math.cube on complex, imaginary part

Percentage Accurate: 82.4% → 99.8%

Time: 4.1s

Alternatives: 8

Speedup: 0.9×

• 55.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + \left|x.im\right| \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-228}:\\ \;\;\;\;-1 \cdot {\left(\left|x.im\right|\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left|x.im\right| \cdot \left(\left(x.re + x.re\right) + \left(\left|x.im\right| + x.re\right) \cdot \left(1 - \frac{\left|x.im\right|}{x.re}\right)\right)\right) \cdot x.re\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<=
      (+
       (* (- (* x.re x.re) (* (fabs x.im) (fabs x.im))) (fabs x.im))
       (* (+ (* x.re (fabs x.im)) (* (fabs x.im) x.re)) x.re))
      -1e-228)
   (* -1.0 (pow (fabs x.im) 3.0))
   (*
    (*
     (fabs x.im)
     (+
      (+ x.re x.re)
      (* (+ (fabs x.im) x.re) (- 1.0 (/ (fabs x.im) x.re)))))
    x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + (fabs(x_46_im) * x_46_re)) * x_46_re)) <= -1e-228) {
		tmp = -1.0 * pow(fabs(x_46_im), 3.0);
	} else {
		tmp = (fabs(x_46_im) * ((x_46_re + x_46_re) + ((fabs(x_46_im) + x_46_re) * (1.0 - (fabs(x_46_im) / x_46_re))))) * x_46_re;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((((x_46_re * x_46_re) - (Math.abs(x_46_im) * Math.abs(x_46_im))) * Math.abs(x_46_im)) + (((x_46_re * Math.abs(x_46_im)) + (Math.abs(x_46_im) * x_46_re)) * x_46_re)) <= -1e-228) {
		tmp = -1.0 * Math.pow(Math.abs(x_46_im), 3.0);
	} else {
		tmp = (Math.abs(x_46_im) * ((x_46_re + x_46_re) + ((Math.abs(x_46_im) + x_46_re) * (1.0 - (Math.abs(x_46_im) / x_46_re))))) * x_46_re;
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((((x_46_re * x_46_re) - (math.fabs(x_46_im) * math.fabs(x_46_im))) * math.fabs(x_46_im)) + (((x_46_re * math.fabs(x_46_im)) + (math.fabs(x_46_im) * x_46_re)) * x_46_re)) <= -1e-228:
		tmp = -1.0 * math.pow(math.fabs(x_46_im), 3.0)
	else:
		tmp = (math.fabs(x_46_im) * ((x_46_re + x_46_re) + ((math.fabs(x_46_im) + x_46_re) * (1.0 - (math.fabs(x_46_im) / x_46_re))))) * x_46_re
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + Float64(abs(x_46_im) * x_46_re)) * x_46_re)) <= -1e-228)
		tmp = Float64(-1.0 * (abs(x_46_im) ^ 3.0));
	else
		tmp = Float64(Float64(abs(x_46_im) * Float64(Float64(x_46_re + x_46_re) + Float64(Float64(abs(x_46_im) + x_46_re) * Float64(1.0 - Float64(abs(x_46_im) / x_46_re))))) * x_46_re);
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - (abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + (((x_46_re * abs(x_46_im)) + (abs(x_46_im) * x_46_re)) * x_46_re)) <= -1e-228)
		tmp = -1.0 * (abs(x_46_im) ^ 3.0);
	else
		tmp = (abs(x_46_im) * ((x_46_re + x_46_re) + ((abs(x_46_im) + x_46_re) * (1.0 - (abs(x_46_im) / x_46_re))))) * x_46_re;
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], -1e-228], N[(-1.0 * N[Power[N[Abs[x$46$im], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(x$46$re + x$46$re), $MachinePrecision] + N[(N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision] * N[(1.0 - N[(N[Abs[x$46$im], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1e-228

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

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

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

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

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

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

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

      21. sub-negate-rev N/A

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

      22. lift--.f64 N/A

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

   3. Applied rewrites 91.6%

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

   4. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 58.9%

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

   6. Applied rewrites 58.9%

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

   if -1e-228 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

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

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

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

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

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

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

      21. sub-negate-rev N/A

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

      22. lift--.f64 N/A

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

   3. Applied rewrites 91.6%

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

   5. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 89.9%

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

   7. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(\left(x.re + x.re\right) + \left(x.im + x.re\right) \cdot \left(1 - \frac{x.im}{x.re}\right)\right)\right) \cdot x.re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot x.re\\ t_1 := \left|x.im\right| \cdot \left|x.im\right|\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - t\_1\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + t\_0\right) \cdot x.re \leq 5 \cdot 10^{+28}:\\ \;\;\;\;3 \cdot \left(t\_0 \cdot x.re\right) - t\_1 \cdot \left|x.im\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\left|x.im\right| \cdot \left(\left(x.re + x.re\right) + \left(\left|x.im\right| + x.re\right) \cdot \left(1 - \frac{\left|x.im\right|}{x.re}\right)\right)\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) x.re)) (t_1 (* (fabs x.im) (fabs x.im))))
  (*
   (copysign 1.0 x.im)
   (if (<=
        (+
         (* (- (* x.re x.re) t_1) (fabs x.im))
         (* (+ (* x.re (fabs x.im)) t_0) x.re))
        5e+28)
     (- (* 3.0 (* t_0 x.re)) (* t_1 (fabs x.im)))
     (*
      (*
       (fabs x.im)
       (+
        (+ x.re x.re)
        (* (+ (fabs x.im) x.re) (- 1.0 (/ (fabs x.im) x.re)))))
      x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * x_46_re;
	double t_1 = fabs(x_46_im) * fabs(x_46_im);
	double tmp;
	if (((((x_46_re * x_46_re) - t_1) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + t_0) * x_46_re)) <= 5e+28) {
		tmp = (3.0 * (t_0 * x_46_re)) - (t_1 * fabs(x_46_im));
	} else {
		tmp = (fabs(x_46_im) * ((x_46_re + x_46_re) + ((fabs(x_46_im) + x_46_re) * (1.0 - (fabs(x_46_im) / x_46_re))))) * x_46_re;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = Math.abs(x_46_im) * x_46_re;
	double t_1 = Math.abs(x_46_im) * Math.abs(x_46_im);
	double tmp;
	if (((((x_46_re * x_46_re) - t_1) * Math.abs(x_46_im)) + (((x_46_re * Math.abs(x_46_im)) + t_0) * x_46_re)) <= 5e+28) {
		tmp = (3.0 * (t_0 * x_46_re)) - (t_1 * Math.abs(x_46_im));
	} else {
		tmp = (Math.abs(x_46_im) * ((x_46_re + x_46_re) + ((Math.abs(x_46_im) + x_46_re) * (1.0 - (Math.abs(x_46_im) / x_46_re))))) * x_46_re;
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = math.fabs(x_46_im) * x_46_re
	t_1 = math.fabs(x_46_im) * math.fabs(x_46_im)
	tmp = 0
	if ((((x_46_re * x_46_re) - t_1) * math.fabs(x_46_im)) + (((x_46_re * math.fabs(x_46_im)) + t_0) * x_46_re)) <= 5e+28:
		tmp = (3.0 * (t_0 * x_46_re)) - (t_1 * math.fabs(x_46_im))
	else:
		tmp = (math.fabs(x_46_im) * ((x_46_re + x_46_re) + ((math.fabs(x_46_im) + x_46_re) * (1.0 - (math.fabs(x_46_im) / x_46_re))))) * x_46_re
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * x_46_re)
	t_1 = Float64(abs(x_46_im) * abs(x_46_im))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - t_1) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + t_0) * x_46_re)) <= 5e+28)
		tmp = Float64(Float64(3.0 * Float64(t_0 * x_46_re)) - Float64(t_1 * abs(x_46_im)));
	else
		tmp = Float64(Float64(abs(x_46_im) * Float64(Float64(x_46_re + x_46_re) + Float64(Float64(abs(x_46_im) + x_46_re) * Float64(1.0 - Float64(abs(x_46_im) / x_46_re))))) * x_46_re);
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = abs(x_46_im) * x_46_re;
	t_1 = abs(x_46_im) * abs(x_46_im);
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - t_1) * abs(x_46_im)) + (((x_46_re * abs(x_46_im)) + t_0) * x_46_re)) <= 5e+28)
		tmp = (3.0 * (t_0 * x_46_re)) - (t_1 * abs(x_46_im));
	else
		tmp = (abs(x_46_im) * ((x_46_re + x_46_re) + ((abs(x_46_im) + x_46_re) * (1.0 - (abs(x_46_im) / x_46_re))))) * x_46_re;
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - t$95$1), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], 5e+28], N[(N[(3.0 * N[(t$95$0 * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(x$46$re + x$46$re), $MachinePrecision] + N[(N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision] * N[(1.0 - N[(N[Abs[x$46$im], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 4.9999999999999996e28

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. associate-+r+ N/A

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

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

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

      15. lower--.f64 N/A

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

   3. Applied rewrites 86.1%

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

   if 4.9999999999999996e28 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

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

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

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

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

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

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

      21. sub-negate-rev N/A

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

      22. lift--.f64 N/A

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

   3. Applied rewrites 91.6%

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

   5. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 89.9%

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

   7. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(\left(x.re + x.re\right) + \left(x.im + x.re\right) \cdot \left(1 - \frac{x.im}{x.re}\right)\right)\right) \cdot x.re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 1.5 \cdot 10^{+101}:\\ \;\;\;\;\left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right| - \left(\left|x.im\right| \cdot x.re\right) \cdot \left(\left(-x.re\right) \cdot 2 - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|x.im\right| \cdot \left(\left(x.re + x.re\right) + \left|x.im\right| \cdot \left(1 - \frac{\left|x.im\right|}{x.re}\right)\right)\right) \cdot x.re\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 1.5e+101)
   (-
    (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im))
    (* (* (fabs x.im) x.re) (- (* (- x.re) 2.0) x.re)))
   (*
    (*
     (fabs x.im)
     (+ (+ x.re x.re) (* (fabs x.im) (- 1.0 (/ (fabs x.im) x.re)))))
    x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 1.5e+101) {
		tmp = ((-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im)) - ((fabs(x_46_im) * x_46_re) * ((-x_46_re * 2.0) - x_46_re));
	} else {
		tmp = (fabs(x_46_im) * ((x_46_re + x_46_re) + (fabs(x_46_im) * (1.0 - (fabs(x_46_im) / x_46_re))))) * x_46_re;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (Math.abs(x_46_im) <= 1.5e+101) {
		tmp = ((-Math.abs(x_46_im) * Math.abs(x_46_im)) * Math.abs(x_46_im)) - ((Math.abs(x_46_im) * x_46_re) * ((-x_46_re * 2.0) - x_46_re));
	} else {
		tmp = (Math.abs(x_46_im) * ((x_46_re + x_46_re) + (Math.abs(x_46_im) * (1.0 - (Math.abs(x_46_im) / x_46_re))))) * x_46_re;
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_im) <= 1.5e+101:
		tmp = ((-math.fabs(x_46_im) * math.fabs(x_46_im)) * math.fabs(x_46_im)) - ((math.fabs(x_46_im) * x_46_re) * ((-x_46_re * 2.0) - x_46_re))
	else:
		tmp = (math.fabs(x_46_im) * ((x_46_re + x_46_re) + (math.fabs(x_46_im) * (1.0 - (math.fabs(x_46_im) / x_46_re))))) * x_46_re
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 1.5e+101)
		tmp = Float64(Float64(Float64(Float64(-abs(x_46_im)) * abs(x_46_im)) * abs(x_46_im)) - Float64(Float64(abs(x_46_im) * x_46_re) * Float64(Float64(Float64(-x_46_re) * 2.0) - x_46_re)));
	else
		tmp = Float64(Float64(abs(x_46_im) * Float64(Float64(x_46_re + x_46_re) + Float64(abs(x_46_im) * Float64(1.0 - Float64(abs(x_46_im) / x_46_re))))) * x_46_re);
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (abs(x_46_im) <= 1.5e+101)
		tmp = ((-abs(x_46_im) * abs(x_46_im)) * abs(x_46_im)) - ((abs(x_46_im) * x_46_re) * ((-x_46_re * 2.0) - x_46_re));
	else
		tmp = (abs(x_46_im) * ((x_46_re + x_46_re) + (abs(x_46_im) * (1.0 - (abs(x_46_im) / x_46_re))))) * x_46_re;
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 1.5e+101], N[(N[(N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision] * N[(N[((-x$46$re) * 2.0), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(x$46$re + x$46$re), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * N[(1.0 - N[(N[Abs[x$46$im], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.5e101

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip-reverse N/A

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

      6. distribute-neg-out N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift--.f64 N/A

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

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

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

   3. Applied rewrites 86.1%

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

   if 1.5e101 < x.im

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

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

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

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

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

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

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

      21. sub-negate-rev N/A

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

      22. lift--.f64 N/A

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

   3. Applied rewrites 91.6%

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

   5. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 89.9%

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in x.re around 0

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

   10. Applied rewrites 75.1%

       \[\leadsto \left(x.im \cdot \left(\left(x.re + x.re\right) + \color{blue}{x.im} \cdot \left(1 - \frac{x.im}{x.re}\right)\right)\right) \cdot x.re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.6% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 1.5 \cdot 10^{+101}:\\ \;\;\;\;3 \cdot \left(\left(\left|x.im\right| \cdot x.re\right) \cdot x.re\right) - \left(\left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\left|x.im\right| \cdot \left(\left(x.re + x.re\right) + \left|x.im\right| \cdot \left(1 - \frac{\left|x.im\right|}{x.re}\right)\right)\right) \cdot x.re\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 1.5e+101)
   (-
    (* 3.0 (* (* (fabs x.im) x.re) x.re))
    (* (* (fabs x.im) (fabs x.im)) (fabs x.im)))
   (*
    (*
     (fabs x.im)
     (+ (+ x.re x.re) (* (fabs x.im) (- 1.0 (/ (fabs x.im) x.re)))))
    x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 1.5e+101) {
		tmp = (3.0 * ((fabs(x_46_im) * x_46_re) * x_46_re)) - ((fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im));
	} else {
		tmp = (fabs(x_46_im) * ((x_46_re + x_46_re) + (fabs(x_46_im) * (1.0 - (fabs(x_46_im) / x_46_re))))) * x_46_re;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (Math.abs(x_46_im) <= 1.5e+101) {
		tmp = (3.0 * ((Math.abs(x_46_im) * x_46_re) * x_46_re)) - ((Math.abs(x_46_im) * Math.abs(x_46_im)) * Math.abs(x_46_im));
	} else {
		tmp = (Math.abs(x_46_im) * ((x_46_re + x_46_re) + (Math.abs(x_46_im) * (1.0 - (Math.abs(x_46_im) / x_46_re))))) * x_46_re;
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_im) <= 1.5e+101:
		tmp = (3.0 * ((math.fabs(x_46_im) * x_46_re) * x_46_re)) - ((math.fabs(x_46_im) * math.fabs(x_46_im)) * math.fabs(x_46_im))
	else:
		tmp = (math.fabs(x_46_im) * ((x_46_re + x_46_re) + (math.fabs(x_46_im) * (1.0 - (math.fabs(x_46_im) / x_46_re))))) * x_46_re
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 1.5e+101)
		tmp = Float64(Float64(3.0 * Float64(Float64(abs(x_46_im) * x_46_re) * x_46_re)) - Float64(Float64(abs(x_46_im) * abs(x_46_im)) * abs(x_46_im)));
	else
		tmp = Float64(Float64(abs(x_46_im) * Float64(Float64(x_46_re + x_46_re) + Float64(abs(x_46_im) * Float64(1.0 - Float64(abs(x_46_im) / x_46_re))))) * x_46_re);
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (abs(x_46_im) <= 1.5e+101)
		tmp = (3.0 * ((abs(x_46_im) * x_46_re) * x_46_re)) - ((abs(x_46_im) * abs(x_46_im)) * abs(x_46_im));
	else
		tmp = (abs(x_46_im) * ((x_46_re + x_46_re) + (abs(x_46_im) * (1.0 - (abs(x_46_im) / x_46_re))))) * x_46_re;
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 1.5e+101], N[(N[(3.0 * N[(N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(x$46$re + x$46$re), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * N[(1.0 - N[(N[Abs[x$46$im], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.5e101

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. associate-+r+ N/A

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

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

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

      15. lower--.f64 N/A

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

   3. Applied rewrites 86.1%

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

   if 1.5e101 < x.im

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

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

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

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

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

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

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

      21. sub-negate-rev N/A

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

      22. lift--.f64 N/A

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

   3. Applied rewrites 91.6%

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

   5. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 89.9%

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in x.re around 0

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

   10. Applied rewrites 75.1%

       \[\leadsto \left(x.im \cdot \left(\left(x.re + x.re\right) + \color{blue}{x.im} \cdot \left(1 - \frac{x.im}{x.re}\right)\right)\right) \cdot x.re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 6 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \left(\left(\left|x.re\right| + \left|x.re\right|\right) \cdot \left|x.re\right| - \left(x.im - \left|x.re\right|\right) \cdot \left(x.im + \left|x.re\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot \left|x.re\right|\right) \cdot \left(3 \cdot \left|x.re\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (if (<= (fabs x.re) 6e+154)
  (*
   x.im
   (-
    (* (+ (fabs x.re) (fabs x.re)) (fabs x.re))
    (* (- x.im (fabs x.re)) (+ x.im (fabs x.re)))))
  (* (* x.im (fabs x.re)) (* 3.0 (fabs x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_re) <= 6e+154) {
		tmp = x_46_im * (((fabs(x_46_re) + fabs(x_46_re)) * fabs(x_46_re)) - ((x_46_im - fabs(x_46_re)) * (x_46_im + fabs(x_46_re))));
	} else {
		tmp = (x_46_im * fabs(x_46_re)) * (3.0 * fabs(x_46_re));
	}
	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_46re) <= 6d+154) then
        tmp = x_46im * (((abs(x_46re) + abs(x_46re)) * abs(x_46re)) - ((x_46im - abs(x_46re)) * (x_46im + abs(x_46re))))
    else
        tmp = (x_46im * abs(x_46re)) * (3.0d0 * abs(x_46re))
    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_re) <= 6e+154) {
		tmp = x_46_im * (((Math.abs(x_46_re) + Math.abs(x_46_re)) * Math.abs(x_46_re)) - ((x_46_im - Math.abs(x_46_re)) * (x_46_im + Math.abs(x_46_re))));
	} else {
		tmp = (x_46_im * Math.abs(x_46_re)) * (3.0 * Math.abs(x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_re) <= 6e+154:
		tmp = x_46_im * (((math.fabs(x_46_re) + math.fabs(x_46_re)) * math.fabs(x_46_re)) - ((x_46_im - math.fabs(x_46_re)) * (x_46_im + math.fabs(x_46_re))))
	else:
		tmp = (x_46_im * math.fabs(x_46_re)) * (3.0 * math.fabs(x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_re) <= 6e+154)
		tmp = Float64(x_46_im * Float64(Float64(Float64(abs(x_46_re) + abs(x_46_re)) * abs(x_46_re)) - Float64(Float64(x_46_im - abs(x_46_re)) * Float64(x_46_im + abs(x_46_re)))));
	else
		tmp = Float64(Float64(x_46_im * abs(x_46_re)) * Float64(3.0 * abs(x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (abs(x_46_re) <= 6e+154)
		tmp = x_46_im * (((abs(x_46_re) + abs(x_46_re)) * abs(x_46_re)) - ((x_46_im - abs(x_46_re)) * (x_46_im + abs(x_46_re))));
	else
		tmp = (x_46_im * abs(x_46_re)) * (3.0 * abs(x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 6e+154], N[(x$46$im * N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$im - N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[(x$46$im + N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[(3.0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.0000000000000005e154

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

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

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

      8. sub-flip-reverse N/A

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

   3. Applied rewrites 90.6%

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

   if 6.0000000000000005e154 < x.re

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 49.5%

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

   4. Applied rewrites 49.5%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.5%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 55.6%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 55.6%

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

   6. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 55.6%

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

   8. Applied rewrites 55.6%

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

Alternative 6: 91.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot x.re\\ t_1 := \left|x.im\right| \cdot \left|x.im\right|\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - t\_1\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + t\_0\right) \cdot x.re \leq -1 \cdot 10^{-323}:\\ \;\;\;\;\left|x.im\right| \cdot \left(\left(x.re + x.re\right) \cdot x.re - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(3 \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) x.re)) (t_1 (* (fabs x.im) (fabs x.im))))
  (*
   (copysign 1.0 x.im)
   (if (<=
        (+
         (* (- (* x.re x.re) t_1) (fabs x.im))
         (* (+ (* x.re (fabs x.im)) t_0) x.re))
        -1e-323)
     (* (fabs x.im) (- (* (+ x.re x.re) x.re) t_1))
     (* t_0 (* 3.0 x.re))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * x_46_re;
	double t_1 = fabs(x_46_im) * fabs(x_46_im);
	double tmp;
	if (((((x_46_re * x_46_re) - t_1) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + t_0) * x_46_re)) <= -1e-323) {
		tmp = fabs(x_46_im) * (((x_46_re + x_46_re) * x_46_re) - t_1);
	} else {
		tmp = t_0 * (3.0 * x_46_re);
	}
	return copysign(1.0, x_46_im) * tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = Math.abs(x_46_im) * x_46_re;
	double t_1 = Math.abs(x_46_im) * Math.abs(x_46_im);
	double tmp;
	if (((((x_46_re * x_46_re) - t_1) * Math.abs(x_46_im)) + (((x_46_re * Math.abs(x_46_im)) + t_0) * x_46_re)) <= -1e-323) {
		tmp = Math.abs(x_46_im) * (((x_46_re + x_46_re) * x_46_re) - t_1);
	} else {
		tmp = t_0 * (3.0 * x_46_re);
	}
	return Math.copySign(1.0, x_46_im) * tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = math.fabs(x_46_im) * x_46_re
	t_1 = math.fabs(x_46_im) * math.fabs(x_46_im)
	tmp = 0
	if ((((x_46_re * x_46_re) - t_1) * math.fabs(x_46_im)) + (((x_46_re * math.fabs(x_46_im)) + t_0) * x_46_re)) <= -1e-323:
		tmp = math.fabs(x_46_im) * (((x_46_re + x_46_re) * x_46_re) - t_1)
	else:
		tmp = t_0 * (3.0 * x_46_re)
	return math.copysign(1.0, x_46_im) * tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * x_46_re)
	t_1 = Float64(abs(x_46_im) * abs(x_46_im))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - t_1) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + t_0) * x_46_re)) <= -1e-323)
		tmp = Float64(abs(x_46_im) * Float64(Float64(Float64(x_46_re + x_46_re) * x_46_re) - t_1));
	else
		tmp = Float64(t_0 * Float64(3.0 * x_46_re));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = abs(x_46_im) * x_46_re;
	t_1 = abs(x_46_im) * abs(x_46_im);
	tmp = 0.0;
	if (((((x_46_re * x_46_re) - t_1) * abs(x_46_im)) + (((x_46_re * abs(x_46_im)) + t_0) * x_46_re)) <= -1e-323)
		tmp = abs(x_46_im) * (((x_46_re + x_46_re) * x_46_re) - t_1);
	else
		tmp = t_0 * (3.0 * x_46_re);
	end
	tmp_2 = (sign(x_46_im) * abs(1.0)) * tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - t$95$1), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], -1e-323], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(3.0 * x$46$re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -9.8813129168249309e-324

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

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

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

      8. sub-flip-reverse N/A

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

   3. Applied rewrites 90.6%

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

   4. Taylor expanded in x.re around 0

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

   6. Applied rewrites 72.0%

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

   7. Taylor expanded in x.re around 0

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

   9. Applied rewrites 72.2%

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

   if -9.8813129168249309e-324 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 49.5%

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

   4. Applied rewrites 49.5%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.5%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 55.6%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 55.6%

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

   6. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 55.6%

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

   8. Applied rewrites 55.6%

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

Alternative 7: 55.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return (x_46_im * x_46_re) * (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_46re) * (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_re) * (3.0 * x_46_re);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.re around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 49.5%

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

4. Applied rewrites 49.5%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 49.5%

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lower-*.f64 55.6%

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 55.6%

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

6. Applied rewrites 55.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 55.6%

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

8. Applied rewrites 55.6%

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

Alternative 8: 55.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.re around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 49.5%

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

4. Applied rewrites 49.5%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 49.5%

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lower-*.f64 55.6%

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 55.6%

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

6. Applied rewrites 55.6%

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

Reproduce

herbie shell --seed 2025258 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64
  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))