math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 97.2%

Time: 2.0s

Alternatives: 10

Speedup: 1.0×

• 45.2% 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: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.7 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(-1, {x.im\_m}^{3}, x.re\_m \cdot \mathsf{fma}\left(x.im\_m, x.im\_m + -1 \cdot x.im\_m, x.re\_m \cdot \left(x.im\_m + 2 \cdot x.im\_m\right)\right)\right)\\ \mathbf{elif}\;x.im\_m \leq 2.95 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x.im\_m, x.re\_m, x.im\_m \cdot x.re\_m\right), x.re\_m, \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.7e+100)
    (fma
     -1.0
     (pow x.im_m 3.0)
     (*
      x.re_m
      (fma
       x.im_m
       (+ x.im_m (* -1.0 x.im_m))
       (* x.re_m (+ x.im_m (* 2.0 x.im_m))))))
    (if (<= x.im_m 2.95e+246)
      (fma
       (fma x.im_m x.re_m (* x.im_m x.re_m))
       x.re_m
       (* (* x.im_m (- x.re_m x.im_m)) x.im_m))
      (- (* (* x.im_m x.im_m) x.im_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.7e+100) {
		tmp = fma(-1.0, pow(x_46_im_m, 3.0), (x_46_re_m * fma(x_46_im_m, (x_46_im_m + (-1.0 * x_46_im_m)), (x_46_re_m * (x_46_im_m + (2.0 * x_46_im_m))))));
	} else if (x_46_im_m <= 2.95e+246) {
		tmp = fma(fma(x_46_im_m, x_46_re_m, (x_46_im_m * x_46_re_m)), x_46_re_m, ((x_46_im_m * (x_46_re_m - x_46_im_m)) * x_46_im_m));
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.7e+100)
		tmp = fma(-1.0, (x_46_im_m ^ 3.0), Float64(x_46_re_m * fma(x_46_im_m, Float64(x_46_im_m + Float64(-1.0 * x_46_im_m)), Float64(x_46_re_m * Float64(x_46_im_m + Float64(2.0 * x_46_im_m))))));
	elseif (x_46_im_m <= 2.95e+246)
		tmp = fma(fma(x_46_im_m, x_46_re_m, Float64(x_46_im_m * x_46_re_m)), x_46_re_m, Float64(Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)) * x_46_im_m));
	else
		tmp = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.7e+100], N[(-1.0 * N[Power[x$46$im$95$m, 3.0], $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$im$95$m + N[(-1.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$im$95$m + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 2.95e+246], N[(N[(x$46$im$95$m * x$46$re$95$m + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m + N[(N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 1.69999999999999997e100

   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 84.9

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

      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 84.9

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

      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 84.9

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

      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 88.1

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in x.re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 86.0

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

   6. Applied rewrites 86.0%

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

   if 1.69999999999999997e100 < x.im < 2.94999999999999988e246

   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 84.9

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

      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 84.9

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

      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 84.9

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

      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 88.1

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in x.re around 0

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

   6. Applied rewrites 74.0%

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

   if 2.94999999999999988e246 < 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 59.5

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   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 59.5

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

      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 59.4

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

   6. Applied rewrites 59.4%

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

Alternative 2: 97.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0 \cdot x.im\_m, x.im\_m, \left(3 \cdot x.im\_m\right) \cdot x.re\_m\right), x.re\_m, t\_0\right)\\ \mathbf{elif}\;x.im\_m \leq 2.95 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x.im\_m, x.re\_m, x.im\_m \cdot x.re\_m\right), x.re\_m, \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (- (* (* x.im_m x.im_m) x.im_m))))
   (*
    x.im_s
    (if (<= x.im_m 4e+73)
      (fma (fma (* 0.0 x.im_m) x.im_m (* (* 3.0 x.im_m) x.re_m)) x.re_m t_0)
      (if (<= x.im_m 2.95e+246)
        (fma
         (fma x.im_m x.re_m (* x.im_m x.re_m))
         x.re_m
         (* (* x.im_m (- x.re_m x.im_m)) x.im_m))
        t_0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	double tmp;
	if (x_46_im_m <= 4e+73) {
		tmp = fma(fma((0.0 * x_46_im_m), x_46_im_m, ((3.0 * x_46_im_m) * x_46_re_m)), x_46_re_m, t_0);
	} else if (x_46_im_m <= 2.95e+246) {
		tmp = fma(fma(x_46_im_m, x_46_re_m, (x_46_im_m * x_46_re_m)), x_46_re_m, ((x_46_im_m * (x_46_re_m - x_46_im_m)) * x_46_im_m));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m))
	tmp = 0.0
	if (x_46_im_m <= 4e+73)
		tmp = fma(fma(Float64(0.0 * x_46_im_m), x_46_im_m, Float64(Float64(3.0 * x_46_im_m) * x_46_re_m)), x_46_re_m, t_0);
	elseif (x_46_im_m <= 2.95e+246)
		tmp = fma(fma(x_46_im_m, x_46_re_m, Float64(x_46_im_m * x_46_re_m)), x_46_re_m, Float64(Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)) * x_46_im_m));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4e+73], N[(N[(N[(0.0 * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m + N[(N[(3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m + t$95$0), $MachinePrecision], If[LessEqual[x$46$im$95$m, 2.95e+246], N[(N[(x$46$im$95$m * x$46$re$95$m + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m + N[(N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.im < 3.99999999999999993e73

   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 84.9

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

      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 84.9

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

      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 84.9

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

      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 88.1

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in x.re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 86.0

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

   6. Applied rewrites 86.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 88.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0 \cdot x.im, x.im, \left(3 \cdot x.im\right) \cdot x.re\right), \color{blue}{x.re}, -\left(x.im \cdot x.im\right) \cdot x.im\right) \]

   if 3.99999999999999993e73 < x.im < 2.94999999999999988e246

   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 84.9

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

      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 84.9

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

      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 84.9

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

      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 88.1

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

   3. Applied rewrites 88.1%

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

   4. Taylor expanded in x.re around 0

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

   6. Applied rewrites 74.0%

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

   if 2.94999999999999988e246 < 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 59.5

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   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 59.5

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

      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 59.4

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

   6. Applied rewrites 59.4%

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

Alternative 3: 97.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m + x.im\_m, \left(x.re\_m - x.im\_m\right) \cdot x.im\_m, t\_0 \cdot x.re\_m\right)\\ \mathbf{elif}\;x.im\_m \leq 2.95 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x.re\_m, \left(\left(x.re\_m + x.im\_m\right) \cdot \left(x.re\_m - x.im\_m\right)\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re_m (+ x.im_m x.im_m))))
   (*
    x.im_s
    (if (<= x.im_m 1e-34)
      (fma (+ x.re_m x.im_m) (* (- x.re_m x.im_m) x.im_m) (* t_0 x.re_m))
      (if (<= x.im_m 2.95e+246)
        (fma t_0 x.re_m (* (* (+ x.re_m x.im_m) (- x.re_m x.im_m)) x.im_m))
        (- (* (* x.im_m x.im_m) x.im_m)))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * (x_46_im_m + x_46_im_m);
	double tmp;
	if (x_46_im_m <= 1e-34) {
		tmp = fma((x_46_re_m + x_46_im_m), ((x_46_re_m - x_46_im_m) * x_46_im_m), (t_0 * x_46_re_m));
	} else if (x_46_im_m <= 2.95e+246) {
		tmp = fma(t_0, x_46_re_m, (((x_46_re_m + x_46_im_m) * (x_46_re_m - x_46_im_m)) * x_46_im_m));
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m))
	tmp = 0.0
	if (x_46_im_m <= 1e-34)
		tmp = fma(Float64(x_46_re_m + x_46_im_m), Float64(Float64(x_46_re_m - x_46_im_m) * x_46_im_m), Float64(t_0 * x_46_re_m));
	elseif (x_46_im_m <= 2.95e+246)
		tmp = fma(t_0, x_46_re_m, Float64(Float64(Float64(x_46_re_m + x_46_im_m) * Float64(x_46_re_m - x_46_im_m)) * x_46_im_m));
	else
		tmp = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re$95$m * 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, 1e-34], N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(t$95$0 * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 2.95e+246], N[(t$95$0 * x$46$re$95$m + N[(N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.im < 9.99999999999999928e-35

   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 84.9

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

      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 84.9

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

      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 84.9

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

      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 88.1

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

   3. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x.im, x.re, x.im \cdot x.re\right), x.re, \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right)} \]
   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 91.6

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

      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 91.6

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

   5. Applied rewrites 91.6%

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

   if 9.99999999999999928e-35 < x.im < 2.94999999999999988e246

   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 84.9

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

      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 84.9

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

      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 84.9

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

      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 88.1

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

   3. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x.im, x.re, x.im \cdot x.re\right), x.re, \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right)} \]
   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 88.1

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

   5. Applied rewrites 88.1%

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

   if 2.94999999999999988e246 < 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 59.5

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   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 59.5

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

      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 59.4

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

   6. Applied rewrites 59.4%

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

Alternative 4: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4.5 \cdot 10^{+178}:\\ \;\;\;\;\left(x.re\_m + x.im\_m\right) \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot x.im\_m\right) + \left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 4.5e+178)
    (+
     (* (+ x.re_m x.im_m) (* (- x.re_m x.im_m) x.im_m))
     (* (* x.re_m (+ x.im_m x.im_m)) x.re_m))
    (- (* (* x.im_m x.im_m) x.im_m)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 4.5e+178) {
		tmp = ((x_46_re_m + x_46_im_m) * ((x_46_re_m - x_46_im_m) * x_46_im_m)) + ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_re_m);
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Fortran

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

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

real(8) function code(x_46im_s, x_46re_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 4.5d+178) then
        tmp = ((x_46re_m + x_46im_m) * ((x_46re_m - x_46im_m) * x_46im_m)) + ((x_46re_m * (x_46im_m + x_46im_m)) * x_46re_m)
    else
        tmp = -((x_46im_m * x_46im_m) * x_46im_m)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 4.5e+178) {
		tmp = ((x_46_re_m + x_46_im_m) * ((x_46_re_m - x_46_im_m) * x_46_im_m)) + ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_re_m);
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 4.5e+178:
		tmp = ((x_46_re_m + x_46_im_m) * ((x_46_re_m - x_46_im_m) * x_46_im_m)) + ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_re_m)
	else:
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 4.5e+178)
		tmp = Float64(Float64(Float64(x_46_re_m + x_46_im_m) * Float64(Float64(x_46_re_m - x_46_im_m) * x_46_im_m)) + Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)) * x_46_re_m));
	else
		tmp = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 4.5e+178)
		tmp = ((x_46_re_m + x_46_im_m) * ((x_46_re_m - x_46_im_m) * x_46_im_m)) + ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_re_m);
	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.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4.5e+178], N[(N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.4999999999999997e178

   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 84.9

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

      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 84.9

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

      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 84.9

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

      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 88.1

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

   3. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x.im, x.re, x.im \cdot x.re\right), x.re, \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right)} \]
   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 91.6%

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

   if 4.4999999999999997e178 < 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 59.5

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   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 59.5

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

      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 59.4

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

   6. Applied rewrites 59.4%

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

Alternative 5: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-321}:\\ \;\;\;\;-{x.im\_m}^{3}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m + x.im\_m, x.im\_m \cdot x.re\_m, \left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (<= t_0 -2e-321)
      (- (pow x.im_m 3.0))
      (if (<= t_0 INFINITY)
        (fma
         (+ x.re_m x.im_m)
         (* x.im_m x.re_m)
         (* (* x.re_m (+ x.im_m x.im_m)) x.re_m))
        (- (* (* x.im_m x.im_m) x.im_m)))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_0 <= -2e-321) {
		tmp = -pow(x_46_im_m, 3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x_46_re_m + x_46_im_m), (x_46_im_m * x_46_re_m), ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_re_m));
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_0 <= -2e-321)
		tmp = Float64(-(x_46_im_m ^ 3.0));
	elseif (t_0 <= Inf)
		tmp = fma(Float64(x_46_re_m + x_46_im_m), Float64(x_46_im_m * x_46_re_m), Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)) * x_46_re_m));
	else
		tmp = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -2e-321], (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.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)) < -2.00097e-321

   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 59.5

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   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 59.5

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

      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 59.4

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

   6. Applied rewrites 59.4%

      \[\leadsto \color{blue}{-\left(x.im \cdot x.im\right) \cdot x.im} \]
   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 59.5

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

   8. Applied rewrites 59.5%

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

   if -2.00097e-321 < (+.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 84.9

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

      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 84.9

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

      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 84.9

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

      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 88.1

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

   3. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x.im, x.re, x.im \cdot x.re\right), x.re, \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right)} \]
   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 91.6

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

      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 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

   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 59.5

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   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 59.5

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

      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 59.4

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

   6. Applied rewrites 59.4%

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

Alternative 6: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x.re\_m, \left(\left(x.re\_m + x.im\_m\right) \cdot \left(x.re\_m - x.im\_m\right)\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m + x.im\_m, x.im\_m \cdot x.re\_m, t\_0 \cdot x.re\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re_m (+ x.im_m x.im_m))))
   (*
    x.im_s
    (if (<= x.re_m 1.2e+154)
      (fma t_0 x.re_m (* (* (+ x.re_m x.im_m) (- x.re_m x.im_m)) x.im_m))
      (fma (+ x.re_m x.im_m) (* x.im_m x.re_m) (* t_0 x.re_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * (x_46_im_m + x_46_im_m);
	double tmp;
	if (x_46_re_m <= 1.2e+154) {
		tmp = fma(t_0, x_46_re_m, (((x_46_re_m + x_46_im_m) * (x_46_re_m - x_46_im_m)) * x_46_im_m));
	} else {
		tmp = fma((x_46_re_m + x_46_im_m), (x_46_im_m * x_46_re_m), (t_0 * x_46_re_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m))
	tmp = 0.0
	if (x_46_re_m <= 1.2e+154)
		tmp = fma(t_0, x_46_re_m, Float64(Float64(Float64(x_46_re_m + x_46_im_m) * Float64(x_46_re_m - x_46_im_m)) * x_46_im_m));
	else
		tmp = fma(Float64(x_46_re_m + x_46_im_m), Float64(x_46_im_m * x_46_re_m), Float64(t_0 * x_46_re_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$re$95$m, 1.2e+154], N[(t$95$0 * x$46$re$95$m + N[(N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(t$95$0 * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.20000000000000007e154

   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 84.9

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

      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 84.9

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

      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 84.9

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

      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 88.1

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

   3. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x.im, x.re, x.im \cdot x.re\right), x.re, \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right)} \]
   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 88.1

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

   5. Applied rewrites 88.1%

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

   if 1.20000000000000007e154 < 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. 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 84.9

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

      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 84.9

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

      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 84.9

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

      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 88.1

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

   3. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x.im, x.re, x.im \cdot x.re\right), x.re, \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right)} \]
   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 91.6

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

      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 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in x.re around inf

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

      1. lower-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

Alternative 7: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-321}:\\ \;\;\;\;-{x.im\_m}^{3}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re\_m \cdot x.re\_m\right) \cdot 2, x.im\_m, x.re\_m \cdot \left(x.im\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (<= t_0 -2e-321)
      (- (pow x.im_m 3.0))
      (if (<= t_0 INFINITY)
        (fma (* (* x.re_m x.re_m) 2.0) x.im_m (* x.re_m (* x.im_m x.re_m)))
        (- (* (* x.im_m x.im_m) x.im_m)))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_0 <= -2e-321) {
		tmp = -pow(x_46_im_m, 3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(((x_46_re_m * x_46_re_m) * 2.0), x_46_im_m, (x_46_re_m * (x_46_im_m * x_46_re_m)));
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_0 <= -2e-321)
		tmp = Float64(-(x_46_im_m ^ 3.0));
	elseif (t_0 <= Inf)
		tmp = fma(Float64(Float64(x_46_re_m * x_46_re_m) * 2.0), x_46_im_m, Float64(x_46_re_m * Float64(x_46_im_m * x_46_re_m)));
	else
		tmp = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -2e-321], (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), If[LessEqual[t$95$0, Infinity], N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * x$46$im$95$m + N[(x$46$re$95$m * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 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)) < -2.00097e-321

   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 59.5

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   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 59.5

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

      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 59.4

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

   6. Applied rewrites 59.4%

      \[\leadsto \color{blue}{-\left(x.im \cdot x.im\right) \cdot x.im} \]
   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 59.5

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

   8. Applied rewrites 59.5%

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

   if -2.00097e-321 < (+.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.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 49.7

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

   4. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 49.7

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 49.7

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

   6. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 49.7

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

   8. Applied rewrites 49.7%

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

   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 59.5

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   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 59.5

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

      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 59.4

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

   6. Applied rewrites 59.4%

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

Alternative 8: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-321}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re\_m \cdot x.re\_m\right) \cdot 2, x.im\_m, x.re\_m \cdot \left(x.im\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (- (* (* x.im_m x.im_m) x.im_m)))
        (t_1
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (<= t_1 -2e-321)
      t_0
      (if (<= t_1 INFINITY)
        (fma (* (* x.re_m x.re_m) 2.0) x.im_m (* x.re_m (* x.im_m x.re_m)))
        t_0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	double t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -2e-321) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(((x_46_re_m * x_46_re_m) * 2.0), x_46_im_m, (x_46_re_m * (x_46_im_m * x_46_re_m)));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m))
	t_1 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_1 <= -2e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = fma(Float64(Float64(x_46_re_m * x_46_re_m) * 2.0), x_46_im_m, Float64(x_46_re_m * Float64(x_46_im_m * x_46_re_m)));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-321], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * x$46$im$95$m + N[(x$46$re$95$m * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $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)) < -2.00097e-321 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 59.5

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   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 59.5

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

      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 59.4

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

   6. Applied rewrites 59.4%

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

   if -2.00097e-321 < (+.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.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 49.7

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

   4. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 49.7

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 49.7

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

   6. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 49.7

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

   8. Applied rewrites 49.7%

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

Alternative 9: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-321}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;3 \cdot \left(\left(x.re\_m \cdot x.re\_m\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (- (* (* x.im_m x.im_m) x.im_m)))
        (t_1
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (<= t_1 -2e-321)
      t_0
      (if (<= t_1 INFINITY) (* 3.0 (* (* x.re_m x.re_m) x.im_m)) t_0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	double t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -2e-321) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 3.0 * ((x_46_re_m * x_46_re_m) * x_46_im_m);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	double t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_1 <= -2e-321) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 3.0 * ((x_46_re_m * x_46_re_m) * x_46_im_m);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m)
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if t_1 <= -2e-321:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = 3.0 * ((x_46_re_m * x_46_re_m) * x_46_im_m)
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m))
	t_1 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_1 <= -2e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(3.0 * Float64(Float64(x_46_re_m * x_46_re_m) * x_46_im_m));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	t_1 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if (t_1 <= -2e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = 3.0 * ((x_46_re_m * x_46_re_m) * x_46_im_m);
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-321], t$95$0, If[LessEqual[t$95$1, Infinity], N[(3.0 * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -2.00097e-321 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 59.5

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

   4. Applied rewrites 59.5%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   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 59.5

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

      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 59.4

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

   6. Applied rewrites 59.4%

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

   if -2.00097e-321 < (+.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.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 49.7

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

   4. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 49.7

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

      4. lift-fma.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 49.7

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 49.7

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

   6. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 49.7

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

   8. Applied rewrites 49.7%

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

Alternative 10: 59.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -((x_46_im_m * x_46_im_m) * x_46_im_m);
}

Fortran

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

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

real(8) function code(x_46im_s, x_46re_m, x_46im_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * -((x_46im_m * x_46im_m) * x_46im_m)
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -((x_46_im_m * x_46_im_m) * x_46_im_m);
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * -((x_46_im_m * x_46_im_m) * x_46_im_m)

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m)))
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * -((x_46_im_m * x_46_im_m) * x_46_im_m);
end

Wolfram

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

Derivation

1. Initial program 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 59.5

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

4. Applied rewrites 59.5%

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
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 59.5

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

   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 59.4

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

6. Applied rewrites 59.4%

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

Developer Target 1: 91.6% 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 2025156 
(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)))