left(-2, x.re, \color{blue}{-1 \cdot x.re}\right), x.re \cdot \left(x.re + -1 \cdot x.re\right)\right), {x.re}^{3}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(x.im, \mathsf{fma}\left(-2, x.re, -1 \cdot \color{blue}{x.re}\right), x.re \cdot \left(x.re + -1 \cdot x.re\right)\right), {x.re}^{3}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(x.im, \mathsf{fma}\left(-2, x.re, -1 \cdot x.re\right), x.re \cdot \left(x.re + -1 \cdot x.re\right)\right), {x.re}^{3}\right) \]

      8. lower-pow.f64 87.9%

         \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(x.im, \mathsf{fma}\left(-2, x.re, -1 \cdot x.re\right), \color{blue}{x.re \cdot \left(x.re + -1 \cdot x.re\right)}\right), {x.re}^{3}\right) \]

   6. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, \mathsf{fma}\left(x.im, \mathsf{fma}\left(-2, x.re, -1 \cdot x.re\right), x.re \cdot \left(x.re + -1 \cdot x.re\right)\right), {x.re}^{3}\right)} \]

   if 9.9999999999999997e98 < x.re

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| + \left|x.re\right|\\ t_1 := \left|x.re\right| - \left|x.im\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\left|x.im\right| \cdot \left|x.re\right|, -2 \cdot \left|x.im\right|, \left(t\_0 \cdot \left|x.re\right|\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(\left|x.im\right| + \left|x.im\right|, -\left|x.im\right|, t\_1 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (+ (fabs x.im) (fabs x.re)))
       (t_1 (- (fabs x.re) (fabs x.im))))
  (*
   (copysign 1.0 x.re)
   (if (<= (fabs x.re) 5e+15)
     (fma
      (* (fabs x.im) (fabs x.re))
      (* -2.0 (fabs x.im))
      (* (* t_0 (fabs x.re)) t_1))
     (*
      (fabs x.re)
      (fma
       (+ (fabs x.im) (fabs x.im))
       (- (fabs x.im))
       (* t_1 t_0)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) + fabs(x_46_re);
	double t_1 = fabs(x_46_re) - fabs(x_46_im);
	double tmp;
	if (fabs(x_46_re) <= 5e+15) {
		tmp = fma((fabs(x_46_im) * fabs(x_46_re)), (-2.0 * fabs(x_46_im)), ((t_0 * fabs(x_46_re)) * t_1));
	} else {
		tmp = fabs(x_46_re) * fma((fabs(x_46_im) + fabs(x_46_im)), -fabs(x_46_im), (t_1 * t_0));
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) + abs(x_46_re))
	t_1 = Float64(abs(x_46_re) - abs(x_46_im))
	tmp = 0.0
	if (abs(x_46_re) <= 5e+15)
		tmp = fma(Float64(abs(x_46_im) * abs(x_46_re)), Float64(-2.0 * abs(x_46_im)), Float64(Float64(t_0 * abs(x_46_re)) * t_1));
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(abs(x_46_im) + abs(x_46_im)), Float64(-abs(x_46_im)), Float64(t_1 * t_0)));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$re], $MachinePrecision] - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 5e+15], N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(N[Abs[x$46$im], $MachinePrecision] + N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * (-N[Abs[x$46$im], $MachinePrecision]) + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < 5e15

   1. Initial program 82.4%

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

   3. Applied rewrites 93.2%

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

   if 5e15 < x.re

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(t\_0 - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + x.im \cdot \left|x.re\right|\right) \cdot x.im \leq -5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left|x.re\right|, \left(-x.im\right) \cdot \left(\left(3 \cdot \left|x.re\right|\right) \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(x.im + x.im, -x.im, \left(\left|x.re\right| - x.im\right) \cdot \left(x.im + \left|x.re\right|\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (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))
        -5e+68)
     (fma t_0 (fabs x.re) (* (- x.im) (* (* 3.0 (fabs x.re)) x.im)))
     (*
      (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 = 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)) <= -5e+68) {
		tmp = fma(t_0, fabs(x_46_re), (-x_46_im * ((3.0 * fabs(x_46_re)) * x_46_im)));
	} 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(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)) <= -5e+68)
		tmp = fma(t_0, abs(x_46_re), Float64(Float64(-x_46_im) * Float64(Float64(3.0 * abs(x_46_re)) * x_46_im)));
	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[(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], -5e+68], N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision] + N[((-x$46$im) * N[(N[(3.0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $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)) < -5.0000000000000004e68

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.2%

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

   5. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. distribute-rgt-out N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-fma.f64 N/A

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

      24. lift-*.f64 N/A

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

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

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

      26. sub-negate-rev N/A

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

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

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

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

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

      29. lift-neg.f64 N/A

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

   7. Applied rewrites 88.1%

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

   if -5.0000000000000004e68 < (-.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 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 4: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.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)) < +inf.0

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.2%

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

   5. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. mul-1-neg N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. distribute-rgt-out N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-fma.f64 N/A

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

      24. lift-*.f64 N/A

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

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

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

      26. sub-negate-rev N/A

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

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

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

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

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

      29. lift-neg.f64 N/A

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

   7. Applied rewrites 88.1%

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

   if +inf.0 < (-.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 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 0

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

   6. Applied rewrites 57.8%

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

Alternative 5: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.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)) < +inf.0

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.2%

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

   5. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 88.2%

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

   7. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 82.4%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 88.2%

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

   9. Applied rewrites 88.2%

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

   if +inf.0 < (-.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 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 0

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

   6. Applied rewrites 57.8%

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

Alternative 6: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := x.im \cdot \left|x.re\right|\\ t_1 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(t\_1 - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + t\_0\right) \cdot x.im \leq -2 \cdot 10^{+235}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \left|x.re\right|, t\_0 \cdot \left(-3 \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(-3 \cdot x.im, x.im, t\_1\right)\\ \end{array} \end{array} \]

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))
        -2e+235)
     (fma t_1 (fabs x.re) (* t_0 (* -3.0 x.im)))
     (* (fabs x.re) (fma (* -3.0 x.im) x.im t_1))))))

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)) <= -2e+235) {
		tmp = fma(t_1, fabs(x_46_re), (t_0 * (-3.0 * x_46_im)));
	} else {
		tmp = fabs(x_46_re) * fma((-3.0 * x_46_im), x_46_im, t_1);
	}
	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)) <= -2e+235)
		tmp = fma(t_1, abs(x_46_re), Float64(t_0 * Float64(-3.0 * x_46_im)));
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(-3.0 * x_46_im), x_46_im, t_1));
	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], -2e+235], N[(t$95$1 * N[Abs[x$46$re], $MachinePrecision] + N[(t$95$0 * N[(-3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$im + t$95$1), $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)) < -2.0000000000000001e235

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.2%

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

   5. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 88.2%

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

   7. Applied rewrites 88.2%

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

   if -2.0000000000000001e235 < (-.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 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 90.7%

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

   5. Applied rewrites 90.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-neg.f64 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 91.1%

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

   7. Applied rewrites 91.1%

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

Alternative 7: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := x.im \cdot \left|x.re\right|\\ t_1 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(t\_1 - x.im \cdot x.im\right) \cdot \left|x.re\right| - \left(\left|x.re\right| \cdot x.im + t\_0\right) \cdot x.im \leq -1 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \left|x.re\right|, \left(-3 \cdot t\_0\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(-3 \cdot x.im, x.im, t\_1\right)\\ \end{array} \end{array} \]

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+135)
     (fma t_1 (fabs x.re) (* (* -3.0 t_0) x.im))
     (* (fabs x.re) (fma (* -3.0 x.im) x.im t_1))))))

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+135) {
		tmp = fma(t_1, fabs(x_46_re), ((-3.0 * t_0) * x_46_im));
	} else {
		tmp = fabs(x_46_re) * fma((-3.0 * x_46_im), x_46_im, t_1);
	}
	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+135)
		tmp = fma(t_1, abs(x_46_re), Float64(Float64(-3.0 * t_0) * x_46_im));
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(-3.0 * x_46_im), x_46_im, t_1));
	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+135], N[(t$95$1 * N[Abs[x$46$re], $MachinePrecision] + N[(N[(-3.0 * t$95$0), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$im + t$95$1), $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)) < -9.9999999999999996e134

   1. Initial program 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.2%

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

   5. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

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

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   if -9.9999999999999996e134 < (-.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 82.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 82.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 90.7%

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

   5. Applied rewrites 90.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-neg.f64 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 91.1%

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

   7. Applied rewrites 91.1%

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

Alternative 8: 91.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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 82.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 90.7%

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

5. Applied rewrites 90.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-neg.f64 N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 91.1%

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

7. Applied rewrites 91.1%

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

Alternative 9: 90.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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 82.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 90.7%

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

5. Applied rewrites 90.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 90.7%

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

7. Applied rewrites 90.7%

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

Developer Target 1: 87.0% 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 2025254 
(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)))