math.cube on complex, imaginary part

Percentage Accurate: 82.0% → 99.8%

Time: 4.4s

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: 82.0% 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: 99.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (+ (* x.re x.im) (* x.im x.re)) x.re)))
   (if (<= (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) t_0) INFINITY)
     (+ (* (+ x.im x.re) (* (- x.re x.im) x.im)) t_0)
     (* (* x.im (fma (/ (- x.im) x.re) (/ x.im x.re) 3.0)) (* x.re x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = ((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re;
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + t_0) <= ((double) INFINITY)) {
		tmp = ((x_46_im + x_46_re) * ((x_46_re - x_46_im) * x_46_im)) + t_0;
	} else {
		tmp = (x_46_im * fma((-x_46_im / x_46_re), (x_46_im / x_46_re), 3.0)) * (x_46_re * x_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + t_0) <= Inf)
		tmp = Float64(Float64(Float64(x_46_im + x_46_re) * Float64(Float64(x_46_re - x_46_im) * x_46_im)) + t_0);
	else
		tmp = Float64(Float64(x_46_im * fma(Float64(Float64(-x_46_im) / x_46_re), Float64(x_46_im / x_46_re), 3.0)) * Float64(x_46_re * x_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + t$95$0), $MachinePrecision], Infinity], N[(N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(N[(x$46$re - x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(x$46$im * N[(N[((-x$46$im) / x$46$re), $MachinePrecision] * N[(x$46$im / x$46$re), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. 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. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x.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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 19.0

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

   5. Applied rewrites 19.0%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 100.0%

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

Alternative 2: 59.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-289} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im x.im)) x.im)
          (* (+ (* x.re x.im) (* x.im x.re)) x.re))))
   (if (or (<= t_0 -1e-289) (not (<= t_0 INFINITY)))
     (* (* x.im x.im) (- x.im))
     (* (* 3.0 x.re) (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (((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);
	double tmp;
	if ((t_0 <= -1e-289) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im * x_46_im) * -x_46_im;
	} else {
		tmp = (3.0 * x_46_re) * (x_46_re * x_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (((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);
	double tmp;
	if ((t_0 <= -1e-289) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im * x_46_im) * -x_46_im;
	} else {
		tmp = (3.0 * x_46_re) * (x_46_re * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (((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)
	tmp = 0
	if (t_0 <= -1e-289) or not (t_0 <= math.inf):
		tmp = (x_46_im * x_46_im) * -x_46_im
	else:
		tmp = (3.0 * x_46_re) * (x_46_re * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = 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))
	tmp = 0.0
	if ((t_0 <= -1e-289) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im * x_46_im) * Float64(-x_46_im));
	else
		tmp = Float64(Float64(3.0 * x_46_re) * Float64(x_46_re * x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (((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);
	tmp = 0.0;
	if ((t_0 <= -1e-289) || ~((t_0 <= Inf)))
		tmp = (x_46_im * x_46_im) * -x_46_im;
	else
		tmp = (3.0 * x_46_re) * (x_46_re * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -1e-289], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im * x$46$im), $MachinePrecision] * (-x$46$im)), $MachinePrecision], N[(N[(3.0 * x$46$re), $MachinePrecision] * N[(x$46$re * x$46$im), $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)) < -1e-289 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 74.5%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      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. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 56.8%

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

   if -1e-289 < (+.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 91.7%

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   7. Applied rewrites 67.2%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-289} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-289} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im x.im)) x.im)
          (* (+ (* x.re x.im) (* x.im x.re)) x.re))))
   (if (or (<= t_0 -1e-289) (not (<= t_0 INFINITY)))
     (* (* x.im x.im) (- x.im))
     (* (* (* 3.0 x.im) x.re) x.re))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (((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);
	double tmp;
	if ((t_0 <= -1e-289) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im * x_46_im) * -x_46_im;
	} else {
		tmp = ((3.0 * x_46_im) * x_46_re) * x_46_re;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (((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);
	double tmp;
	if ((t_0 <= -1e-289) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im * x_46_im) * -x_46_im;
	} else {
		tmp = ((3.0 * x_46_im) * x_46_re) * x_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (((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)
	tmp = 0
	if (t_0 <= -1e-289) or not (t_0 <= math.inf):
		tmp = (x_46_im * x_46_im) * -x_46_im
	else:
		tmp = ((3.0 * x_46_im) * x_46_re) * x_46_re
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = 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))
	tmp = 0.0
	if ((t_0 <= -1e-289) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im * x_46_im) * Float64(-x_46_im));
	else
		tmp = Float64(Float64(Float64(3.0 * x_46_im) * x_46_re) * x_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (((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);
	tmp = 0.0;
	if ((t_0 <= -1e-289) || ~((t_0 <= Inf)))
		tmp = (x_46_im * x_46_im) * -x_46_im;
	else
		tmp = ((3.0 * x_46_im) * x_46_re) * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -1e-289], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im * x$46$im), $MachinePrecision] * (-x$46$im)), $MachinePrecision], N[(N[(N[(3.0 * x$46$im), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $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)) < -1e-289 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 74.5%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      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. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 56.8%

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

   if -1e-289 < (+.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 91.7%

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 67.2%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-289} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-289} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im x.im)) x.im)
          (* (+ (* x.re x.im) (* x.im x.re)) x.re))))
   (if (or (<= t_0 -1e-289) (not (<= t_0 INFINITY)))
     (* (* x.im x.im) (- x.im))
     (* (* (* x.re x.re) x.im) 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (((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);
	double tmp;
	if ((t_0 <= -1e-289) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im * x_46_im) * -x_46_im;
	} else {
		tmp = ((x_46_re * x_46_re) * x_46_im) * 3.0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (((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);
	double tmp;
	if ((t_0 <= -1e-289) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im * x_46_im) * -x_46_im;
	} else {
		tmp = ((x_46_re * x_46_re) * x_46_im) * 3.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (((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)
	tmp = 0
	if (t_0 <= -1e-289) or not (t_0 <= math.inf):
		tmp = (x_46_im * x_46_im) * -x_46_im
	else:
		tmp = ((x_46_re * x_46_re) * x_46_im) * 3.0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = 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))
	tmp = 0.0
	if ((t_0 <= -1e-289) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im * x_46_im) * Float64(-x_46_im));
	else
		tmp = Float64(Float64(Float64(x_46_re * x_46_re) * x_46_im) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (((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);
	tmp = 0.0;
	if ((t_0 <= -1e-289) || ~((t_0 <= Inf)))
		tmp = (x_46_im * x_46_im) * -x_46_im;
	else
		tmp = ((x_46_re * x_46_re) * x_46_im) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -1e-289], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im * x$46$im), $MachinePrecision] * (-x$46$im)), $MachinePrecision], N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision] * 3.0), $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)) < -1e-289 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 74.5%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      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. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 56.8%

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

   if -1e-289 < (+.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 91.7%

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 59.1%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-289} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\\ \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + t\_0 \leq \infty:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im \cdot \mathsf{fma}\left(\frac{x.im}{x.re}, x.re, x.re\right), 0 \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (+ (* x.re x.im) (* x.im x.re)) x.re)))
   (if (<= (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) t_0) INFINITY)
     (+ (* (+ x.im x.re) (* (- x.re x.im) x.im)) t_0)
     (fma (- x.re x.im) (* x.im (fma (/ x.im x.re) x.re x.re)) (* 0.0 x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = ((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re;
	double tmp;
	if (((((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + t_0) <= ((double) INFINITY)) {
		tmp = ((x_46_im + x_46_re) * ((x_46_re - x_46_im) * x_46_im)) + t_0;
	} else {
		tmp = fma((x_46_re - x_46_im), (x_46_im * fma((x_46_im / x_46_re), x_46_re, x_46_re)), (0.0 * x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + t_0) <= Inf)
		tmp = Float64(Float64(Float64(x_46_im + x_46_re) * Float64(Float64(x_46_re - x_46_im) * x_46_im)) + t_0);
	else
		tmp = fma(Float64(x_46_re - x_46_im), Float64(x_46_im * fma(Float64(x_46_im / x_46_re), x_46_re, x_46_re)), Float64(0.0 * x_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + t$95$0), $MachinePrecision], Infinity], N[(N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(N[(x$46$re - x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im * N[(N[(x$46$im / x$46$re), $MachinePrecision] * x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision] + N[(0.0 * x$46$re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. 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. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

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

      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. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 14.3

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

   4. Applied rewrites 14.3%

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

   5. Taylor expanded in x.re around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 14.3

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

   7. Applied rewrites 14.3%

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

   9. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, x.im \cdot \mathsf{fma}\left(\frac{x.im}{x.re}, x.re, x.re\right), 0 \cdot x.re\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;\left(-\mathsf{fma}\left(-3 \cdot x.re, x.re, x.im \cdot x.im\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 7.8e+153)
   (* (- (fma (* -3.0 x.re) x.re (* x.im x.im))) x.im)
   (* (* 3.0 x.re) (* x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7.8e+153) {
		tmp = -fma((-3.0 * x_46_re), x_46_re, (x_46_im * x_46_im)) * x_46_im;
	} else {
		tmp = (3.0 * x_46_re) * (x_46_re * x_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 7.8e+153)
		tmp = Float64(Float64(-fma(Float64(-3.0 * x_46_re), x_46_re, Float64(x_46_im * x_46_im))) * x_46_im);
	else
		tmp = Float64(Float64(3.0 * x_46_re) * Float64(x_46_re * x_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 7.8e+153], N[((-N[(N[(-3.0 * x$46$re), $MachinePrecision] * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]) * x$46$im), $MachinePrecision], N[(N[(3.0 * x$46$re), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.79999999999999966e153

   1. Initial program 88.9%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-lft-in N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      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. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 95.2%

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

   if 7.79999999999999966e153 < x.re

   1. Initial program 48.6%

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

   3. Taylor expanded in x.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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 93.7

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

   5. Applied rewrites 93.7%

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

   7. Applied rewrites 93.8%

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

Alternative 7: 59.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return (x_46_im * x_46_im) * -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_46im * x_46im) * -x_46im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in x.re around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-lft-in N/A

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

   7. unpow3 N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   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. *-commutative N/A

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

   14. lower-*.f64 N/A

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

5. Applied rewrites 90.0%

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

6. Taylor expanded in x.re around 0

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

8. Applied rewrites 57.5%

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

9. Final simplification 57.5%

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

Developer Target 1: 91.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025017 
(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)))