math.cube on complex, imaginary part

Percentage Accurate: 82.5% → 98.1%

Time: 3.4s

Alternatives: 7

Speedup: 1.3×

• 44.9% 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: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m + x.re, \left(x.re - x.im\_m\right) \cdot x.im\_m, \left(\left(2 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.re\right)\\ \mathbf{elif}\;x.im\_m \leq 3.5 \cdot 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(-x.im\_m, x.im\_m, \left(x.re \cdot x.re\right) \cdot 3\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 4e+102)
    (fma
     (+ x.im_m x.re)
     (* (- x.re x.im_m) x.im_m)
     (* (* (* 2.0 x.im_m) x.re) x.re))
    (if (<= x.im_m 3.5e+230)
      (* (fma (- x.im_m) x.im_m (* (* x.re x.re) 3.0)) x.im_m)
      (* (* (- x.im_m) x.im_m) x.im_m)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 4e+102) {
		tmp = fma((x_46_im_m + x_46_re), ((x_46_re - x_46_im_m) * x_46_im_m), (((2.0 * x_46_im_m) * x_46_re) * x_46_re));
	} else if (x_46_im_m <= 3.5e+230) {
		tmp = fma(-x_46_im_m, x_46_im_m, ((x_46_re * x_46_re) * 3.0)) * x_46_im_m;
	} else {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 4e+102)
		tmp = fma(Float64(x_46_im_m + x_46_re), Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m), Float64(Float64(Float64(2.0 * x_46_im_m) * x_46_re) * x_46_re));
	elseif (x_46_im_m <= 3.5e+230)
		tmp = Float64(fma(Float64(-x_46_im_m), x_46_im_m, Float64(Float64(x_46_re * x_46_re) * 3.0)) * x_46_im_m);
	else
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4e+102], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(2.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 3.5e+230], N[(N[((-x$46$im$95$m) * x$46$im$95$m + N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 3.99999999999999991e102

   1. Initial program 86.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. lower--.f64 90.1

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

   4. Applied rewrites 90.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. count-2-rev N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 94.7%

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

   if 3.99999999999999991e102 < x.im < 3.5e230

   1. Initial program 67.6%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 97.1

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

   7. Applied rewrites 97.1%

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

   if 3.5e230 < x.im

   1. Initial program 51.9%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-1 \cdot {x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 96.3

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

   8. Applied rewrites 96.3%

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

Alternative 2: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-323} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -2e-323) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* (* 3.0 x.re) (* x.im_m x.re))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-323) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re);
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-323) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -2e-323) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -2e-323) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(3.0 * x_46_re) * Float64(x_46_im_m * x_46_re));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -2e-323) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -2e-323], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(3.0 * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.97626e-323 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 63.4%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-1 \cdot {x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 64.3

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

   8. Applied rewrites 64.3%

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

   if -1.97626e-323 < (+.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 96.2%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 59.3

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

   7. Applied rewrites 59.3%

      \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.re\right) \cdot x.im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 62.9

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

   9. Applied rewrites 62.9%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -2 \cdot 10^{-323} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-323} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.im\_m\right) \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -2e-323) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* (* 3.0 x.im_m) (* x.re x.re))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-323) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (3.0 * x_46_im_m) * (x_46_re * x_46_re);
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-323) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (3.0 * x_46_im_m) * (x_46_re * x_46_re);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -2e-323) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = (3.0 * x_46_im_m) * (x_46_re * x_46_re)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -2e-323) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(3.0 * x_46_im_m) * Float64(x_46_re * x_46_re));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -2e-323) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = (3.0 * x_46_im_m) * (x_46_re * x_46_re);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -2e-323], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(3.0 * x$46$im$95$m), $MachinePrecision] * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.97626e-323 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 63.4%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-1 \cdot {x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 64.3

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

   8. Applied rewrites 64.3%

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

   if -1.97626e-323 < (+.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 96.2%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -2 \cdot 10^{-323} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(-x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x.im\_m, \left(x.re \cdot \left(x.im\_m \cdot x.re\right)\right) \cdot 3\right)\\ \mathbf{elif}\;x.im\_m \leq 3.5 \cdot 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(-x.im\_m, x.im\_m, \left(x.re \cdot x.re\right) \cdot 3\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* (- x.im_m) x.im_m)))
   (*
    x.im_s
    (if (<= x.im_m 4e+102)
      (fma t_0 x.im_m (* (* x.re (* x.im_m x.re)) 3.0))
      (if (<= x.im_m 3.5e+230)
        (* (fma (- x.im_m) x.im_m (* (* x.re x.re) 3.0)) x.im_m)
        (* t_0 x.im_m))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -x_46_im_m * x_46_im_m;
	double tmp;
	if (x_46_im_m <= 4e+102) {
		tmp = fma(t_0, x_46_im_m, ((x_46_re * (x_46_im_m * x_46_re)) * 3.0));
	} else if (x_46_im_m <= 3.5e+230) {
		tmp = fma(-x_46_im_m, x_46_im_m, ((x_46_re * x_46_re) * 3.0)) * x_46_im_m;
	} else {
		tmp = t_0 * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(-x_46_im_m) * x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 4e+102)
		tmp = fma(t_0, x_46_im_m, Float64(Float64(x_46_re * Float64(x_46_im_m * x_46_re)) * 3.0));
	elseif (x_46_im_m <= 3.5e+230)
		tmp = Float64(fma(Float64(-x_46_im_m), x_46_im_m, Float64(Float64(x_46_re * x_46_re) * 3.0)) * x_46_im_m);
	else
		tmp = Float64(t_0 * x_46_im_m);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4e+102], N[(t$95$0 * x$46$im$95$m + N[(N[(x$46$re * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 3.5e+230], N[(N[((-x$46$im$95$m) * x$46$im$95$m + N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(t$95$0 * x$46$im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.im < 3.99999999999999991e102

   1. Initial program 86.0%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft1-in N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

   7. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.6

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

   9. Applied rewrites 93.6%

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

   if 3.99999999999999991e102 < x.im < 3.5e230

   1. Initial program 67.6%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 97.1

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

   7. Applied rewrites 97.1%

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

   if 3.5e230 < x.im

   1. Initial program 51.9%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-1 \cdot {x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 96.3

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

   8. Applied rewrites 96.3%

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

Alternative 5: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 6 \cdot 10^{-96}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \mathbf{elif}\;x.im\_m \leq 3.5 \cdot 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(-x.im\_m, x.im\_m, \left(x.re \cdot x.re\right) \cdot 3\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 6e-96)
    (* (* 3.0 x.re) (* x.im_m x.re))
    (if (<= x.im_m 3.5e+230)
      (* (fma (- x.im_m) x.im_m (* (* x.re x.re) 3.0)) x.im_m)
      (* (* (- x.im_m) x.im_m) x.im_m)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 6e-96) {
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re);
	} else if (x_46_im_m <= 3.5e+230) {
		tmp = fma(-x_46_im_m, x_46_im_m, ((x_46_re * x_46_re) * 3.0)) * x_46_im_m;
	} else {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 6e-96)
		tmp = Float64(Float64(3.0 * x_46_re) * Float64(x_46_im_m * x_46_re));
	elseif (x_46_im_m <= 3.5e+230)
		tmp = Float64(fma(Float64(-x_46_im_m), x_46_im_m, Float64(Float64(x_46_re * x_46_re) * 3.0)) * x_46_im_m);
	else
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 6e-96], N[(N[(3.0 * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 3.5e+230], N[(N[((-x$46$im$95$m) * x$46$im$95$m + N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 6e-96

   1. Initial program 81.8%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 51.2

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

   5. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 51.2

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

   7. Applied rewrites 51.2%

      \[\leadsto \left(\left(3 \cdot x.re\right) \cdot x.re\right) \cdot x.im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.2

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

   9. Applied rewrites 61.2%

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

   if 6e-96 < x.im < 3.5e230

   1. Initial program 85.9%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if 3.5e230 < x.im

   1. Initial program 51.9%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-1 \cdot {x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 96.3

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

   8. Applied rewrites 96.3%

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

Alternative 6: 92.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 1.9 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(3, x.re \cdot x.re, \left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\_m\right) \cdot 3\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 1.9e+152)
    (* (fma 3.0 (* x.re x.re) (* (- x.im_m) x.im_m)) x.im_m)
    (* (* (* x.re x.im_m) 3.0) x.re))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1.9e+152) {
		tmp = fma(3.0, (x_46_re * x_46_re), (-x_46_im_m * x_46_im_m)) * x_46_im_m;
	} else {
		tmp = ((x_46_re * x_46_im_m) * 3.0) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 1.9e+152)
		tmp = Float64(fma(3.0, Float64(x_46_re * x_46_re), Float64(Float64(-x_46_im_m) * x_46_im_m)) * x_46_im_m);
	else
		tmp = Float64(Float64(Float64(x_46_re * x_46_im_m) * 3.0) * x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 1.9e+152], N[(N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision] + N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.9e152

   1. Initial program 81.3%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 1.9e152 < x.re

   1. Initial program 70.5%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 73.5

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

   5. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 87.8

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

   7. Applied rewrites 87.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 87.9

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

   9. Applied rewrites 87.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.9 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(3, x.re \cdot x.re, \left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\right) \cdot 3\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in x.im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 88.5

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

5. Applied rewrites 88.5%

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

6. Taylor expanded in x.re around 0

   \[\leadsto \left(-1 \cdot {x.im}^{2}\right) \cdot x.im \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. pow2 N/A

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

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

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

   4. mul-1-neg N/A

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

   5. lower-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f64 62.2

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

8. Applied rewrites 62.2%

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

Developer Target 1: 91.4% 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 2025064 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* 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)))