math.cube on complex, imaginary part

Percentage Accurate: 83.1% → 98.1%

Time: 3.5s

Alternatives: 5

Speedup: 1.4×

• 44.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

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: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

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: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 900000000:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(3 \cdot x.re\right) - \left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{elif}\;x.im\_m \leq 1.5 \cdot 10^{+212}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.re - x.im\_m, x.im\_m + x.re, \left(x.re + x.re\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 900000000.0)
    (- (* (* x.re x.im_m) (* 3.0 x.re)) (* (* x.im_m x.im_m) x.im_m))
    (if (<= x.im_m 1.5e+212)
      (* x.im_m (fma (- x.re x.im_m) (+ x.im_m x.re) (* (+ x.re x.re) x.re)))
      (* (* (- x.im_m) x.im_m) x.im_m)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 900000000.0) {
		tmp = ((x_46_re * x_46_im_m) * (3.0 * x_46_re)) - ((x_46_im_m * x_46_im_m) * x_46_im_m);
	} else if (x_46_im_m <= 1.5e+212) {
		tmp = x_46_im_m * fma((x_46_re - x_46_im_m), (x_46_im_m + x_46_re), ((x_46_re + x_46_re) * x_46_re));
	} else {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 900000000.0)
		tmp = Float64(Float64(Float64(x_46_re * x_46_im_m) * Float64(3.0 * x_46_re)) - Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m));
	elseif (x_46_im_m <= 1.5e+212)
		tmp = Float64(x_46_im_m * fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m + x_46_re), Float64(Float64(x_46_re + x_46_re) * x_46_re)));
	else
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 900000000.0], N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(3.0 * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 1.5e+212], N[(x$46$im$95$m * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision] + N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 9e8

   1. Initial program 83.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 86.4%

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 86.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.4

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-lft1-in N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if 9e8 < x.im < 1.5e212

   1. Initial program 83.1%

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 91.6%

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

   if 1.5e212 < x.im

   1. Initial program 83.1%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 58.8

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

   4. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 58.7

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

   6. Applied rewrites 58.7%

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

Alternative 2: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-313}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(3 \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* (* (- x.im_m) x.im_m) x.im_m))
        (t_1
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_1 -2e-313)
      t_0
      (if (<= t_1 INFINITY) (* (* x.re x.im_m) (* 3.0 x.re)) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-313) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_46_re * x_46_im_m) * (3.0 * x_46_re);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-313) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re * x_46_im_m) * (3.0 * x_46_re);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m
	t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if t_1 <= -2e-313:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (x_46_re * x_46_im_m) * (3.0 * x_46_re)
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m)
	t_1 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_1 <= -2e-313)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(3.0 * x_46_re));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if (t_1 <= -2e-313)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (x_46_re * x_46_im_m) * (3.0 * x_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-313], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(3.0 * x$46$re), $MachinePrecision]), $MachinePrecision], t$95$0]]), $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)) < -1.99999999998e-313 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 83.1%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 58.8

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

   4. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 58.7

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

   6. Applied rewrites 58.7%

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

   if -1.99999999998e-313 < (+.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 83.1%

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

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

   3. Applied rewrites 88.4%

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

   4. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re + 2 \cdot x.re\right)\right)} \]
   5. 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 50.5

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

   6. Applied rewrites 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 56.1

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

   8. Applied rewrites 56.1%

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

Alternative 3: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 2.05 \cdot 10^{+151}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.re \cdot x.re, 3, \left(-x.im\_m\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 2.05e+151)
    (* x.im_m (fma (* x.re x.re) 3.0 (* (- x.im_m) x.im_m)))
    (* (* (* 3.0 x.im_m) x.re) x.re))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 2.05e+151) {
		tmp = x_46_im_m * fma((x_46_re * x_46_re), 3.0, (-x_46_im_m * x_46_im_m));
	} else {
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 2.05e+151)
		tmp = Float64(x_46_im_m * fma(Float64(x_46_re * x_46_re), 3.0, Float64(Float64(-x_46_im_m) * x_46_im_m)));
	else
		tmp = Float64(Float64(Float64(3.0 * x_46_im_m) * x_46_re) * x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 2.05e+151], N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0 + N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.0499999999999999e151

   1. Initial program 83.1%

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   3. Applied rewrites 91.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift--.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r- N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. count-2 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. lift-neg.f64 N/A

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 88.2%

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

   if 2.0499999999999999e151 < x.re

   1. Initial program 83.1%

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

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

   3. Applied rewrites 88.4%

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

   4. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re + 2 \cdot x.re\right)\right)} \]
   5. 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 50.5

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

   6. Applied rewrites 50.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 56.1

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 56.1

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

   8. Applied rewrites 56.1%

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

Alternative 4: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-313}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* (* (- x.im_m) x.im_m) x.im_m))
        (t_1
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_1 -2e-313)
      t_0
      (if (<= t_1 INFINITY) (* x.im_m (* (* x.re x.re) 3.0)) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-313) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_im_m * ((x_46_re * x_46_re) * 3.0);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-313) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_im_m * ((x_46_re * x_46_re) * 3.0);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m
	t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if t_1 <= -2e-313:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = x_46_im_m * ((x_46_re * x_46_re) * 3.0)
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m)
	t_1 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_1 <= -2e-313)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) * 3.0));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if (t_1 <= -2e-313)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = x_46_im_m * ((x_46_re * x_46_re) * 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-313], t$95$0, If[LessEqual[t$95$1, Infinity], N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]), $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)) < -1.99999999998e-313 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 83.1%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 58.8

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

   4. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 58.7

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

   6. Applied rewrites 58.7%

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

   if -1.99999999998e-313 < (+.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 83.1%

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

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

   3. Applied rewrites 88.4%

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

   4. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re + 2 \cdot x.re\right)\right)} \]
   5. 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 50.5

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

   6. Applied rewrites 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 50.5

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

   8. Applied rewrites 50.5%

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

Alternative 5: 58.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \left(\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\right) \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (* x.im_s (* (* (- x.im_m) x.im_m) x.im_m)))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * ((-x_46_im_m * x_46_im_m) * x_46_im_m);
}

Fortran

x.im\_m =     private
x.im\_s =     private
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_46im_s, x_46re, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * ((-x_46im_m * x_46im_m) * x_46im_m)
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * ((-x_46_im_m * x_46_im_m) * x_46_im_m);
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	return x_46_im_s * ((-x_46_im_m * x_46_im_m) * x_46_im_m)

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m))
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * ((-x_46_im_m * x_46_im_m) * x_46_im_m);
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

2. Taylor expanded in x.re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 58.8

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

4. Applied rewrites 58.8%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow3 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-neg.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 58.7

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

6. Applied rewrites 58.7%

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

Developer Target 1: 91.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025155 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform c (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))