math.cube on complex, imaginary part

Percentage Accurate: 83.1% → 97.9%

Time: 2.5s

Alternatives: 8

Speedup: 1.4×

• 44.5% 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: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.5 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.im\_m + x.im\_m\right) \cdot x.re\_m, x.re\_m, \left(x.im\_m + x.re\_m\right) \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 3.5e+196)
    (fma
     (* (+ x.im_m x.im_m) x.re_m)
     x.re_m
     (* (+ x.im_m x.re_m) (* (- x.re_m x.im_m) x.im_m)))
    (- (* (* x.im_m x.im_m) x.im_m)))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3.5e+196) {
		tmp = fma(((x_46_im_m + x_46_im_m) * x_46_re_m), x_46_re_m, ((x_46_im_m + x_46_re_m) * ((x_46_re_m - x_46_im_m) * x_46_im_m)));
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 3.5e+196)
		tmp = fma(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re_m), x_46_re_m, Float64(Float64(x_46_im_m + x_46_re_m) * Float64(Float64(x_46_re_m - x_46_im_m) * x_46_im_m)));
	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.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 3.5e+196], N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m + N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $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 2 regimes

2. if x.im < 3.4999999999999998e196

   1. Initial program 89.1%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. difference-of-squares N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 N/A

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

      22. lower--.f64 92.2

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

   3. Applied rewrites 92.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 3.4999999999999998e196 < x.im

   1. Initial program 55.9%

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

   2. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -{x.im}^{3} \]

      3. unpow3 N/A

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

      4. pow2 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      5. lower-*.f64 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      6. pow2 N/A

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

      7. lift-*.f64 91.8

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

   4. Applied rewrites 91.8%

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

Alternative 2: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m + x.im\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m \cdot x.im\_m, t\_0 \cdot x.re\_m\right)\\ \mathbf{elif}\;x.im\_m \leq 3.5 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x.re\_m, \left(\left(x.re\_m + x.im\_m\right) \cdot \left(x.re\_m - x.im\_m\right)\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* (+ x.im_m x.im_m) x.re_m)))
   (*
    x.im_s
    (if (<= x.im_m 7.2e-142)
      (fma x.re_m (* x.re_m x.im_m) (* t_0 x.re_m))
      (if (<= x.im_m 3.5e+196)
        (fma t_0 x.re_m (* (* (+ x.re_m x.im_m) (- x.re_m x.im_m)) x.im_m))
        (- (* (* x.im_m x.im_m) x.im_m)))))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double t_0 = (x_46_im_m + x_46_im_m) * x_46_re_m;
	double tmp;
	if (x_46_im_m <= 7.2e-142) {
		tmp = fma(x_46_re_m, (x_46_re_m * x_46_im_m), (t_0 * x_46_re_m));
	} else if (x_46_im_m <= 3.5e+196) {
		tmp = fma(t_0, x_46_re_m, (((x_46_re_m + x_46_im_m) * (x_46_re_m - x_46_im_m)) * x_46_im_m));
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re_m)
	tmp = 0.0
	if (x_46_im_m <= 7.2e-142)
		tmp = fma(x_46_re_m, Float64(x_46_re_m * x_46_im_m), Float64(t_0 * x_46_re_m));
	elseif (x_46_im_m <= 3.5e+196)
		tmp = fma(t_0, x_46_re_m, Float64(Float64(Float64(x_46_re_m + x_46_im_m) * Float64(x_46_re_m - x_46_im_m)) * x_46_im_m));
	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.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 7.2e-142], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(t$95$0 * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 3.5e+196], N[(t$95$0 * x$46$re$95$m + N[(N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $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 < 7.20000000000000001e-142

   1. Initial program 83.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 83.2

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

   6. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt1-in N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-+.f64 N/A

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

      22. count-2-rev N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 99.8%

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

   if 7.20000000000000001e-142 < x.im < 3.4999999999999998e196

   1. Initial program 92.0%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. difference-of-squares N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 N/A

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

      22. lower--.f64 96.7

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

   3. Applied rewrites 96.7%

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

   if 3.4999999999999998e196 < x.im

   1. Initial program 55.9%

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

   2. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -{x.im}^{3} \]

      3. unpow3 N/A

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

      4. pow2 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      5. lower-*.f64 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      6. pow2 N/A

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

      7. lift-*.f64 91.8

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

   4. Applied rewrites 91.8%

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

Alternative 3: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 8 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m \cdot x.im\_m, \left(\left(x.im\_m + x.im\_m\right) \cdot x.re\_m\right) \cdot x.re\_m\right)\\ \mathbf{elif}\;x.im\_m \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(-x.im\_m, x.im\_m, \left(x.re\_m \cdot x.re\_m\right) \cdot 3\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 8e-142)
    (fma x.re_m (* x.re_m x.im_m) (* (* (+ x.im_m x.im_m) x.re_m) x.re_m))
    (if (<= x.im_m 5e+195)
      (* (fma (- x.im_m) x.im_m (* (* x.re_m x.re_m) 3.0)) x.im_m)
      (- (* (* x.im_m x.im_m) x.im_m))))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 8e-142) {
		tmp = fma(x_46_re_m, (x_46_re_m * x_46_im_m), (((x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_re_m));
	} else if (x_46_im_m <= 5e+195) {
		tmp = fma(-x_46_im_m, x_46_im_m, ((x_46_re_m * x_46_re_m) * 3.0)) * x_46_im_m;
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 8e-142)
		tmp = fma(x_46_re_m, Float64(x_46_re_m * x_46_im_m), Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_re_m));
	elseif (x_46_im_m <= 5e+195)
		tmp = Float64(fma(Float64(-x_46_im_m), x_46_im_m, Float64(Float64(x_46_re_m * x_46_re_m) * 3.0)) * x_46_im_m);
	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.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 8e-142], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 5e+195], N[(N[((-x$46$im$95$m) * x$46$im$95$m + N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $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 < 8.0000000000000003e-142

   1. Initial program 83.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 83.2

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

   6. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt1-in N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-+.f64 N/A

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

      22. count-2-rev N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 99.8%

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

   if 8.0000000000000003e-142 < x.im < 4.9999999999999998e195

   1. Initial program 92.2%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

   if 4.9999999999999998e195 < x.im

   1. Initial program 55.8%

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

   2. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -{x.im}^{3} \]

      3. unpow3 N/A

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

      4. pow2 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      5. lower-*.f64 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      6. pow2 N/A

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

      7. lift-*.f64 91.5

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

   4. Applied rewrites 91.5%

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

Alternative 4: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;\left(x.re\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right) \cdot 3\\ \mathbf{elif}\;x.im\_m \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(-x.im\_m, x.im\_m, \left(x.re\_m \cdot x.re\_m\right) \cdot 3\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 7.2e-142)
    (* (* x.re_m (* x.re_m x.im_m)) 3.0)
    (if (<= x.im_m 5e+195)
      (* (fma (- x.im_m) x.im_m (* (* x.re_m x.re_m) 3.0)) x.im_m)
      (- (* (* x.im_m x.im_m) x.im_m))))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7.2e-142) {
		tmp = (x_46_re_m * (x_46_re_m * x_46_im_m)) * 3.0;
	} else if (x_46_im_m <= 5e+195) {
		tmp = fma(-x_46_im_m, x_46_im_m, ((x_46_re_m * x_46_re_m) * 3.0)) * x_46_im_m;
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 7.2e-142)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_re_m * x_46_im_m)) * 3.0);
	elseif (x_46_im_m <= 5e+195)
		tmp = Float64(fma(Float64(-x_46_im_m), x_46_im_m, Float64(Float64(x_46_re_m * x_46_re_m) * 3.0)) * x_46_im_m);
	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.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 7.2e-142], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[x$46$im$95$m, 5e+195], N[(N[((-x$46$im$95$m) * x$46$im$95$m + N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $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 < 7.20000000000000001e-142

   1. Initial program 83.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 83.2

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

   6. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if 7.20000000000000001e-142 < x.im < 4.9999999999999998e195

   1. Initial program 92.2%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

   if 4.9999999999999998e195 < x.im

   1. Initial program 55.8%

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

   2. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -{x.im}^{3} \]

      3. unpow3 N/A

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

      4. pow2 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      5. lower-*.f64 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      6. pow2 N/A

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

      7. lift-*.f64 91.5

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

   4. Applied rewrites 91.5%

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

Alternative 5: 96.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 7.1 \cdot 10^{+152}:\\ \;\;\;\;\left(\left(x.re\_m \cdot x.re\_m\right) \cdot 3 - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re_m 7.1e+152)
    (* (- (* (* x.re_m x.re_m) 3.0) (* x.im_m x.im_m)) x.im_m)
    (* (* x.re_m (* x.re_m x.im_m)) 3.0))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 7.1e+152) {
		tmp = (((x_46_re_m * x_46_re_m) * 3.0) - (x_46_im_m * x_46_im_m)) * x_46_im_m;
	} else {
		tmp = (x_46_re_m * (x_46_re_m * x_46_im_m)) * 3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m =     private
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_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 7.1d+152) then
        tmp = (((x_46re_m * x_46re_m) * 3.0d0) - (x_46im_m * x_46im_m)) * x_46im_m
    else
        tmp = (x_46re_m * (x_46re_m * x_46im_m)) * 3.0d0
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 7.1e+152) {
		tmp = (((x_46_re_m * x_46_re_m) * 3.0) - (x_46_im_m * x_46_im_m)) * x_46_im_m;
	} else {
		tmp = (x_46_re_m * (x_46_re_m * x_46_im_m)) * 3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 7.1e+152:
		tmp = (((x_46_re_m * x_46_re_m) * 3.0) - (x_46_im_m * x_46_im_m)) * x_46_im_m
	else:
		tmp = (x_46_re_m * (x_46_re_m * x_46_im_m)) * 3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 7.1e+152)
		tmp = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) * 3.0) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_re_m * x_46_im_m)) * 3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 7.1e+152)
		tmp = (((x_46_re_m * x_46_re_m) * 3.0) - (x_46_im_m * x_46_im_m)) * x_46_im_m;
	else
		tmp = (x_46_re_m * (x_46_re_m * x_46_im_m)) * 3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re$95$m, 7.1e+152], N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.10000000000000017e152

   1. Initial program 93.1%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.re around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 7.10000000000000017e152 < x.re

   1. Initial program 53.9%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 64.8

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

   4. Applied rewrites 64.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 64.9

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

   6. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 87.6

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

   8. Applied rewrites 87.6%

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

Alternative 6: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-296}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x.re\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (- (* (* x.im_m x.im_m) x.im_m)))
        (t_1
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (<= t_1 -4e-296)
      t_0
      (if (<= t_1 INFINITY) (* (* x.re_m (* x.re_m x.im_m)) 3.0) t_0)))))

C

x.re_m = fabs(x_46_re);
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_m, 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_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -4e-296) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_46_re_m * (x_46_re_m * x_46_im_m)) * 3.0;
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.re_m = Math.abs(x_46_re);
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_m, 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_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -4e-296) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re_m * (x_46_re_m * x_46_im_m)) * 3.0;
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m)
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if t_1 <= -4e-296:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (x_46_re_m * (x_46_re_m * x_46_im_m)) * 3.0
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, 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_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_1 <= -4e-296)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_re_m * x_46_im_m)) * 3.0);
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if (t_1 <= -4e-296)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (x_46_re_m * (x_46_re_m * x_46_im_m)) * 3.0;
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, 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$95$m * x$46$re$95$m), $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$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -4e-296], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * 3.0), $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)) < -4e-296 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 76.4%

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

   2. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -{x.im}^{3} \]

      3. unpow3 N/A

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

      4. pow2 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      5. lower-*.f64 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      6. pow2 N/A

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

      7. lift-*.f64 93.4

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

   4. Applied rewrites 93.4%

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

   if -4e-296 < (+.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 88.9%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 88.0

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

   4. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 88.0

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

   6. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.8

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

   8. Applied rewrites 98.8%

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

Alternative 7: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-296}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.re\_m \cdot \left(\left(3 \cdot x.im\_m\right) \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (- (* (* x.im_m x.im_m) x.im_m)))
        (t_1
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (<= t_1 -4e-296)
      t_0
      (if (<= t_1 INFINITY) (* x.re_m (* (* 3.0 x.im_m) x.re_m)) t_0)))))

C

x.re_m = fabs(x_46_re);
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_m, 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_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -4e-296) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_re_m * ((3.0 * x_46_im_m) * x_46_re_m);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.re_m = Math.abs(x_46_re);
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_m, 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_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -4e-296) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re_m * ((3.0 * x_46_im_m) * x_46_re_m);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m)
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if t_1 <= -4e-296:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = x_46_re_m * ((3.0 * x_46_im_m) * x_46_re_m)
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, 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_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_1 <= -4e-296)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_re_m * Float64(Float64(3.0 * x_46_im_m) * x_46_re_m));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if (t_1 <= -4e-296)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = x_46_re_m * ((3.0 * x_46_im_m) * x_46_re_m);
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, 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$95$m * x$46$re$95$m), $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$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -4e-296], t$95$0, If[LessEqual[t$95$1, Infinity], N[(x$46$re$95$m * N[(N[(3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $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)) < -4e-296 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 76.4%

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

   2. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -{x.im}^{3} \]

      3. unpow3 N/A

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

      4. pow2 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      5. lower-*.f64 N/A

         \[\leadsto -{x.im}^{2} \cdot x.im \]

      6. pow2 N/A

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

      7. lift-*.f64 93.4

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

   4. Applied rewrites 93.4%

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

   if -4e-296 < (+.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 88.9%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 88.0

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

   4. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 88.0

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

   6. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 98.8

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

   8. Applied rewrites 98.8%

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

Alternative 8: 58.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * -((x_46_im_m * x_46_im_m) * x_46_im_m);
}

Fortran

x.re_m =     private
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_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * -((x_46im_m * x_46im_m) * x_46im_m)
end function

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * -((x_46_im_m * x_46_im_m) * x_46_im_m);
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	return x_46_im_s * -((x_46_im_m * x_46_im_m) * x_46_im_m)

Julia

x.re_m = abs(x_46_re)
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_m, 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.re_m = abs(x_46_re);
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_m, x_46_im_m)
	tmp = x_46_im_s * -((x_46_im_m * x_46_im_m) * x_46_im_m);
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, 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. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

      \[\leadsto -{x.im}^{3} \]

   3. unpow3 N/A

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

   4. pow2 N/A

      \[\leadsto -{x.im}^{2} \cdot x.im \]

   5. lower-*.f64 N/A

      \[\leadsto -{x.im}^{2} \cdot x.im \]

   6. pow2 N/A

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

   7. lift-*.f64 58.4

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

4. Applied rewrites 58.4%

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

Developer Target 1: 91.7% 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 2025122 
(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)))