math.cube on complex, real part

Percentage Accurate: 83.0% → 97.9%

Time: 3.2s

Alternatives: 6

Speedup: N/A×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, N/A× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 7.8 \cdot 10^{+219}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(x.im\_m \cdot x.re\_m\right), x.im\_m, \left(x.im\_m + x.re\_m\right) \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 7.8e+219)
    (fma
     (* -2.0 (* x.im_m x.re_m))
     x.im_m
     (* (+ x.im_m x.re_m) (* (- x.re_m x.im_m) x.re_m)))
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 7.8e+219) {
		tmp = fma((-2.0 * (x_46_im_m * x_46_re_m)), x_46_im_m, ((x_46_im_m + x_46_re_m) * ((x_46_re_m - x_46_im_m) * x_46_re_m)));
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 7.8e+219)
		tmp = fma(Float64(-2.0 * Float64(x_46_im_m * x_46_re_m)), x_46_im_m, Float64(Float64(x_46_im_m + x_46_re_m) * Float64(Float64(x_46_re_m - x_46_im_m) * x_46_re_m)));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 7.8e+219], N[(N[(-2.0 * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$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$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.7999999999999998e219

   1. Initial program 87.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. difference-of-squares N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 90.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift--.f64 97.3

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

   6. Applied rewrites 97.3%

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

   if 7.7999999999999998e219 < x.re

   1. Initial program 33.3%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. unpow3 N/A

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

      2. pow2 N/A

         \[\leadsto {x.re}^{2} \cdot x.re \]

      3. lower-*.f64 N/A

         \[\leadsto {x.re}^{2} \cdot \color{blue}{x.re} \]

      4. pow2 N/A

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

      5. lift-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

Alternative 2: 96.4% accurate, N/A× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3 \cdot x.re\_m\right) \cdot x.im\_m, x.im\_m, t\_0\right)\\ \mathbf{elif}\;x.re\_m \leq 1.2 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left({x.re\_m}^{1.5}, {x.re\_m}^{0.75} \cdot {x.re\_m}^{0.75}, \left(x.re\_m \cdot -3\right) \cdot \left(x.im\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* (* x.re_m x.re_m) x.re_m)))
   (*
    x.re_s
    (if (<= x.re_m 2e+94)
      (fma (* (* -3.0 x.re_m) x.im_m) x.im_m t_0)
      (if (<= x.re_m 1.2e+175)
        (fma
         (pow x.re_m 1.5)
         (* (pow x.re_m 0.75) (pow x.re_m 0.75))
         (* (* x.re_m -3.0) (* x.im_m x.im_m)))
        t_0)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (x_46_re_m * x_46_re_m) * x_46_re_m;
	double tmp;
	if (x_46_re_m <= 2e+94) {
		tmp = fma(((-3.0 * x_46_re_m) * x_46_im_m), x_46_im_m, t_0);
	} else if (x_46_re_m <= 1.2e+175) {
		tmp = fma(pow(x_46_re_m, 1.5), (pow(x_46_re_m, 0.75) * pow(x_46_re_m, 0.75)), ((x_46_re_m * -3.0) * (x_46_im_m * x_46_im_m)));
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m)
	tmp = 0.0
	if (x_46_re_m <= 2e+94)
		tmp = fma(Float64(Float64(-3.0 * x_46_re_m) * x_46_im_m), x_46_im_m, t_0);
	elseif (x_46_re_m <= 1.2e+175)
		tmp = fma((x_46_re_m ^ 1.5), Float64((x_46_re_m ^ 0.75) * (x_46_re_m ^ 0.75)), Float64(Float64(x_46_re_m * -3.0) * Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = t_0;
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2e+94], N[(N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m + t$95$0), $MachinePrecision], If[LessEqual[x$46$re$95$m, 1.2e+175], N[(N[Power[x$46$re$95$m, 1.5], $MachinePrecision] * N[(N[Power[x$46$re$95$m, 0.75], $MachinePrecision] * N[Power[x$46$re$95$m, 0.75], $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re$95$m * -3.0), $MachinePrecision] * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.re < 2e94

   1. Initial program 87.9%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. sqr-pow N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 42.3

          \[\leadsto \mathsf{fma}\left({x.re}^{1.5}, {x.re}^{1.5}, \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\right) \]

   5. Applied rewrites 42.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.re}^{1.5}, {x.re}^{1.5}, \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\right)} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow-prod-up N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if 2e94 < x.re < 1.2e175

   1. Initial program 86.7%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. sqr-pow N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 93.3

          \[\leadsto \mathsf{fma}\left({x.re}^{1.5}, {x.re}^{1.5}, \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\right) \]

   5. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.re}^{1.5}, {x.re}^{1.5}, \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\right)} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x.re}^{\frac{3}{2}}, {x.re}^{\frac{3}{4}} \cdot {x.re}^{\left(\frac{\frac{3}{2}}{2}\right)}, \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left({x.re}^{\frac{3}{2}}, {x.re}^{\frac{3}{4}} \cdot {x.re}^{\color{blue}{\left(\frac{\frac{3}{2}}{2}\right)}}, \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\right) \]

      7. metadata-eval 93.3

         \[\leadsto \mathsf{fma}\left({x.re}^{1.5}, {x.re}^{0.75} \cdot {x.re}^{0.75}, \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\right) \]

   7. Applied rewrites 93.3%

      \[\leadsto \mathsf{fma}\left({x.re}^{1.5}, {x.re}^{0.75} \cdot \color{blue}{{x.re}^{0.75}}, \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\right) \]

   if 1.2e175 < x.re

   1. Initial program 43.8%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. unpow3 N/A

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

      2. pow2 N/A

         \[\leadsto {x.re}^{2} \cdot x.re \]

      3. lower-*.f64 N/A

         \[\leadsto {x.re}^{2} \cdot \color{blue}{x.re} \]

      4. pow2 N/A

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

      5. lift-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

Alternative 3: 93.9% accurate, N/A× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot \left(-1 \cdot x.im\_m\right) \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3 \cdot x.re\_m\right) \cdot x.im\_m, x.im\_m, \left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (+
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) (* -1.0 x.im_m)))
       2e-124)
    (fma (* (* -3.0 x.re_m) x.im_m) x.im_m (* (* x.re_m x.re_m) x.re_m))
    (pow x.re_m 3.0))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * (-1.0 * x_46_im_m))) <= 2e-124) {
		tmp = fma(((-3.0 * x_46_re_m) * x_46_im_m), x_46_im_m, ((x_46_re_m * x_46_re_m) * x_46_re_m));
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * Float64(-1.0 * x_46_im_m))) <= 2e-124)
		tmp = fma(Float64(Float64(-3.0 * x_46_re_m) * x_46_im_m), x_46_im_m, Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m));
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[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$46$re$95$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] * N[(-1.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-124], N[(N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m + N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.99999999999999987e-124

   1. Initial program 95.3%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. sqr-pow N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 49.2

          \[\leadsto \mathsf{fma}\left({x.re}^{1.5}, {x.re}^{1.5}, \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\right) \]

   5. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.re}^{1.5}, {x.re}^{1.5}, \left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\right)} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow-prod-up N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if 1.99999999999999987e-124 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 65.7%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. unpow3 N/A

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

      2. pow2 N/A

         \[\leadsto {x.re}^{2} \cdot x.re \]

      3. lower-*.f64 N/A

         \[\leadsto {x.re}^{2} \cdot \color{blue}{x.re} \]

      4. pow2 N/A

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

      5. lift-*.f64 53.4

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

   5. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

         \[\leadsto {x.re}^{\color{blue}{3}} \]

      4. lower-pow.f64 53.5

         \[\leadsto {x.re}^{\color{blue}{3}} \]

   7. Applied rewrites 53.5%

      \[\leadsto {x.re}^{\color{blue}{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(-1 \cdot x.im\right) \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3 \cdot x.re\right) \cdot x.im, x.im, \left(x.re \cdot x.re\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.1% accurate, N/A× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot \left(-1 \cdot x.im\_m\right) \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\left(\mathsf{fma}\left(x.im\_m \cdot x.im\_m, -1, x.re\_m \cdot x.re\_m\right) - \left(x.im\_m \cdot x.im\_m\right) \cdot 2\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3}\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (+
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) (* -1.0 x.im_m)))
       2e-124)
    (*
     (-
      (fma (* x.im_m x.im_m) -1.0 (* x.re_m x.re_m))
      (* (* x.im_m x.im_m) 2.0))
     x.re_m)
    (pow x.re_m 3.0))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * (-1.0 * x_46_im_m))) <= 2e-124) {
		tmp = (fma((x_46_im_m * x_46_im_m), -1.0, (x_46_re_m * x_46_re_m)) - ((x_46_im_m * x_46_im_m) * 2.0)) * x_46_re_m;
	} else {
		tmp = pow(x_46_re_m, 3.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * Float64(-1.0 * x_46_im_m))) <= 2e-124)
		tmp = Float64(Float64(fma(Float64(x_46_im_m * x_46_im_m), -1.0, Float64(x_46_re_m * x_46_re_m)) - Float64(Float64(x_46_im_m * x_46_im_m) * 2.0)) * x_46_re_m);
	else
		tmp = x_46_re_m ^ 3.0;
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[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$46$re$95$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] * N[(-1.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-124], N[(N[(N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -1.0 + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[Power[x$46$re$95$m, 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.99999999999999987e-124

   1. Initial program 95.3%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if 1.99999999999999987e-124 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 65.7%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. unpow3 N/A

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

      2. pow2 N/A

         \[\leadsto {x.re}^{2} \cdot x.re \]

      3. lower-*.f64 N/A

         \[\leadsto {x.re}^{2} \cdot \color{blue}{x.re} \]

      4. pow2 N/A

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

      5. lift-*.f64 53.4

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

   5. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

         \[\leadsto {x.re}^{\color{blue}{3}} \]

      4. lower-pow.f64 53.5

         \[\leadsto {x.re}^{\color{blue}{3}} \]

   7. Applied rewrites 53.5%

      \[\leadsto {x.re}^{\color{blue}{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(-1 \cdot x.im\right) \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\left(\mathsf{fma}\left(x.im \cdot x.im, -1, x.re \cdot x.re\right) - \left(x.im \cdot x.im\right) \cdot 2\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, N/A× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot \left(-1 \cdot x.im\_m\right) \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\left(\mathsf{fma}\left(x.im\_m \cdot x.im\_m, -1, x.re\_m \cdot x.re\_m\right) - \left(x.im\_m \cdot x.im\_m\right) \cdot 2\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (+
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) (* -1.0 x.im_m)))
       2e-124)
    (*
     (-
      (fma (* x.im_m x.im_m) -1.0 (* x.re_m x.re_m))
      (* (* x.im_m x.im_m) 2.0))
     x.re_m)
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * (-1.0 * x_46_im_m))) <= 2e-124) {
		tmp = (fma((x_46_im_m * x_46_im_m), -1.0, (x_46_re_m * x_46_re_m)) - ((x_46_im_m * x_46_im_m) * 2.0)) * x_46_re_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * Float64(-1.0 * x_46_im_m))) <= 2e-124)
		tmp = Float64(Float64(fma(Float64(x_46_im_m * x_46_im_m), -1.0, Float64(x_46_re_m * x_46_re_m)) - Float64(Float64(x_46_im_m * x_46_im_m) * 2.0)) * x_46_re_m);
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[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$46$re$95$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] * N[(-1.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-124], N[(N[(N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -1.0 + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.99999999999999987e-124

   1. Initial program 95.3%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if 1.99999999999999987e-124 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 65.7%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. unpow3 N/A

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

      2. pow2 N/A

         \[\leadsto {x.re}^{2} \cdot x.re \]

      3. lower-*.f64 N/A

         \[\leadsto {x.re}^{2} \cdot \color{blue}{x.re} \]

      4. pow2 N/A

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

      5. lift-*.f64 53.4

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

   5. Applied rewrites 53.4%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(-1 \cdot x.im\right) \leq 2 \cdot 10^{-124}:\\ \;\;\;\;\left(\mathsf{fma}\left(x.im \cdot x.im, -1, x.re \cdot x.re\right) - \left(x.im \cdot x.im\right) \cdot 2\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.9% accurate, N/A× speedup

Math

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

FPCore

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

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m);
}

Fortran

x.im_m =     private
x.re\_m =     private
x.re\_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_46re_s, x_46re_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * ((x_46re_m * x_46re_m) * x_46re_m)
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m);
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m)

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m);
end

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Taylor expanded in x.re around inf

   \[\leadsto \color{blue}{{x.re}^{3}} \]
4. Step-by-step derivation

   1. unpow3 N/A

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

   2. pow2 N/A

      \[\leadsto {x.re}^{2} \cdot x.re \]

   3. lower-*.f64 N/A

      \[\leadsto {x.re}^{2} \cdot \color{blue}{x.re} \]

   4. pow2 N/A

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

   5. lift-*.f64 58.4

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

5. Applied rewrites 58.4%

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

Reproduce

herbie shell --seed 2025066 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform c (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

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