math.cube on complex, imaginary part

Percentage Accurate: 82.3% → 99.7%

Time: 2.0s

Alternatives: 6

Speedup: 1.4×

• 45.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.3% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+235}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re + x.im\_m\right) \cdot \left(x.re - x.im\_m\right), x.im\_m, \left(\left(x.im\_m + x.im\_m\right) \cdot x.re\right) \cdot x.re\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, \frac{x.re}{x.im\_m} \cdot \frac{x.re}{x.im\_m}, -1\right) \cdot \left(\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (<= t_0 1e+235)
      (fma
       (* (+ x.re x.im_m) (- x.re x.im_m))
       x.im_m
       (* (* (+ x.im_m x.im_m) x.re) x.re))
      (if (<= t_0 INFINITY)
        (* x.re (* x.re (* 3.0 x.im_m)))
        (*
         (fma 3.0 (* (/ x.re x.im_m) (/ x.re x.im_m)) -1.0)
         (* (* x.im_m x.im_m) x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= 1e+235) {
		tmp = fma(((x_46_re + x_46_im_m) * (x_46_re - x_46_im_m)), x_46_im_m, (((x_46_im_m + x_46_im_m) * x_46_re) * x_46_re));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im_m));
	} else {
		tmp = fma(3.0, ((x_46_re / x_46_im_m) * (x_46_re / x_46_im_m)), -1.0) * ((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= 1e+235)
		tmp = fma(Float64(Float64(x_46_re + x_46_im_m) * Float64(x_46_re - x_46_im_m)), x_46_im_m, Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re) * x_46_re));
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(3.0 * x_46_im_m)));
	else
		tmp = Float64(fma(3.0, Float64(Float64(x_46_re / x_46_im_m) * Float64(x_46_re / x_46_im_m)), -1.0) * Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 1e+235], N[(N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m + N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(x$46$re * N[(x$46$re * N[(3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * N[(N[(x$46$re / x$46$im$95$m), $MachinePrecision] * N[(x$46$re / x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 1.0000000000000001e235

   1. Initial program 82.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. difference-of-squares N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. count-2-rev N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 85.5

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

   3. Applied rewrites 85.5%

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

   if 1.0000000000000001e235 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 50.0

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

   6. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 55.7

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

   8. Applied rewrites 55.7%

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

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

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow3 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 62.9

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

   4. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 77.4

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

   6. Applied rewrites 77.4%

      \[\leadsto \mathsf{fma}\left(3, \frac{x.re}{x.im} \cdot \frac{x.re}{x.im}, -1\right) \cdot \left(\left(x.im \cdot \color{blue}{x.im}\right) \cdot x.im\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\left(\left(x.re \cdot x.re\right) \cdot 3 - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_0 5e+101)
      (* (- (* (* x.re x.re) 3.0) (* x.im_m x.im_m)) x.im_m)
      (if (<= t_0 INFINITY)
        (* x.re (* (* 3.0 x.re) 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 t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= 5e+101) {
		tmp = (((x_46_re * x_46_re) * 3.0) - (x_46_im_m * x_46_im_m)) * x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re * ((3.0 * x_46_re) * x_46_im_m);
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= 5e+101) {
		tmp = (((x_46_re * x_46_re) * 3.0) - (x_46_im_m * x_46_im_m)) * x_46_im_m;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re * ((3.0 * x_46_re) * x_46_im_m);
	} else {
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if t_0 <= 5e+101:
		tmp = (((x_46_re * x_46_re) * 3.0) - (x_46_im_m * x_46_im_m)) * x_46_im_m
	elif t_0 <= math.inf:
		tmp = x_46_re * ((3.0 * x_46_re) * 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)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= 5e+101)
		tmp = Float64(Float64(Float64(Float64(x_46_re * x_46_re) * 3.0) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m);
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re * Float64(Float64(3.0 * x_46_re) * 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

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if (t_0 <= 5e+101)
		tmp = (((x_46_re * x_46_re) * 3.0) - (x_46_im_m * x_46_im_m)) * x_46_im_m;
	elseif (t_0 <= Inf)
		tmp = x_46_re * ((3.0 * x_46_re) * x_46_im_m);
	else
		tmp = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 5e+101], N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(x$46$re * N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], (-N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 4.99999999999999989e101

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 87.6

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

   4. Applied rewrites 87.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 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. sub-flip N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

   if 4.99999999999999989e101 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 50.0

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

   6. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 55.7

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

   8. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 55.7

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

   10. Applied rewrites 55.7%

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

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

   1. Initial program 82.3%

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

   2. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 58.4

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

   4. Applied rewrites 58.4%

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

Alternative 3: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 7 \cdot 10^{-154}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3 \cdot x.re, x.re, \left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

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 7e-154)
    (* x.re (* x.re (* 3.0 x.im_m)))
    (* (fma (* 3.0 x.re) x.re (* (- x.im_m) x.im_m)) x.im_m))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7e-154) {
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im_m));
	} else {
		tmp = fma((3.0 * x_46_re), x_46_re, (-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 <= 7e-154)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(3.0 * x_46_im_m)));
	else
		tmp = Float64(fma(Float64(3.0 * x_46_re), x_46_re, 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, 7e-154], N[(x$46$re * N[(x$46$re * N[(3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$re + N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.0000000000000001e-154

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 50.0

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

   6. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 55.7

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

   8. Applied rewrites 55.7%

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

   if 7.0000000000000001e-154 < x.im

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 87.6

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

   4. Applied rewrites 87.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. mul-1-neg N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 90.7

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

   6. Applied rewrites 90.7%

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

Alternative 4: 95.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-314}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* (* x.im_m x.im_m) x.im_m)))
        (t_1
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_1 -2e-314)
      t_0
      (if (<= t_1 INFINITY) (* x.re (* x.re (* 3.0 x.im_m))) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -((x_46_im_m * x_46_im_m) * x_46_im_m);
	double t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-314) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im_m));
	} else {
		tmp = t_0;
	}
	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_im_m * x_46_im_m) * x_46_im_m);
	double t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= -2e-314) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im_m));
	} else {
		tmp = t_0;
	}
	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_im_m * x_46_im_m) * x_46_im_m)
	t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if t_1 <= -2e-314:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im_m))
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(-Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m))
	t_1 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_1 <= -2e-314)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(3.0 * x_46_im_m)));
	else
		tmp = t_0;
	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_im_m * x_46_im_m) * x_46_im_m);
	t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if (t_1 <= -2e-314)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im_m));
	else
		tmp = t_0;
	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[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision])}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-314], t$95$0, If[LessEqual[t$95$1, Infinity], N[(x$46$re * N[(x$46$re * N[(3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.9999999999e-314 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 82.3%

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

   2. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 58.4

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

   4. Applied rewrites 58.4%

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

   if -1.9999999999e-314 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 50.0

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

   6. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 55.7

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

   8. Applied rewrites 55.7%

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

Alternative 5: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 2.8 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\_m\right)\right)\\ \end{array} \end{array} \]

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 2.8e+153)
    (* (- (* (* 3.0 x.re) x.re) (* x.im_m x.im_m)) x.im_m)
    (* x.re (* x.re (* 3.0 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 tmp;
	if (x_46_re <= 2.8e+153) {
		tmp = (((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_re * (3.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

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
    real(8) :: tmp
    if (x_46re <= 2.8d+153) then
        tmp = (((3.0d0 * x_46re) * x_46re) - (x_46im_m * x_46im_m)) * x_46im_m
    else
        tmp = x_46re * (x_46re * (3.0d0 * x_46im_m))
    end if
    code = x_46im_s * tmp
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) {
	double tmp;
	if (x_46_re <= 2.8e+153) {
		tmp = (((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_re * (3.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 2.8e+153:
		tmp = (((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_re * (3.0 * 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)
	tmp = 0.0
	if (x_46_re <= 2.8e+153)
		tmp = Float64(Float64(Float64(Float64(3.0 * x_46_re) * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m);
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(3.0 * x_46_im_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 2.8e+153)
		tmp = (((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_re * (3.0 * x_46_im_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 2.8e+153], N[(N[(N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(x$46$re * N[(x$46$re * N[(3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.79999999999999985e153

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 87.6

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

   4. Applied rewrites 87.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 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. sub-flip N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 87.5

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

   8. Applied rewrites 87.5%

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

   if 2.79999999999999985e153 < x.re

   1. Initial program 82.3%

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

   2. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 50.0

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

   6. Applied rewrites 50.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 55.7

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

   8. Applied rewrites 55.7%

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

Alternative 6: 58.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

2. Taylor expanded in x.re around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. unpow3 N/A

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

   4. pow2 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 58.4

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

4. Applied rewrites 58.4%

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

Developer Target 1: 91.2% 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 2025136 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform c (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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