math.cube on complex, imaginary part

Percentage Accurate: 83.0% → 100.0%

Time: 37.1s

Alternatives: 5

Speedup: 1.6×

• 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: 83.0% 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: 100.0% accurate, 1.4× speedup

Math

\[\mathsf{134\_z0z1z2z3z4}\left(x.im, \left(x.re + x.re\right), x.re, \left(x.im - x.re\right), \left(x.im + x.re\right)\right) \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (134-z0z1z2z3z4 x.im (+ x.re x.re) x.re (- x.im x.re) (+ x.im x.re)))

Derivation

1. Initial program 83.0%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
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 100.0%

   \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x.im, \left(x.re + x.re\right), x.re, \left(x.im - x.re\right), \left(x.im + x.re\right)\right)} \]
4. Add Preprocessing

Alternative 2: 99.8% accurate, 1.6× speedup

Math

\[\mathsf{134\_z0z1z2z3z4}\left(x.im, x.re, \left(3 \cdot x.re\right), x.im, x.im\right) \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (134-z0z1z2z3z4 x.im x.re (* 3 x.re) x.im x.im))

Derivation

1. Initial program 83.0%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
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. lift-*.f64 N/A

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

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

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

   8. distribute-rgt-in N/A

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

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

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. associate-+r+ N/A

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

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

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

   16. 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 85.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

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

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. remove-double-neg N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

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

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-134-z0z1z2z3z4 N/A

       \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x.im, x.re, \left(3 \cdot x.re\right), x.im, x.im\right)} \]

   20. lower-*.f64 99.8%

       \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(x.im, x.re, \color{blue}{\left(3 \cdot x.re\right)}, x.im, x.im\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(x.im, x.re, \left(3 \cdot x.re\right), x.im, x.im\right)} \]
6. Add Preprocessing

Alternative 3: 91.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 579999999999999970346306021154662620410815719091006025691049679413018715111325154758617509342024965348628887254730200477171420651107780304619988345189308120000626688:\\ \;\;\;\;x.im \cdot \left(\left(3 \cdot \left|x.re\right|\right) \cdot \left|x.re\right| - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left|x.re\right| \cdot \left(\left|x.re\right| + 2 \cdot \left|x.re\right|\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (if (<=
     (fabs x.re)
     579999999999999970346306021154662620410815719091006025691049679413018715111325154758617509342024965348628887254730200477171420651107780304619988345189308120000626688)
  (* x.im (- (* (* 3 (fabs x.re)) (fabs x.re)) (* x.im x.im)))
  (* x.im (* (fabs x.re) (+ (fabs x.re) (* 2 (fabs x.re)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_re) <= 5.8e+164) {
		tmp = x_46_im * (((3.0 * fabs(x_46_re)) * fabs(x_46_re)) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_im * (fabs(x_46_re) * (fabs(x_46_re) + (2.0 * 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) <= 5.8d+164) then
        tmp = x_46im * (((3.0d0 * abs(x_46re)) * abs(x_46re)) - (x_46im * x_46im))
    else
        tmp = x_46im * (abs(x_46re) * (abs(x_46re) + (2.0d0 * 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) <= 5.8e+164) {
		tmp = x_46_im * (((3.0 * Math.abs(x_46_re)) * Math.abs(x_46_re)) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_im * (Math.abs(x_46_re) * (Math.abs(x_46_re) + (2.0 * Math.abs(x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if math.fabs(x_46_re) <= 5.8e+164:
		tmp = x_46_im * (((3.0 * math.fabs(x_46_re)) * math.fabs(x_46_re)) - (x_46_im * x_46_im))
	else:
		tmp = x_46_im * (math.fabs(x_46_re) * (math.fabs(x_46_re) + (2.0 * math.fabs(x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_re) <= 5.8e+164)
		tmp = Float64(x_46_im * Float64(Float64(Float64(3.0 * abs(x_46_re)) * abs(x_46_re)) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(abs(x_46_re) * Float64(abs(x_46_re) + Float64(2.0 * 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) <= 5.8e+164)
		tmp = x_46_im * (((3.0 * abs(x_46_re)) * abs(x_46_re)) - (x_46_im * x_46_im));
	else
		tmp = x_46_im * (abs(x_46_re) * (abs(x_46_re) + (2.0 * 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], 579999999999999970346306021154662620410815719091006025691049679413018715111325154758617509342024965348628887254730200477171420651107780304619988345189308120000626688], N[(x$46$im * N[(N[(N[(3 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] + N[(2 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.7999999999999997e164

   1. Initial program 83.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. lift-*.f64 N/A

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

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

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

      8. distribute-rgt-in N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. associate-+r+ N/A

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

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

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

      16. 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 85.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. remove-double-neg N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 88.2%

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

   5. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.2%

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

   7. Applied rewrites 88.2%

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

   if 5.7999999999999997e164 < x.re

   1. Initial program 83.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 91.2%

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

   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 72.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. sub-negate N/A

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

      7. add-flip N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. associate--l+ N/A

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

      15. add-flip N/A

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in x.im around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 50.2%

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

   11. Applied rewrites 50.2%

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

Alternative 4: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot \left|x.im\right|\\ t_1 := \left|x.im\right| \cdot \left(\left|x.re\right| \cdot \left(-1 \cdot \left|x.im\right|\right) - t\_0\right)\\ t_2 := \left(\left|x.re\right| \cdot \left|x.re\right| - t\_0\right) \cdot \left|x.im\right| + \left(\left|x.re\right| \cdot \left|x.im\right| + \left|x.im\right| \cdot \left|x.re\right|\right) \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq \frac{-809609013}{202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left|x.im\right| \cdot \left(\left|x.re\right| \cdot \left(\left|x.re\right| + 2 \cdot \left|x.re\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) (fabs x.im)))
       (t_1
        (* (fabs x.im) (- (* (fabs x.re) (* -1 (fabs x.im))) t_0)))
       (t_2
        (+
         (* (- (* (fabs x.re) (fabs x.re)) t_0) (fabs x.im))
         (*
          (+ (* (fabs x.re) (fabs x.im)) (* (fabs x.im) (fabs x.re)))
          (fabs x.re)))))
  (*
   (copysign 1 x.im)
   (if (<=
        t_2
        -809609013/202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784)
     t_1
     (if (<= t_2 INFINITY)
       (*
        (fabs x.im)
        (* (fabs x.re) (+ (fabs x.re) (* 2 (fabs x.re)))))
       t_1)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * fabs(x_46_im);
	double t_1 = fabs(x_46_im) * ((fabs(x_46_re) * (-1.0 * fabs(x_46_im))) - t_0);
	double t_2 = (((fabs(x_46_re) * fabs(x_46_re)) - t_0) * fabs(x_46_im)) + (((fabs(x_46_re) * fabs(x_46_im)) + (fabs(x_46_im) * fabs(x_46_re))) * fabs(x_46_re));
	double tmp;
	if (t_2 <= -4e-315) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fabs(x_46_im) * (fabs(x_46_re) * (fabs(x_46_re) + (2.0 * fabs(x_46_re))));
	} else {
		tmp = t_1;
	}
	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) * Math.abs(x_46_im);
	double t_1 = Math.abs(x_46_im) * ((Math.abs(x_46_re) * (-1.0 * Math.abs(x_46_im))) - t_0);
	double t_2 = (((Math.abs(x_46_re) * Math.abs(x_46_re)) - t_0) * Math.abs(x_46_im)) + (((Math.abs(x_46_re) * Math.abs(x_46_im)) + (Math.abs(x_46_im) * Math.abs(x_46_re))) * Math.abs(x_46_re));
	double tmp;
	if (t_2 <= -4e-315) {
		tmp = t_1;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.abs(x_46_im) * (Math.abs(x_46_re) * (Math.abs(x_46_re) + (2.0 * Math.abs(x_46_re))));
	} else {
		tmp = t_1;
	}
	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) * math.fabs(x_46_im)
	t_1 = math.fabs(x_46_im) * ((math.fabs(x_46_re) * (-1.0 * math.fabs(x_46_im))) - t_0)
	t_2 = (((math.fabs(x_46_re) * math.fabs(x_46_re)) - t_0) * math.fabs(x_46_im)) + (((math.fabs(x_46_re) * math.fabs(x_46_im)) + (math.fabs(x_46_im) * math.fabs(x_46_re))) * math.fabs(x_46_re))
	tmp = 0
	if t_2 <= -4e-315:
		tmp = t_1
	elif t_2 <= math.inf:
		tmp = math.fabs(x_46_im) * (math.fabs(x_46_re) * (math.fabs(x_46_re) + (2.0 * math.fabs(x_46_re))))
	else:
		tmp = t_1
	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) * abs(x_46_im))
	t_1 = Float64(abs(x_46_im) * Float64(Float64(abs(x_46_re) * Float64(-1.0 * abs(x_46_im))) - t_0))
	t_2 = Float64(Float64(Float64(Float64(abs(x_46_re) * abs(x_46_re)) - t_0) * abs(x_46_im)) + Float64(Float64(Float64(abs(x_46_re) * abs(x_46_im)) + Float64(abs(x_46_im) * abs(x_46_re))) * abs(x_46_re)))
	tmp = 0.0
	if (t_2 <= -4e-315)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(abs(x_46_im) * Float64(abs(x_46_re) * Float64(abs(x_46_re) + Float64(2.0 * abs(x_46_re)))));
	else
		tmp = t_1;
	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) * abs(x_46_im);
	t_1 = abs(x_46_im) * ((abs(x_46_re) * (-1.0 * abs(x_46_im))) - t_0);
	t_2 = (((abs(x_46_re) * abs(x_46_re)) - t_0) * abs(x_46_im)) + (((abs(x_46_re) * abs(x_46_im)) + (abs(x_46_im) * abs(x_46_re))) * abs(x_46_re));
	tmp = 0.0;
	if (t_2 <= -4e-315)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = abs(x_46_im) * (abs(x_46_re) * (abs(x_46_re) + (2.0 * abs(x_46_re))));
	else
		tmp = t_1;
	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] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[(N[Abs[x$46$re], $MachinePrecision] * N[(-1 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -809609013/202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784], t$95$1, If[LessEqual[t$95$2, Infinity], N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] + N[(2 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $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)) < -3.9999999988673917e-315 or +inf.0 < (+.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 83.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 91.2%

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

   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 72.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. sub-negate N/A

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

      7. add-flip N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. associate--l+ N/A

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

      15. add-flip N/A

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in x.re around 0

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

       1. lower-*.f64 59.1%

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

   11. Applied rewrites 59.1%

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

   if -3.9999999988673917e-315 < (+.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)) < +inf.0

   1. Initial program 83.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 91.2%

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

   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 72.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. sub-negate N/A

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

      7. add-flip N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. associate--l+ N/A

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

      15. add-flip N/A

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in x.im around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 50.2%

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

   11. Applied rewrites 50.2%

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

Alternative 5: 50.2% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (* x.im (* x.re (+ x.re (* 2 x.re)))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_re * (x_46_re + (2.0 * 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 * (x_46re + (2.0d0 * x_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_re * (x_46_re + (2.0 * x_46_re)));
}

Python

def code(x_46_re, x_46_im):
	return x_46_im * (x_46_re * (x_46_re + (2.0 * x_46_re)))

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_im * Float64(x_46_re * Float64(x_46_re + Float64(2.0 * x_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_im * (x_46_re * (x_46_re + (2.0 * x_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$im * N[(x$46$re * N[(x$46$re + N[(2 * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
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 91.2%

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

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 72.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. add-flip N/A

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

   5. sub-negate-rev N/A

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

   6. sub-negate N/A

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

   7. add-flip N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. lift-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. distribute-lft-in N/A

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

   14. associate--l+ N/A

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

   15. add-flip N/A

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

8. Applied rewrites 72.8%

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

9. Taylor expanded in x.im around 0

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. lower-*.f64 50.2%

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

11. Applied rewrites 50.2%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(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)))