math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 99.3%

Time: 7.0s

Alternatives: 8

Speedup: 1.0×

• 44.6% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im)
    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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-307} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(\left(x.re - x.im\_m\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\_m\right) \cdot x.re\\ \end{array} \end{array} \end{array} \]

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 -5e-307) (not (<= t_0 INFINITY)))
      (* (+ x.im_m x.re) (* (- x.re 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 <= -5e-307) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im_m + x_46_re) * ((x_46_re - 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 <= -5e-307) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im_m + x_46_re) * ((x_46_re - 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 <= -5e-307) or not (t_0 <= math.inf):
		tmp = (x_46_im_m + x_46_re) * ((x_46_re - 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 <= -5e-307) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im_m + x_46_re) * Float64(Float64(x_46_re - x_46_im_m) * 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 <= -5e-307) || ~((t_0 <= Inf)))
		tmp = (x_46_im_m + x_46_re) * ((x_46_re - 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, -5e-307], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $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)) < -5.00000000000000014e-307 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 72.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. 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 83.1

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

   4. Applied rewrites 83.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. +-inverses N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\color{blue}{0}} \]

      10. +-inverses N/A

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

      11. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \frac{\color{blue}{0}}{x.im \cdot x.im - x.im \cdot x.im} \]

      12. +-inverses N/A

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

      13. associate-*r/ N/A

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

      14. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \frac{x.re \cdot \color{blue}{0}}{x.im \cdot x.im - x.im \cdot x.im} \]

      15. +-inverses N/A

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

      16. distribute-lft-out-- N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \frac{\color{blue}{0}}{x.im \cdot x.im - x.im \cdot x.im} \]

      20. +-inverses N/A

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

      21. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\color{blue}{0}} \]

      22. +-inverses N/A

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

      23. flip-+ N/A

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

   8. Applied rewrites 78.7%

      \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{0} \]

   if -5.00000000000000014e-307 < (+.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 92.8%

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

   3. Taylor expanded in x.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 62.5

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

   5. Applied rewrites 62.5%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-307} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-307} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-x.im\_m\right) \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\_m\right) \cdot x.re\\ \end{array} \end{array} \end{array} \]

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 -5e-307) (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 <= -5e-307) || !(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 <= -5e-307) || !(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 <= -5e-307) 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 <= -5e-307) || !(t_0 <= Inf))
		tmp = Float64(Float64(-x_46_im_m) * Float64(x_46_im_m * 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 <= -5e-307) || ~((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, -5e-307], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[((-x$46$im$95$m) * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $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)) < -5.00000000000000014e-307 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 72.6%

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

   3. Taylor expanded in x.re around 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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

      13. distribute-lft-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 57.6%

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

   if -5.00000000000000014e-307 < (+.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 92.8%

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

   3. Taylor expanded in x.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 62.5

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

   5. Applied rewrites 62.5%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-307} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-307} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-x.im\_m\right) \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(3 \cdot x.re\right)\\ \end{array} \end{array} \end{array} \]

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 -5e-307) (not (<= t_0 INFINITY)))
      (* (- x.im_m) (* x.im_m x.im_m))
      (* (* x.re x.im_m) (* 3.0 x.re))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -5e-307) || !(t_0 <= ((double) INFINITY))) {
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	} else {
		tmp = (x_46_re * x_46_im_m) * (3.0 * x_46_re);
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -5e-307) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	} else {
		tmp = (x_46_re * x_46_im_m) * (3.0 * x_46_re);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -5e-307) or not (t_0 <= math.inf):
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m)
	else:
		tmp = (x_46_re * x_46_im_m) * (3.0 * x_46_re)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -5e-307) || !(t_0 <= Inf))
		tmp = Float64(Float64(-x_46_im_m) * Float64(x_46_im_m * x_46_im_m));
	else
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(3.0 * 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 <= -5e-307) || ~((t_0 <= Inf)))
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	else
		tmp = (x_46_re * x_46_im_m) * (3.0 * x_46_re);
	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, -5e-307], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[((-x$46$im$95$m) * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(3.0 * 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)) < -5.00000000000000014e-307 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 72.6%

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

   3. Taylor expanded in x.re around 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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

      13. distribute-lft-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 57.6%

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

   if -5.00000000000000014e-307 < (+.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 92.8%

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

   3. Taylor expanded in x.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 62.5

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

   5. Applied rewrites 62.5%

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

   7. Applied rewrites 62.4%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-307} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(3 \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-307} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-x.im\_m\right) \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -5e-307) (not (<= t_0 INFINITY)))
      (* (- x.im_m) (* x.im_m x.im_m))
      (* 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_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -5e-307) || !(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_re) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -5e-307) || !(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_re) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -5e-307) 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_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(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -5e-307) || !(t_0 <= Inf))
		tmp = Float64(Float64(-x_46_im_m) * Float64(x_46_im_m * x_46_im_m));
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -5e-307) || ~((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_re) * x_46_im_m);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -5e-307], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[((-x$46$im$95$m) * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im$95$m), $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)) < -5.00000000000000014e-307 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 72.6%

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

   3. Taylor expanded in x.re around 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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

      13. distribute-lft-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 57.6%

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

   if -5.00000000000000014e-307 < (+.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 92.8%

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

   3. Taylor expanded in x.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 62.5

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

   5. Applied rewrites 62.5%

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

   7. Applied rewrites 55.5%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-307} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(\left(x.re - x.im\_m\right) \cdot x.im\_m\right) + \left(x.re \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(\left(\left(x.re - x.im\_m\right) \cdot \sqrt{x.im\_m}\right) \cdot \sqrt{x.im\_m}\right)\\ \end{array} \end{array} \]

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.im_m 5e+84)
    (+
     (* (+ x.im_m x.re) (* (- x.re x.im_m) x.im_m))
     (* (* x.re (+ x.im_m x.im_m)) x.re))
    (* (+ x.im_m x.re) (* (* (- x.re x.im_m) (sqrt x.im_m)) (sqrt x.im_m))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+84) {
		tmp = ((x_46_im_m + x_46_re) * ((x_46_re - x_46_im_m) * x_46_im_m)) + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	} else {
		tmp = (x_46_im_m + x_46_re) * (((x_46_re - x_46_im_m) * sqrt(x_46_im_m)) * sqrt(x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 5d+84) then
        tmp = ((x_46im_m + x_46re) * ((x_46re - x_46im_m) * x_46im_m)) + ((x_46re * (x_46im_m + x_46im_m)) * x_46re)
    else
        tmp = (x_46im_m + x_46re) * (((x_46re - x_46im_m) * sqrt(x_46im_m)) * sqrt(x_46im_m))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+84) {
		tmp = ((x_46_im_m + x_46_re) * ((x_46_re - x_46_im_m) * x_46_im_m)) + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	} else {
		tmp = (x_46_im_m + x_46_re) * (((x_46_re - x_46_im_m) * Math.sqrt(x_46_im_m)) * Math.sqrt(x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 5e+84:
		tmp = ((x_46_im_m + x_46_re) * ((x_46_re - x_46_im_m) * x_46_im_m)) + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re)
	else:
		tmp = (x_46_im_m + x_46_re) * (((x_46_re - x_46_im_m) * math.sqrt(x_46_im_m)) * math.sqrt(x_46_im_m))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+84)
		tmp = Float64(Float64(Float64(x_46_im_m + x_46_re) * Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m)) + Float64(Float64(x_46_re * Float64(x_46_im_m + x_46_im_m)) * x_46_re));
	else
		tmp = Float64(Float64(x_46_im_m + x_46_re) * Float64(Float64(Float64(x_46_re - x_46_im_m) * sqrt(x_46_im_m)) * sqrt(x_46_im_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 5e+84)
		tmp = ((x_46_im_m + x_46_re) * ((x_46_re - x_46_im_m) * x_46_im_m)) + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	else
		tmp = (x_46_im_m + x_46_re) * (((x_46_re - x_46_im_m) * sqrt(x_46_im_m)) * sqrt(x_46_im_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+84], N[(N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[Sqrt[x$46$im$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[x$46$im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000001e84

   1. Initial program 85.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. 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 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 93.6

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

   6. Applied rewrites 93.6%

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

   if 5.0000000000000001e84 < x.im

   1. Initial program 73.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. 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 84.4

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

   4. Applied rewrites 84.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 82.2

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

   6. Applied rewrites 82.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. +-inverses N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\color{blue}{0}} \]

      10. +-inverses N/A

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

      11. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \frac{\color{blue}{0}}{x.im \cdot x.im - x.im \cdot x.im} \]

      12. +-inverses N/A

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

      13. associate-*r/ N/A

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

      14. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \frac{x.re \cdot \color{blue}{0}}{x.im \cdot x.im - x.im \cdot x.im} \]

      15. +-inverses N/A

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

      16. distribute-lft-out-- N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \frac{\color{blue}{0}}{x.im \cdot x.im - x.im \cdot x.im} \]

      20. +-inverses N/A

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

      21. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\color{blue}{0}} \]

      22. +-inverses N/A

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

      23. flip-+ N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{0} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot x.im\right)} + 0 \]

      2. rem-square-sqrt N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot \color{blue}{\left(\sqrt{x.im} \cdot \sqrt{x.im}\right)}\right) + 0 \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot \left(\color{blue}{\sqrt{x.im}} \cdot \sqrt{x.im}\right)\right) + 0 \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot \left(\sqrt{x.im} \cdot \color{blue}{\sqrt{x.im}}\right)\right) + 0 \]

      5. associate-*r* N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \color{blue}{\left(\left(\left(x.re - x.im\right) \cdot \sqrt{x.im}\right) \cdot \sqrt{x.im}\right)} + 0 \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(x.re - x.im\right) \cdot \color{blue}{\left(\sqrt{\sqrt{x.im}} \cdot \sqrt{\sqrt{x.im}}\right)}\right) \cdot \sqrt{x.im}\right) + 0 \]

      7. sqrt-prod N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(x.re - x.im\right) \cdot \color{blue}{\sqrt{\sqrt{x.im} \cdot \sqrt{x.im}}}\right) \cdot \sqrt{x.im}\right) + 0 \]

      8. sqr-neg-rev N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(x.re - x.im\right) \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\sqrt{x.im}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{x.im}\right)\right)}}\right) \cdot \sqrt{x.im}\right) + 0 \]

      9. sqrt-prod N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(x.re - x.im\right) \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(\sqrt{x.im}\right)} \cdot \sqrt{\mathsf{neg}\left(\sqrt{x.im}\right)}\right)}\right) \cdot \sqrt{x.im}\right) + 0 \]

      10. rem-square-sqrt N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(\left(x.re - x.im\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{x.im}\right)\right)}\right) \cdot \sqrt{x.im}\right) + 0 \]

      11. lower-*.f64 N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \color{blue}{\left(\left(\left(x.re - x.im\right) \cdot \left(\mathsf{neg}\left(\sqrt{x.im}\right)\right)\right) \cdot \sqrt{x.im}\right)} + 0 \]

   10. Applied rewrites 99.9%

       \[\leadsto \left(x.im + x.re\right) \cdot \color{blue}{\left(\left(\left(x.re - x.im\right) \cdot \sqrt{x.im}\right) \cdot \sqrt{x.im}\right)} + 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(\left(\left(x.re - x.im\right) \cdot \sqrt{x.im}\right) \cdot \sqrt{x.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m + x.re\right) \cdot \left(\left(x.re - x.im\_m\right) \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+62}:\\ \;\;\;\;t\_0 + \left(x.re \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* (+ x.im_m x.re) (* (- x.re x.im_m) x.im_m))))
   (*
    x.im_s
    (if (<= x.im_m 5e+62) (+ t_0 (* (* x.re (+ x.im_m x.im_m)) x.re)) t_0))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_im_m + x_46_re) * ((x_46_re - x_46_im_m) * x_46_im_m);
	double tmp;
	if (x_46_im_m <= 5e+62) {
		tmp = t_0 + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im_m + x_46re) * ((x_46re - x_46im_m) * x_46im_m)
    if (x_46im_m <= 5d+62) then
        tmp = t_0 + ((x_46re * (x_46im_m + x_46im_m)) * x_46re)
    else
        tmp = t_0
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_im_m + x_46_re) * ((x_46_re - x_46_im_m) * x_46_im_m);
	double tmp;
	if (x_46_im_m <= 5e+62) {
		tmp = t_0 + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (x_46_im_m + x_46_re) * ((x_46_re - x_46_im_m) * x_46_im_m)
	tmp = 0
	if x_46_im_m <= 5e+62:
		tmp = t_0 + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re)
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m + x_46_re) * Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m))
	tmp = 0.0
	if (x_46_im_m <= 5e+62)
		tmp = Float64(t_0 + Float64(Float64(x_46_re * Float64(x_46_im_m + x_46_im_m)) * x_46_re));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (x_46_im_m + x_46_re) * ((x_46_re - x_46_im_m) * x_46_im_m);
	tmp = 0.0;
	if (x_46_im_m <= 5e+62)
		tmp = t_0 + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+62], N[(t$95$0 + N[(N[(x$46$re * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.00000000000000029e62

   1. Initial program 84.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. 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 93.3

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

   4. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 93.3

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

   6. Applied rewrites 93.3%

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

   if 5.00000000000000029e62 < x.im

   1. Initial program 77.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. 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 86.7

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

   4. Applied rewrites 86.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 84.8

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

   6. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. +-inverses N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\color{blue}{0}} \]

      10. +-inverses N/A

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

      11. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + x.re \cdot \frac{\color{blue}{0}}{x.im \cdot x.im - x.im \cdot x.im} \]

      12. +-inverses N/A

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

      13. associate-*r/ N/A

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

      14. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \frac{x.re \cdot \color{blue}{0}}{x.im \cdot x.im - x.im \cdot x.im} \]

      15. +-inverses N/A

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

      16. distribute-lft-out-- N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \frac{\color{blue}{0}}{x.im \cdot x.im - x.im \cdot x.im} \]

      20. +-inverses N/A

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

      21. +-inverses N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\color{blue}{0}} \]

      22. +-inverses N/A

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

      23. flip-+ N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 4.5 \cdot 10^{+183}:\\ \;\;\;\;\left(-x.im\_m\right) \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.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 4.5e+183)
    (* (- x.im_m) (* x.im_m x.im_m))
    (* (* x.im_m x.im_m) x.im_m))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 4.5e+183) {
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	} else {
		tmp = (x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 4.5d+183) then
        tmp = -x_46im_m * (x_46im_m * x_46im_m)
    else
        tmp = (x_46im_m * x_46im_m) * x_46im_m
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 4.5e+183) {
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	} else {
		tmp = (x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 4.5e+183:
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m)
	else:
		tmp = (x_46_im_m * x_46_im_m) * x_46_im_m
	return x_46_im_s * tmp

Julia

x.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 <= 4.5e+183)
		tmp = Float64(Float64(-x_46_im_m) * Float64(x_46_im_m * x_46_im_m));
	else
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * x_46_im_m);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 4.5e+183)
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	else
		tmp = (x_46_im_m * x_46_im_m) * x_46_im_m;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 4.5e+183], N[((-x$46$im$95$m) * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.50000000000000017e183

   1. Initial program 86.2%

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

   3. Taylor expanded in x.re around 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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

      13. distribute-lft-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 68.7%

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

   if 4.50000000000000017e183 < x.re

   1. Initial program 55.7%

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

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

      13. distribute-lft-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 5.1%

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

   10. Applied rewrites 0.5%

       \[\leadsto \left(-\sqrt{x.im} \cdot \sqrt{x.im}\right) \cdot \left(x.im \cdot x.im\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 19.1%

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

Alternative 8: 21.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    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) * 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 83.3%

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

3. Taylor expanded in x.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

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

   2. distribute-rgt-in N/A

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

   3. associate-*r* N/A

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

   4. count-2-rev N/A

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

   5. distribute-lft-in N/A

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

   6. count-2-rev N/A

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

   7. distribute-lft-in N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. mul-1-neg N/A

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

   10. cube-mult N/A

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

   11. unpow2 N/A

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

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

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

   13. distribute-lft-out-- N/A

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

   14. lower-*.f64 N/A

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

   15. lower-neg.f64 N/A

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

5. Applied rewrites 91.5%

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

6. Taylor expanded in x.re around 0

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

8. Applied rewrites 62.7%

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

10. Applied rewrites 31.8%

    \[\leadsto \left(-\sqrt{x.im} \cdot \sqrt{x.im}\right) \cdot \left(x.im \cdot x.im\right) \]
11. Step-by-step derivation

12. Applied rewrites 20.2%

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

Developer Target 1: 91.8% 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

real(8) function code(x_46re, x_46im)
    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 2024339 
(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)))