math.cube on complex, imaginary part

Percentage Accurate: 82.9% → 99.8%

Time: 5.9s

Alternatives: 8

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: 82.9% 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, 0.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ t_1 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + t\_0\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+226}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x.im\_m \cdot x.re\right) \cdot x.re + t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{x.im\_m}{x.re}, \frac{x.im\_m}{x.re}, -3\right) \cdot x.re\right) \cdot x.re\right) \cdot \left(-x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

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.im_m) (* x.im_m x.re)) x.re))
        (t_1 (+ (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m) t_0)))
   (*
    x.im_s
    (if (<= t_1 2e+226)
      t_1
      (if (<= t_1 INFINITY)
        (+ (* (* x.im_m x.re) x.re) t_0)
        (*
         (* (* (fma (/ x.im_m x.re) (/ x.im_m x.re) -3.0) x.re) x.re)
         (- x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = ((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re;
	double t_1 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + t_0;
	double tmp;
	if (t_1 <= 2e+226) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x_46_im_m * x_46_re) * x_46_re) + t_0;
	} else {
		tmp = ((fma((x_46_im_m / x_46_re), (x_46_im_m / x_46_re), -3.0) * x_46_re) * x_46_re) * -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(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re)
	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) + t_0)
	tmp = 0.0
	if (t_1 <= 2e+226)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re) * x_46_re) + t_0);
	else
		tmp = Float64(Float64(Float64(fma(Float64(x_46_im_m / x_46_re), Float64(x_46_im_m / x_46_re), -3.0) * x_46_re) * x_46_re) * Float64(-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[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $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] + t$95$0), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, 2e+226], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(x$46$im$95$m / x$46$re), $MachinePrecision] * N[(x$46$im$95$m / x$46$re), $MachinePrecision] + -3.0), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $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)) < 1.99999999999999992e226

   1. Initial program 97.0%

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

   if 1.99999999999999992e226 < (+.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 87.5%

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

   3. Taylor expanded in x.re around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 45.3

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

   5. Applied rewrites 45.3%

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

   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 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 71.0%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq 2 \cdot 10^{+226}:\\ \;\;\;\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\\ \mathbf{elif}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{x.im}{x.re}, \frac{x.im}{x.re}, -3\right) \cdot x.re\right) \cdot x.re\right) \cdot \left(-x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\_m\right) \cdot x.re\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -2e-317) (not (<= t_0 INFINITY)))
      (* (* x.im_m x.im_m) (- x.im_m))
      (* (* (* 3.0 x.re) x.im_m) x.re)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -2e-317) or not (t_0 <= math.inf):
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m
	else:
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -2e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m));
	else
		tmp = Float64(Float64(Float64(3.0 * x_46_re) * x_46_im_m) * x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -2e-317) || ~((t_0 <= Inf)))
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	else
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -2e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision], N[(N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * 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)) < -1.99999997e-317 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 71.4%

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

   3. Taylor expanded in x.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 88.9%

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

   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 55.7%

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

   if -1.99999997e-317 < (+.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 94.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 61.0

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

   5. Applied rewrites 61.0%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -2 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -2e-317) (not (<= t_0 INFINITY)))
      (* (* x.im_m x.im_m) (- x.im_m))
      (* (* (* 3.0 x.im_m) x.re) x.re)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -2e-317) or not (t_0 <= math.inf):
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m
	else:
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -2e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m));
	else
		tmp = Float64(Float64(Float64(3.0 * x_46_im_m) * x_46_re) * x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -2e-317) || ~((t_0 <= Inf)))
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	else
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -2e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision], N[(N[(N[(3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * 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)) < -1.99999997e-317 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 71.4%

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

   3. Taylor expanded in x.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 88.9%

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

   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 55.7%

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

   if -1.99999997e-317 < (+.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 94.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 61.0

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

   5. Applied rewrites 61.0%

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

   7. Applied rewrites 60.9%

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

   9. Applied rewrites 61.0%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -2 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \left(x.im\_m \cdot x.re\right)\right) \cdot x.re\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -2e-317) (not (<= t_0 INFINITY)))
      (* (* x.im_m x.im_m) (- x.im_m))
      (* (* 3.0 (* x.im_m x.re)) x.re)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -2e-317) or not (t_0 <= math.inf):
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m
	else:
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -2e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m));
	else
		tmp = Float64(Float64(3.0 * Float64(x_46_im_m * x_46_re)) * x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -2e-317) || ~((t_0 <= Inf)))
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	else
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -2e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision], N[(N[(3.0 * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * 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)) < -1.99999997e-317 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 71.4%

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

   3. Taylor expanded in x.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 88.9%

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

   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 55.7%

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

   if -1.99999997e-317 < (+.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 94.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 61.0

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

   5. Applied rewrites 61.0%

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

   7. Applied rewrites 60.9%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -2 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -2e-317) (not (<= t_0 INFINITY)))
      (* (* x.im_m x.im_m) (- x.im_m))
      (* (* 3.0 x.re) (* x.im_m x.re))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re);
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -2e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -2e-317) or not (t_0 <= math.inf):
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m
	else:
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -2e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m));
	else
		tmp = Float64(Float64(3.0 * x_46_re) * Float64(x_46_im_m * x_46_re));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -2e-317) || ~((t_0 <= Inf)))
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	else
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -2e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision], N[(N[(3.0 * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.99999997e-317 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 71.4%

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

   3. Taylor expanded in x.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 88.9%

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

   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 55.7%

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

   if -1.99999997e-317 < (+.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 94.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 61.0

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

   5. Applied rewrites 61.0%

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

   7. Applied rewrites 60.9%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -2 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 6.2 \cdot 10^{+177}:\\ \;\;\;\;\left(-\mathsf{fma}\left(-3 \cdot x.re, x.re, x.im\_m \cdot x.im\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 6.2e+177)
    (* (- (fma (* -3.0 x.re) x.re (* x.im_m x.im_m))) x.im_m)
    (+
     (* (* x.im_m x.re) x.re)
     (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 6.2e+177) {
		tmp = -fma((-3.0 * x_46_re), x_46_re, (x_46_im_m * x_46_im_m)) * x_46_im_m;
	} else {
		tmp = ((x_46_im_m * x_46_re) * x_46_re) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 6.2e+177)
		tmp = Float64(Float64(-fma(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(Float64(Float64(x_46_im_m * x_46_re) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 6.2e+177], 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], N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * x$46$re), $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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.1999999999999998e177

   1. Initial program 85.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 94.2%

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

   if 6.1999999999999998e177 < x.re

   1. Initial program 64.6%

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

   3. Taylor expanded in x.re around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 90.8

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

   5. Applied rewrites 90.8%

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

Alternative 7: 93.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 6.2 \cdot 10^{+177}:\\ \;\;\;\;\left(-\mathsf{fma}\left(-3 \cdot x.re, x.re, x.im\_m \cdot x.im\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \end{array} \end{array} \]

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 6.2e+177)
    (* (- (fma (* -3.0 x.re) x.re (* x.im_m x.im_m))) x.im_m)
    (* (* 3.0 x.re) (* x.im_m x.re)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 6.2e+177) {
		tmp = -fma((-3.0 * x_46_re), x_46_re, (x_46_im_m * x_46_im_m)) * x_46_im_m;
	} else {
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 6.2e+177)
		tmp = Float64(Float64(-fma(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(Float64(3.0 * x_46_re) * Float64(x_46_im_m * x_46_re));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 6.2e+177], 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], N[(N[(3.0 * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.1999999999999998e177

   1. Initial program 85.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 94.2%

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

   if 6.1999999999999998e177 < x.re

   1. Initial program 64.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

   7. Applied rewrites 90.8%

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

Alternative 8: 58.3% accurate, 3.1× speedup

Math

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

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(x_46_im_m * x_46_im_m) * Float64(-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 83.0%

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

3. Taylor expanded in x.re around 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 91.6%

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

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 58.5%

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

9. Final simplification 58.5%

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

Developer Target 1: 91.9% 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 2025010 
(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)))