math.cube on complex, real part

Percentage Accurate: 83.0% → 99.8%

Time: 1.9s

Alternatives: 7

Speedup: 1.1×

• 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: 83.0% 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, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 1700000000000000023149577102140402537782413642223660022978012006712853548765339960860481327185241571328:\\ \;\;\;\;\left(\left|x.re\right| \cdot \left|x.re\right|\right) \cdot \left|x.re\right| + 3 \cdot \left(\left(\left(-x.im\right) \cdot \left|x.re\right|\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \left(x.im \cdot \left(\left(x.im + \left|x.re\right|\right) \cdot \frac{\left|x.re\right| - x.im}{x.im} - \left(x.im + x.im\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1 x.re)
 (if (<=
      (fabs x.re)
      1700000000000000023149577102140402537782413642223660022978012006712853548765339960860481327185241571328)
   (+
    (* (* (fabs x.re) (fabs x.re)) (fabs x.re))
    (* 3 (* (* (- x.im) (fabs x.re)) x.im)))
   (*
    (fabs x.re)
    (*
     x.im
     (-
      (* (+ x.im (fabs x.re)) (/ (- (fabs x.re) x.im) x.im))
      (+ x.im x.im)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_re) <= 1.7e+102) {
		tmp = ((fabs(x_46_re) * fabs(x_46_re)) * fabs(x_46_re)) + (3.0 * ((-x_46_im * fabs(x_46_re)) * x_46_im));
	} else {
		tmp = fabs(x_46_re) * (x_46_im * (((x_46_im + fabs(x_46_re)) * ((fabs(x_46_re) - x_46_im) / x_46_im)) - (x_46_im + x_46_im)));
	}
	return copysign(1.0, x_46_re) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (Math.abs(x_46_re) <= 1.7e+102) {
		tmp = ((Math.abs(x_46_re) * Math.abs(x_46_re)) * Math.abs(x_46_re)) + (3.0 * ((-x_46_im * Math.abs(x_46_re)) * x_46_im));
	} else {
		tmp = Math.abs(x_46_re) * (x_46_im * (((x_46_im + Math.abs(x_46_re)) * ((Math.abs(x_46_re) - x_46_im) / x_46_im)) - (x_46_im + x_46_im)));
	}
	return Math.copySign(1.0, x_46_re) * tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_re) <= 1.7e+102:
		tmp = ((math.fabs(x_46_re) * math.fabs(x_46_re)) * math.fabs(x_46_re)) + (3.0 * ((-x_46_im * math.fabs(x_46_re)) * x_46_im))
	else:
		tmp = math.fabs(x_46_re) * (x_46_im * (((x_46_im + math.fabs(x_46_re)) * ((math.fabs(x_46_re) - x_46_im) / x_46_im)) - (x_46_im + x_46_im)))
	return math.copysign(1.0, x_46_re) * tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_re) <= 1.7e+102)
		tmp = Float64(Float64(Float64(abs(x_46_re) * abs(x_46_re)) * abs(x_46_re)) + Float64(3.0 * Float64(Float64(Float64(-x_46_im) * abs(x_46_re)) * x_46_im)));
	else
		tmp = Float64(abs(x_46_re) * Float64(x_46_im * Float64(Float64(Float64(x_46_im + abs(x_46_re)) * Float64(Float64(abs(x_46_re) - x_46_im) / x_46_im)) - Float64(x_46_im + x_46_im))));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (abs(x_46_re) <= 1.7e+102)
		tmp = ((abs(x_46_re) * abs(x_46_re)) * abs(x_46_re)) + (3.0 * ((-x_46_im * abs(x_46_re)) * x_46_im));
	else
		tmp = abs(x_46_re) * (x_46_im * (((x_46_im + abs(x_46_re)) * ((abs(x_46_re) - x_46_im) / x_46_im)) - (x_46_im + x_46_im)));
	end
	tmp_2 = (sign(x_46_re) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 1700000000000000023149577102140402537782413642223660022978012006712853548765339960860481327185241571328], N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] + N[(3 * N[(N[((-x$46$im) * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(x$46$im * N[(N[(N[(x$46$im + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[x$46$re], $MachinePrecision] - x$46$im), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.7e102

   1. Initial program 83.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. fp-cancel-sub-sign-inv N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. associate--l+ N/A

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

   3. Applied rewrites 80.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. lower-*.f64 86.1%

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

   5. Applied rewrites 86.1%

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

   if 1.7e102 < x.re

   1. Initial program 83.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 91.1%

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

   5. Applied rewrites 91.6%

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

Alternative 2: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq \frac{578960446186581}{57896044618658097711785492504343953926634992332820282019728792003956564819968}:\\ \;\;\;\;t\_0 \cdot \left|x.re\right| + 3 \cdot \left(\left(\left(-x.im\right) \cdot \left|x.re\right|\right) \cdot x.im\right)\\ \mathbf{elif}\;\left|x.re\right| \leq 200000000000000016531517668251747580868252952853088870140921275623123251200204950421777121660801104008620977885871710627547264408583791539263482088984782477300371894320431629895715109375821874825666256654733483033231360:\\ \;\;\;\;\left|x.re\right| \cdot \left(\left(\left|x.re\right| - x.im\right) \cdot \left(x.im + \left|x.re\right|\right) - \left(x.im + x.im\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{t\_0}{x.im} - x.im\right) - \left(x.im + x.im\right)\right) \cdot \left|x.re\right|\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.re) (fabs x.re))))
  (*
   (copysign 1 x.re)
   (if (<=
        (fabs x.re)
        578960446186581/57896044618658097711785492504343953926634992332820282019728792003956564819968)
     (+ (* t_0 (fabs x.re)) (* 3 (* (* (- x.im) (fabs x.re)) x.im)))
     (if (<=
          (fabs x.re)
          200000000000000016531517668251747580868252952853088870140921275623123251200204950421777121660801104008620977885871710627547264408583791539263482088984782477300371894320431629895715109375821874825666256654733483033231360)
       (*
        (fabs x.re)
        (-
         (* (- (fabs x.re) x.im) (+ x.im (fabs x.re)))
         (* (+ x.im x.im) x.im)))
       (*
        (* (- (- (/ t_0 x.im) x.im) (+ x.im x.im)) (fabs x.re))
        x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) * fabs(x_46_re);
	double tmp;
	if (fabs(x_46_re) <= 1e-62) {
		tmp = (t_0 * fabs(x_46_re)) + (3.0 * ((-x_46_im * fabs(x_46_re)) * x_46_im));
	} else if (fabs(x_46_re) <= 2e+218) {
		tmp = fabs(x_46_re) * (((fabs(x_46_re) - x_46_im) * (x_46_im + fabs(x_46_re))) - ((x_46_im + x_46_im) * x_46_im));
	} else {
		tmp = ((((t_0 / x_46_im) - x_46_im) - (x_46_im + x_46_im)) * fabs(x_46_re)) * x_46_im;
	}
	return copysign(1.0, x_46_re) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = Math.abs(x_46_re) * Math.abs(x_46_re);
	double tmp;
	if (Math.abs(x_46_re) <= 1e-62) {
		tmp = (t_0 * Math.abs(x_46_re)) + (3.0 * ((-x_46_im * Math.abs(x_46_re)) * x_46_im));
	} else if (Math.abs(x_46_re) <= 2e+218) {
		tmp = Math.abs(x_46_re) * (((Math.abs(x_46_re) - x_46_im) * (x_46_im + Math.abs(x_46_re))) - ((x_46_im + x_46_im) * x_46_im));
	} else {
		tmp = ((((t_0 / x_46_im) - x_46_im) - (x_46_im + x_46_im)) * Math.abs(x_46_re)) * x_46_im;
	}
	return Math.copySign(1.0, x_46_re) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = math.fabs(x_46_re) * math.fabs(x_46_re)
	tmp = 0
	if math.fabs(x_46_re) <= 1e-62:
		tmp = (t_0 * math.fabs(x_46_re)) + (3.0 * ((-x_46_im * math.fabs(x_46_re)) * x_46_im))
	elif math.fabs(x_46_re) <= 2e+218:
		tmp = math.fabs(x_46_re) * (((math.fabs(x_46_re) - x_46_im) * (x_46_im + math.fabs(x_46_re))) - ((x_46_im + x_46_im) * x_46_im))
	else:
		tmp = ((((t_0 / x_46_im) - x_46_im) - (x_46_im + x_46_im)) * math.fabs(x_46_re)) * x_46_im
	return math.copysign(1.0, x_46_re) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_re) * abs(x_46_re))
	tmp = 0.0
	if (abs(x_46_re) <= 1e-62)
		tmp = Float64(Float64(t_0 * abs(x_46_re)) + Float64(3.0 * Float64(Float64(Float64(-x_46_im) * abs(x_46_re)) * x_46_im)));
	elseif (abs(x_46_re) <= 2e+218)
		tmp = Float64(abs(x_46_re) * Float64(Float64(Float64(abs(x_46_re) - x_46_im) * Float64(x_46_im + abs(x_46_re))) - Float64(Float64(x_46_im + x_46_im) * x_46_im)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 / x_46_im) - x_46_im) - Float64(x_46_im + x_46_im)) * abs(x_46_re)) * x_46_im);
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = abs(x_46_re) * abs(x_46_re);
	tmp = 0.0;
	if (abs(x_46_re) <= 1e-62)
		tmp = (t_0 * abs(x_46_re)) + (3.0 * ((-x_46_im * abs(x_46_re)) * x_46_im));
	elseif (abs(x_46_re) <= 2e+218)
		tmp = abs(x_46_re) * (((abs(x_46_re) - x_46_im) * (x_46_im + abs(x_46_re))) - ((x_46_im + x_46_im) * x_46_im));
	else
		tmp = ((((t_0 / x_46_im) - x_46_im) - (x_46_im + x_46_im)) * abs(x_46_re)) * x_46_im;
	end
	tmp_2 = (sign(x_46_re) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 578960446186581/57896044618658097711785492504343953926634992332820282019728792003956564819968], N[(N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] + N[(3 * N[(N[((-x$46$im) * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 200000000000000016531517668251747580868252952853088870140921275623123251200204950421777121660801104008620977885871710627547264408583791539263482088984782477300371894320431629895715109375821874825666256654733483033231360], N[(N[Abs[x$46$re], $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] - N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t$95$0 / x$46$im), $MachinePrecision] - x$46$im), $MachinePrecision] - N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.re < 1e-62

   1. Initial program 83.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. fp-cancel-sub-sign-inv N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. associate--l+ N/A

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

   3. Applied rewrites 80.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. lower-*.f64 86.1%

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

   5. Applied rewrites 86.1%

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

   if 1e-62 < x.re < 2.0000000000000002e218

   1. Initial program 83.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 91.1%

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

   if 2.0000000000000002e218 < x.re

   1. Initial program 83.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 91.1%

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

   5. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 96.0%

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

   7. Applied rewrites 92.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 92.7%

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

Alternative 3: 95.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 95000000000000009373330514286885868742778266785675655959220971840194703074788250542986739317996208152821440321781914968597276393472:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - 3 \cdot \left(\left|x.im\right| \cdot \left|x.im\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|x.im\right| \cdot x.re\right) \cdot 3\right) \cdot \left(-\left|x.im\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (if (<=
     (fabs x.im)
     95000000000000009373330514286885868742778266785675655959220971840194703074788250542986739317996208152821440321781914968597276393472)
  (* x.re (- (* x.re x.re) (* 3 (* (fabs x.im) (fabs x.im)))))
  (* (* (* (fabs x.im) x.re) 3) (- (fabs x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 9.5e+130) {
		tmp = x_46_re * ((x_46_re * x_46_re) - (3.0 * (fabs(x_46_im) * fabs(x_46_im))));
	} else {
		tmp = ((fabs(x_46_im) * x_46_re) * 3.0) * -fabs(x_46_im);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (abs(x_46im) <= 9.5d+130) then
        tmp = x_46re * ((x_46re * x_46re) - (3.0d0 * (abs(x_46im) * abs(x_46im))))
    else
        tmp = ((abs(x_46im) * x_46re) * 3.0d0) * -abs(x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (Math.abs(x_46_im) <= 9.5e+130) {
		tmp = x_46_re * ((x_46_re * x_46_re) - (3.0 * (Math.abs(x_46_im) * Math.abs(x_46_im))));
	} else {
		tmp = ((Math.abs(x_46_im) * x_46_re) * 3.0) * -Math.abs(x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_im) <= 9.5e+130:
		tmp = x_46_re * ((x_46_re * x_46_re) - (3.0 * (math.fabs(x_46_im) * math.fabs(x_46_im))))
	else:
		tmp = ((math.fabs(x_46_im) * x_46_re) * 3.0) * -math.fabs(x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 9.5e+130)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(3.0 * Float64(abs(x_46_im) * abs(x_46_im)))));
	else
		tmp = Float64(Float64(Float64(abs(x_46_im) * x_46_re) * 3.0) * Float64(-abs(x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (abs(x_46_im) <= 9.5e+130)
		tmp = x_46_re * ((x_46_re * x_46_re) - (3.0 * (abs(x_46_im) * abs(x_46_im))));
	else
		tmp = ((abs(x_46_im) * x_46_re) * 3.0) * -abs(x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 95000000000000009373330514286885868742778266785675655959220971840194703074788250542986739317996208152821440321781914968597276393472], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(3 * N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision] * 3), $MachinePrecision] * (-N[Abs[x$46$im], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 9.5000000000000009e130

   1. Initial program 83.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. fp-cancel-sub-sign-inv N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. associate--l+ N/A

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

   3. Applied rewrites 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 86.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 86.1%

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

   5. Applied rewrites 86.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 88.0%

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

   7. Applied rewrites 88.0%

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

   if 9.5000000000000009e130 < x.im

   1. Initial program 83.0%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 50.3%

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-div N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 50.3%

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

      19. lift-/.f64 N/A

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

      20. lift-pow.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow-div N/A

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

      23. metadata-eval N/A

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

      24. pow2 N/A

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

      25. lower-*.f64 50.3%

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

   6. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 55.8%

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

   8. Applied rewrites 55.8%

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

Alternative 4: 55.8% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (* (* (* x.im x.re) x.im) -3))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46im * x_46re) * x_46im) * (-3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$im * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision] * -3), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

2. Taylor expanded in x.im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 50.3%

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

4. Applied rewrites 50.3%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-div N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow1 N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 55.7%

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

   15. lift--.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

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

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

   19. metadata-eval N/A

       \[\leadsto \frac{\left(x.re \cdot -3\right) \cdot x.im}{{x.im}^{-1}} \]

   20. metadata-eval N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 N/A

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

   23. metadata-eval 55.7%

       \[\leadsto \frac{\left(-3 \cdot x.re\right) \cdot x.im}{{x.im}^{-1}} \]

   24. lift-pow.f64 N/A

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

   25. unpow-1 N/A

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

   26. lower-/.f64 55.7%

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

6. Applied rewrites 55.7%

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

   1. lift-/.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(-3 \cdot x.re\right) \cdot x.im}{1 \cdot \frac{1}{x.im}} \]

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. mult-flip-rev N/A

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

   11. lift-/.f64 N/A

       \[\leadsto \left(\left(x.re \cdot x.im\right) \cdot \frac{1}{\frac{1}{x.im}}\right) \cdot \frac{-3}{1} \]

   12. remove-double-div N/A

       \[\leadsto \left(\left(x.re \cdot x.im\right) \cdot x.im\right) \cdot \frac{-3}{1} \]

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 55.8%

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

8. Applied rewrites 55.8%

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

Alternative 5: 55.8% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (* (* x.im x.re) (* -3 x.im)))

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(x$46$im * x$46$re), $MachinePrecision] * N[(-3 * x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

2. Taylor expanded in x.im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 50.3%

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

4. Applied rewrites 50.3%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-div N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow1 N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 55.7%

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

   15. lift--.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

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

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

   19. metadata-eval N/A

       \[\leadsto \frac{\left(x.re \cdot -3\right) \cdot x.im}{{x.im}^{-1}} \]

   20. metadata-eval N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 N/A

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

   23. metadata-eval 55.7%

       \[\leadsto \frac{\left(-3 \cdot x.re\right) \cdot x.im}{{x.im}^{-1}} \]

   24. lift-pow.f64 N/A

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

   25. unpow-1 N/A

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

   26. lower-/.f64 55.7%

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

6. Applied rewrites 55.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(-3 \cdot x.re\right) \cdot x.im}{\frac{1}{x.im}} \]

   4. associate-*l* N/A

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

   5. *-commutative N/A

      \[\leadsto \frac{-3 \cdot \left(x.im \cdot x.re\right)}{\frac{1}{x.im}} \]

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(-3 \cdot x.im\right) \cdot x.re}{\frac{1}{x.im}} \]

   8. associate-/l* N/A

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

   9. mult-flip-rev N/A

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

   10. lift-/.f64 N/A

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

   11. remove-double-div N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 55.8%

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

8. Applied rewrites 55.8%

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

Alternative 6: 50.3% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (* (* x.im x.im) (* -3 x.re)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (x_46im * x_46im) * ((-3.0d0) * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(-3 * x$46$re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

2. Taylor expanded in x.im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 50.3%

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

4. Applied rewrites 50.3%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-div N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval 50.3%

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

   19. lift-/.f64 N/A

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

   20. lift-pow.f64 N/A

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

   21. lift-pow.f64 N/A

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

   22. pow-div N/A

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

   23. metadata-eval N/A

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

   24. pow2 N/A

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

   25. lower-*.f64 50.3%

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

6. Applied rewrites 50.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 50.3%

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

8. Applied rewrites 50.3%

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

Alternative 7: 50.3% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (* x.re (* -3 (* x.im x.im))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = x_46re * ((-3.0d0) * (x_46im * x_46im))
end function

Java

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

Python

def code(x_46_re, x_46_im):
	return 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 * Float64(-3.0 * Float64(x_46_im * x_46_im)))
end

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$re * N[(-3 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

2. Taylor expanded in x.im around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 50.3%

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

4. Applied rewrites 50.3%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-div N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval 50.3%

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

   19. lift-/.f64 N/A

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

   20. lift-pow.f64 N/A

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

   21. lift-pow.f64 N/A

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

   22. pow-div N/A

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

   23. metadata-eval N/A

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

   24. pow2 N/A

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

   25. lower-*.f64 50.3%

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

6. Applied rewrites 50.3%

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

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64
  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))