math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 96.7%

Time: 2.7s

Alternatives: 7

Speedup: 0.9×

• 44.2% 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: 82.7% 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: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re \cdot \left(x.im\_m + x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4.8 \cdot 10^{-176}:\\ \;\;\;\;\left(x.re + x.im\_m\right) \cdot \left(x.im\_m \cdot x.re\right) + t\_0 \cdot x.re\\ \mathbf{elif}\;x.im\_m \leq 1.55 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x.re, \left(\left(x.re + x.im\_m\right) \cdot \left(x.re - x.im\_m\right)\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-e^{3 \cdot \log x.im\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re (+ x.im_m x.im_m))))
   (*
    x.im_s
    (if (<= x.im_m 4.8e-176)
      (+ (* (+ x.re x.im_m) (* x.im_m x.re)) (* t_0 x.re))
      (if (<= x.im_m 1.55e+202)
        (fma t_0 x.re (* (* (+ x.re x.im_m) (- x.re x.im_m)) x.im_m))
        (- (exp (* 3.0 (log x.im_m)))))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_re * (x_46_im_m + x_46_im_m);
	double tmp;
	if (x_46_im_m <= 4.8e-176) {
		tmp = ((x_46_re + x_46_im_m) * (x_46_im_m * x_46_re)) + (t_0 * x_46_re);
	} else if (x_46_im_m <= 1.55e+202) {
		tmp = fma(t_0, x_46_re, (((x_46_re + x_46_im_m) * (x_46_re - x_46_im_m)) * x_46_im_m));
	} else {
		tmp = -exp((3.0 * log(x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(x_46_re * Float64(x_46_im_m + x_46_im_m))
	tmp = 0.0
	if (x_46_im_m <= 4.8e-176)
		tmp = Float64(Float64(Float64(x_46_re + x_46_im_m) * Float64(x_46_im_m * x_46_re)) + Float64(t_0 * x_46_re));
	elseif (x_46_im_m <= 1.55e+202)
		tmp = fma(t_0, x_46_re, Float64(Float64(Float64(x_46_re + x_46_im_m) * Float64(x_46_re - x_46_im_m)) * x_46_im_m));
	else
		tmp = Float64(-exp(Float64(3.0 * log(x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4.8e-176], N[(N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 1.55e+202], N[(t$95$0 * x$46$re + N[(N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], (-N[Exp[N[(3.0 * N[Log[x$46$im$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.im < 4.80000000000000012e-176

   1. Initial program 82.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 84.6

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 84.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 84.6

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. lower--.f64 87.7

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

   3. Applied rewrites 87.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. difference-of-squares-rev N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-+.f64 82.7

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-*.f64 60.1

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

   8. Applied rewrites 60.1%

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

   if 4.80000000000000012e-176 < x.im < 1.54999999999999996e202

   1. Initial program 82.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 84.6

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 84.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 84.6

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. lower--.f64 87.7

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

   3. Applied rewrites 87.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if 1.54999999999999996e202 < x.im

   1. Initial program 82.7%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 59.6

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

   4. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 59.6

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 59.5

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

   6. Applied rewrites 59.5%

      \[\leadsto -\left(x.im \cdot x.im\right) \cdot x.im \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. exp-to-pow N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-exp.f64 58.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 58.0

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

   8. Applied rewrites 58.0%

      \[\leadsto -e^{3 \cdot \log x.im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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)) < +inf.0

   1. Initial program 82.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 84.6

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 84.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 84.6

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. lower--.f64 87.7

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

   3. Applied rewrites 87.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 91.4

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if +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 82.7%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 59.6

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

   4. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 59.6

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 59.5

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

   6. Applied rewrites 59.5%

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

Alternative 3: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;-{x.im\_m}^{3}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re + x.im\_m\right) \cdot \left(x.im\_m \cdot x.re\right) + \left(x.re \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_0 -5e-324)
      (- (pow x.im_m 3.0))
      (if (<= t_0 INFINITY)
        (+
         (* (+ x.re x.im_m) (* x.im_m x.re))
         (* (* x.re (+ x.im_m x.im_m)) x.re))
        (- (* (* x.im_m x.im_m) x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= -5e-324) {
		tmp = -pow(x_46_im_m, 3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((x_46_re + x_46_im_m) * (x_46_im_m * x_46_re)) + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= -5e-324) {
		tmp = -Math.pow(x_46_im_m, 3.0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_re + x_46_im_m) * (x_46_im_m * x_46_re)) + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if t_0 <= -5e-324:
		tmp = -math.pow(x_46_im_m, 3.0)
	elif t_0 <= math.inf:
		tmp = ((x_46_re + x_46_im_m) * (x_46_im_m * x_46_re)) + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re)
	else:
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= -5e-324)
		tmp = Float64(-(x_46_im_m ^ 3.0));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(x_46_re + x_46_im_m) * Float64(x_46_im_m * x_46_re)) + Float64(Float64(x_46_re * Float64(x_46_im_m + x_46_im_m)) * x_46_re));
	else
		tmp = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if (t_0 <= -5e-324)
		tmp = -(x_46_im_m ^ 3.0);
	elseif (t_0 <= Inf)
		tmp = ((x_46_re + x_46_im_m) * (x_46_im_m * x_46_re)) + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	else
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -5e-324], (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), If[LessEqual[t$95$0, Infinity], N[(N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.94066e-324

   1. Initial program 82.7%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 59.6

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

   4. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 59.6

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 59.5

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

   6. Applied rewrites 59.5%

      \[\leadsto -\left(x.im \cdot x.im\right) \cdot x.im \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 59.6

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

   8. Applied rewrites 59.6%

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

   if -4.94066e-324 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 84.6

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 84.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 84.6

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. lower--.f64 87.7

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

   3. Applied rewrites 87.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. difference-of-squares-rev N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-+.f64 82.7

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-*.f64 60.1

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

   8. Applied rewrites 60.1%

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

   if +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 82.7%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 59.6

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

   4. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 59.6

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 59.5

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

   6. Applied rewrites 59.5%

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

Alternative 4: 74.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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)) < +inf.0

   1. Initial program 82.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 84.6

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 84.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 84.6

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

      15. lift--.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. lower--.f64 87.7

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

   3. Applied rewrites 87.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 91.4

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 68.0%

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

   if +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 82.7%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 59.6

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

   4. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 59.6

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 59.5

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

   6. Applied rewrites 59.5%

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

Alternative 5: 74.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;-{x.im\_m}^{3}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m, x.im\_m \cdot x.re, \left(\left(x.re \cdot x.re\right) \cdot x.im\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_0 -5e-324)
      (- (pow x.im_m 3.0))
      (if (<= t_0 INFINITY)
        (fma x.im_m (* x.im_m x.re) (* (* (* x.re x.re) x.im_m) 2.0))
        (- (* (* x.im_m x.im_m) x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= -5e-324) {
		tmp = -pow(x_46_im_m, 3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x_46_im_m, (x_46_im_m * x_46_re), (((x_46_re * x_46_re) * x_46_im_m) * 2.0));
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= -5e-324)
		tmp = Float64(-(x_46_im_m ^ 3.0));
	elseif (t_0 <= Inf)
		tmp = fma(x_46_im_m, Float64(x_46_im_m * x_46_re), Float64(Float64(Float64(x_46_re * x_46_re) * x_46_im_m) * 2.0));
	else
		tmp = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -5e-324], (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), If[LessEqual[t$95$0, Infinity], N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * 2.0), $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 (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.94066e-324

   1. Initial program 82.7%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 59.6

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

   4. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 59.6

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 59.5

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

   6. Applied rewrites 59.5%

      \[\leadsto -\left(x.im \cdot x.im\right) \cdot x.im \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 59.6

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

   8. Applied rewrites 59.6%

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

   if -4.94066e-324 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.7%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 82.6

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

   4. Applied rewrites 82.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 85.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 85.8

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 85.8

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

   6. Applied rewrites 85.8%

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

   7. Taylor expanded in x.re around 0

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

   9. Applied rewrites 67.2%

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

   10. Taylor expanded in x.re around inf

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

       1. lower-*.f64 36.5

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

   12. Applied rewrites 36.5%

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

   if +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 82.7%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 59.6

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

   4. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 59.6

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 59.5

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

   6. Applied rewrites 59.5%

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

Alternative 6: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m, x.im\_m \cdot x.re, \left(\left(x.re \cdot x.re\right) \cdot x.im\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* (* x.im_m x.im_m) x.im_m)))
        (t_1
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_1 -5e-324)
      t_0
      (if (<= t_1 INFINITY)
        (fma x.im_m (* x.im_m x.re) (* (* (* x.re x.re) x.im_m) 2.0))
        t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	double t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -5e-324) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(x_46_im_m, (x_46_im_m * x_46_re), (((x_46_re * x_46_re) * x_46_im_m) * 2.0));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m))
	t_1 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_1 <= -5e-324)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = fma(x_46_im_m, Float64(x_46_im_m * x_46_re), Float64(Float64(Float64(x_46_re * x_46_re) * x_46_im_m) * 2.0));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -5e-324], t$95$0, If[LessEqual[t$95$1, Infinity], N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.94066e-324 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 82.7%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 59.6

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

   4. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 59.6

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 59.5

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

   6. Applied rewrites 59.5%

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

   if -4.94066e-324 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.7%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 82.6

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

   4. Applied rewrites 82.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 85.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 85.8

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 85.8

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

   6. Applied rewrites 85.8%

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

   7. Taylor expanded in x.re around 0

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

   9. Applied rewrites 67.2%

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

   10. Taylor expanded in x.re around inf

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

       1. lower-*.f64 36.5

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

   12. Applied rewrites 36.5%

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

Alternative 7: 59.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

x.im\_m =     private
x.im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46im_s, x_46re, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * -((x_46im_m * x_46im_m) * x_46im_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

2. Taylor expanded in x.re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 59.6

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

4. Applied rewrites 59.6%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 59.6

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

   4. lift-pow.f64 N/A

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

   5. unpow3 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 59.5

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

6. Applied rewrites 59.5%

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

Developer Target 1: 91.3% 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 2025149 
(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)))