math.cube on complex, real part

Percentage Accurate: 82.5% → 99.8%

Time: 20.8s

Alternatives: 5

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ t_1 := t\_0 - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;t\_0 - x.im \cdot \left(x.re\_m \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im \cdot \left(\frac{x.re\_m}{\frac{x.im}{x.re\_m}} + x.im \cdot -3\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im))))
        (t_1 (- t_0 (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))))
   (*
    x.re_s
    (if (<= t_1 (- INFINITY))
      (* -3.0 (* x.im (* x.re_m x.im)))
      (if (<= t_1 1e+308)
        (- t_0 (* x.im (* x.re_m (+ x.im x.im))))
        (* x.re_m (* x.im (+ (/ x.re_m (/ x.im x.re_m)) (* x.im -3.0)))))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double t_1 = t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	} else if (t_1 <= 1e+308) {
		tmp = t_0 - (x_46_im * (x_46_re_m * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_re_m * (x_46_im * ((x_46_re_m / (x_46_im / x_46_re_m)) + (x_46_im * -3.0)));
	}
	return x_46_re_s * tmp;
}

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double t_1 = t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	} else if (t_1 <= 1e+308) {
		tmp = t_0 - (x_46_im * (x_46_re_m * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_re_m * (x_46_im * ((x_46_re_m / (x_46_im / x_46_re_m)) + (x_46_im * -3.0)));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))
	t_1 = t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im))
	elif t_1 <= 1e+308:
		tmp = t_0 - (x_46_im * (x_46_re_m * (x_46_im + x_46_im)))
	else:
		tmp = x_46_re_m * (x_46_im * ((x_46_re_m / (x_46_im / x_46_re_m)) + (x_46_im * -3.0)))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)))
	t_1 = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im)));
	elseif (t_1 <= 1e+308)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im * Float64(Float64(x_46_re_m / Float64(x_46_im / x_46_re_m)) + Float64(x_46_im * -3.0))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	t_1 = t_0 - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	elseif (t_1 <= 1e+308)
		tmp = t_0 - (x_46_im * (x_46_re_m * (x_46_im + x_46_im)));
	else
		tmp = x_46_re_m * (x_46_im * ((x_46_re_m / (x_46_im / x_46_re_m)) + (x_46_im * -3.0)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], N[(t$95$0 - N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im * N[(N[(x$46$re$95$m / N[(x$46$im / x$46$re$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 96.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re + \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right) + \color{blue}{\left(x.im + x.im\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot x.re + \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)} + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(\mathsf{neg}\left(\left(x.im + x.im\right) \cdot x.im\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \color{blue}{\left(x.im + x.im\right) \cdot x.im}\right)\right)\right) \]

      17. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \left(2 \cdot x.im\right) \cdot x.im\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - 2 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right)\right) \]

   3. Simplified 96.0%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \color{blue}{\left({x.im}^{2} \cdot x.re\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \left(\left(x.im \cdot x.im\right) \cdot x.re\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(x.im, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      5. *-lowering-*.f64 41.0%

         \[\leadsto \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 41.0%

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

   if -inf.0 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1e308

   1. Initial program 99.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right), \mathsf{*.f64}\left(\left(x.im \cdot x.re + x.im \cdot x.re\right), x.im\right)\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right), \mathsf{*.f64}\left(\left(x.re \cdot \left(x.im + x.im\right)\right), x.im\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.im + x.im\right)\right), x.im\right)\right) \]

      4. +-lowering-+.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(x.im, x.im\right)\right), x.im\right)\right) \]

   4. Applied egg-rr 99.7%

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

   if 1e308 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 45.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re + \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right) + \color{blue}{\left(x.im + x.im\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot x.re + \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)} + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(\mathsf{neg}\left(\left(x.im + x.im\right) \cdot x.im\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \color{blue}{\left(x.im + x.im\right) \cdot x.im}\right)\right)\right) \]

      17. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \left(2 \cdot x.im\right) \cdot x.im\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - 2 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right)\right) \]

   3. Simplified 69.9%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.im around inf

      \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left({x.im}^{2} \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} - 3\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\left({x.im}^{2}\right), \color{blue}{\left(\frac{{x.re}^{2}}{{x.im}^{2}} - 3\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\left(x.im \cdot x.im\right), \left(\color{blue}{\frac{{x.re}^{2}}{{x.im}^{2}}} - 3\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\color{blue}{\frac{{x.re}^{2}}{{x.im}^{2}}} - 3\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\frac{{x.re}^{2}}{{x.im}^{2}} + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\frac{{x.re}^{2}}{{x.im}^{2}} + -3\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(-3 + \color{blue}{\frac{{x.re}^{2}}{{x.im}^{2}}}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \color{blue}{\left(\frac{{x.re}^{2}}{{x.im}^{2}}\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left({x.re}^{2}\right), \color{blue}{\left({x.im}^{2}\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(x.re \cdot x.re\right), \left({\color{blue}{x.im}}^{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left({\color{blue}{x.im}}^{2}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot \color{blue}{x.im}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 51.1%

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right)\right)\right) \]

   7. Simplified 51.1%

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

      1. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \left(\frac{x.re}{x.im} \cdot \color{blue}{\frac{x.re}{x.im}}\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \left(\frac{\frac{x.re}{x.im} \cdot x.re}{\color{blue}{x.im}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(\frac{x.re}{x.im} \cdot x.re\right), \color{blue}{x.im}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{x.re}{x.im}\right), x.re\right), x.im\right)\right)\right)\right) \]

      5. /-lowering-/.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x.re, x.im\right), x.re\right), x.im\right)\right)\right)\right) \]

   9. Applied egg-rr 75.7%

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x.im \cdot x.im\right) \cdot \left(-3 + \frac{\frac{x.re}{x.im} \cdot x.re}{x.im}\right)\right), \color{blue}{x.re}\right) \]

   11. Applied egg-rr 69.9%

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

       1. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x.re}{\frac{x.im}{x.re}} \cdot x.im + \left(x.im \cdot x.im\right) \cdot -3\right), x.re\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot \frac{x.re}{\frac{x.im}{x.re}} + \left(x.im \cdot x.im\right) \cdot -3\right), x.re\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot \frac{x.re}{\frac{x.im}{x.re}} + x.im \cdot \left(x.im \cdot -3\right)\right), x.re\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot \left(\frac{x.re}{\frac{x.im}{x.re}} + x.im \cdot -3\right)\right), x.re\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{x.re}{\frac{x.im}{x.re}} + x.im \cdot -3\right)\right), x.re\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(\left(\frac{x.re}{\frac{x.im}{x.re}}\right), \left(x.im \cdot -3\right)\right)\right), x.re\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x.re, \left(\frac{x.im}{x.re}\right)\right), \left(x.im \cdot -3\right)\right)\right), x.re\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(x.im, x.re\right)\right), \left(x.im \cdot -3\right)\right)\right), x.re\right) \]

       9. *-lowering-*.f64 94.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(x.im, x.re\right)\right), \mathsf{*.f64}\left(x.im, -3\right)\right)\right), x.re\right) \]

   13. Applied egg-rr 94.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -\infty:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+308}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(\frac{x.re}{\frac{x.im}{x.re}} + x.im \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5 \cdot 10^{-107}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re\_m \leq 5 \cdot 10^{+102}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m + \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im \cdot \left(\frac{x.re\_m}{\frac{x.im}{x.re\_m}} + x.im \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5e-107)
    (* -3.0 (* x.im (* x.re_m x.im)))
    (if (<= x.re_m 5e+102)
      (* x.re_m (+ (* x.re_m x.re_m) (* (* x.im x.im) -3.0)))
      (* x.re_m (* x.im (+ (/ x.re_m (/ x.im x.re_m)) (* x.im -3.0))))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5e-107) {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	} else if (x_46_re_m <= 5e+102) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + ((x_46_im * x_46_im) * -3.0));
	} else {
		tmp = x_46_re_m * (x_46_im * ((x_46_re_m / (x_46_im / x_46_re_m)) + (x_46_im * -3.0)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 5d-107) then
        tmp = (-3.0d0) * (x_46im * (x_46re_m * x_46im))
    else if (x_46re_m <= 5d+102) then
        tmp = x_46re_m * ((x_46re_m * x_46re_m) + ((x_46im * x_46im) * (-3.0d0)))
    else
        tmp = x_46re_m * (x_46im * ((x_46re_m / (x_46im / x_46re_m)) + (x_46im * (-3.0d0))))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5e-107) {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	} else if (x_46_re_m <= 5e+102) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + ((x_46_im * x_46_im) * -3.0));
	} else {
		tmp = x_46_re_m * (x_46_im * ((x_46_re_m / (x_46_im / x_46_re_m)) + (x_46_im * -3.0)));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 5e-107:
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im))
	elif x_46_re_m <= 5e+102:
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + ((x_46_im * x_46_im) * -3.0))
	else:
		tmp = x_46_re_m * (x_46_im * ((x_46_re_m / (x_46_im / x_46_re_m)) + (x_46_im * -3.0)))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 5e-107)
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im)));
	elseif (x_46_re_m <= 5e+102)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) + Float64(Float64(x_46_im * x_46_im) * -3.0)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im * Float64(Float64(x_46_re_m / Float64(x_46_im / x_46_re_m)) + Float64(x_46_im * -3.0))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 5e-107)
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	elseif (x_46_re_m <= 5e+102)
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + ((x_46_im * x_46_im) * -3.0));
	else
		tmp = x_46_re_m * (x_46_im * ((x_46_re_m / (x_46_im / x_46_re_m)) + (x_46_im * -3.0)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 5e-107], N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 5e+102], N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im * N[(N[(x$46$re$95$m / N[(x$46$im / x$46$re$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 4.99999999999999971e-107

   1. Initial program 85.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re + \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right) + \color{blue}{\left(x.im + x.im\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot x.re + \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)} + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(\mathsf{neg}\left(\left(x.im + x.im\right) \cdot x.im\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \color{blue}{\left(x.im + x.im\right) \cdot x.im}\right)\right)\right) \]

      17. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \left(2 \cdot x.im\right) \cdot x.im\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - 2 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right)\right) \]

   3. Simplified 92.5%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \color{blue}{\left({x.im}^{2} \cdot x.re\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \left(\left(x.im \cdot x.im\right) \cdot x.re\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(x.im, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      5. *-lowering-*.f64 56.6%

         \[\leadsto \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 56.6%

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

   if 4.99999999999999971e-107 < x.re < 5e102

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re + \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right) + \color{blue}{\left(x.im + x.im\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot x.re + \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)} + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(\mathsf{neg}\left(\left(x.im + x.im\right) \cdot x.im\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \color{blue}{\left(x.im + x.im\right) \cdot x.im}\right)\right)\right) \]

      17. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \left(2 \cdot x.im\right) \cdot x.im\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - 2 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)} \]
   4. Add Preprocessing

   if 5e102 < x.re

   1. Initial program 57.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re + \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right) + \color{blue}{\left(x.im + x.im\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot x.re + \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)} + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(\mathsf{neg}\left(\left(x.im + x.im\right) \cdot x.im\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \color{blue}{\left(x.im + x.im\right) \cdot x.im}\right)\right)\right) \]

      17. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \left(2 \cdot x.im\right) \cdot x.im\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - 2 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right)\right) \]

   3. Simplified 71.4%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.im around inf

      \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left({x.im}^{2} \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} - 3\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\left({x.im}^{2}\right), \color{blue}{\left(\frac{{x.re}^{2}}{{x.im}^{2}} - 3\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\left(x.im \cdot x.im\right), \left(\color{blue}{\frac{{x.re}^{2}}{{x.im}^{2}}} - 3\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\color{blue}{\frac{{x.re}^{2}}{{x.im}^{2}}} - 3\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\frac{{x.re}^{2}}{{x.im}^{2}} + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\frac{{x.re}^{2}}{{x.im}^{2}} + -3\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(-3 + \color{blue}{\frac{{x.re}^{2}}{{x.im}^{2}}}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \color{blue}{\left(\frac{{x.re}^{2}}{{x.im}^{2}}\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left({x.re}^{2}\right), \color{blue}{\left({x.im}^{2}\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(x.re \cdot x.re\right), \left({\color{blue}{x.im}}^{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left({\color{blue}{x.im}}^{2}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot \color{blue}{x.im}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 34.3%

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right)\right)\right) \]

   7. Simplified 34.3%

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

      1. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \left(\frac{x.re}{x.im} \cdot \color{blue}{\frac{x.re}{x.im}}\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \left(\frac{\frac{x.re}{x.im} \cdot x.re}{\color{blue}{x.im}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(\frac{x.re}{x.im} \cdot x.re\right), \color{blue}{x.im}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{x.re}{x.im}\right), x.re\right), x.im\right)\right)\right)\right) \]

      5. /-lowering-/.f64 62.9%

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x.re, x.im\right), x.re\right), x.im\right)\right)\right)\right) \]

   9. Applied egg-rr 62.9%

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x.im \cdot x.im\right) \cdot \left(-3 + \frac{\frac{x.re}{x.im} \cdot x.re}{x.im}\right)\right), \color{blue}{x.re}\right) \]

   11. Applied egg-rr 71.4%

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

       1. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x.re}{\frac{x.im}{x.re}} \cdot x.im + \left(x.im \cdot x.im\right) \cdot -3\right), x.re\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot \frac{x.re}{\frac{x.im}{x.re}} + \left(x.im \cdot x.im\right) \cdot -3\right), x.re\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot \frac{x.re}{\frac{x.im}{x.re}} + x.im \cdot \left(x.im \cdot -3\right)\right), x.re\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot \left(\frac{x.re}{\frac{x.im}{x.re}} + x.im \cdot -3\right)\right), x.re\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{x.re}{\frac{x.im}{x.re}} + x.im \cdot -3\right)\right), x.re\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(\left(\frac{x.re}{\frac{x.im}{x.re}}\right), \left(x.im \cdot -3\right)\right)\right), x.re\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x.re, \left(\frac{x.im}{x.re}\right)\right), \left(x.im \cdot -3\right)\right)\right), x.re\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(x.im, x.re\right)\right), \left(x.im \cdot -3\right)\right)\right), x.re\right) \]

       9. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(x.im, x.re\right)\right), \mathsf{*.f64}\left(x.im, -3\right)\right)\right), x.re\right) \]

   13. Applied egg-rr 100.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5 \cdot 10^{-107}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+102}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(\frac{x.re}{\frac{x.im}{x.re}} + x.im \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 7 \cdot 10^{+149}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m + \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 7e+149)
    (* x.re_m (+ (* x.re_m x.re_m) (* (* x.im x.im) -3.0)))
    (* -3.0 (* x.im (* x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 7e+149) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + ((x_46_im * x_46_im) * -3.0));
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 7d+149) then
        tmp = x_46re_m * ((x_46re_m * x_46re_m) + ((x_46im * x_46im) * (-3.0d0)))
    else
        tmp = (-3.0d0) * (x_46im * (x_46re_m * x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 7e+149) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + ((x_46_im * x_46_im) * -3.0));
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 7e+149:
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + ((x_46_im * x_46_im) * -3.0))
	else:
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 7e+149)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) + Float64(Float64(x_46_im * x_46_im) * -3.0)));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 7e+149)
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + ((x_46_im * x_46_im) * -3.0));
	else
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 7e+149], N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.00000000000000023e149

   1. Initial program 87.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re + \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right) + \color{blue}{\left(x.im + x.im\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot x.re + \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)} + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(\mathsf{neg}\left(\left(x.im + x.im\right) \cdot x.im\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \color{blue}{\left(x.im + x.im\right) \cdot x.im}\right)\right)\right) \]

      17. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \left(2 \cdot x.im\right) \cdot x.im\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - 2 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right)\right) \]

   3. Simplified 95.0%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)} \]
   4. Add Preprocessing

   if 7.00000000000000023e149 < x.im

   1. Initial program 60.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re + \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right) + \color{blue}{\left(x.im + x.im\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot x.re + \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)} + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(\mathsf{neg}\left(\left(x.im + x.im\right) \cdot x.im\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \color{blue}{\left(x.im + x.im\right) \cdot x.im}\right)\right)\right) \]

      17. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \left(2 \cdot x.im\right) \cdot x.im\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - 2 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right)\right) \]

   3. Simplified 60.3%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \color{blue}{\left({x.im}^{2} \cdot x.re\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \left(\left(x.im \cdot x.im\right) \cdot x.re\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(x.im, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      5. *-lowering-*.f64 86.7%

         \[\leadsto \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 86.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7 \cdot 10^{+149}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 800000000000:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 800000000000.0)
    (* x.re_m (* x.re_m x.re_m))
    (* -3.0 (* x.im (* x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 800000000000.0) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 800000000000.0d0) then
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    else
        tmp = (-3.0d0) * (x_46im * (x_46re_m * x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 800000000000.0) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 800000000000.0:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	else:
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 800000000000.0)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 800000000000.0)
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	else
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 800000000000.0], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 8e11

   1. Initial program 90.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re + \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right) + \color{blue}{\left(x.im + x.im\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot x.re + \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)} + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(\mathsf{neg}\left(\left(x.im + x.im\right) \cdot x.im\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \color{blue}{\left(x.im + x.im\right) \cdot x.im}\right)\right)\right) \]

      17. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \left(2 \cdot x.im\right) \cdot x.im\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - 2 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right)\right) \]

   3. Simplified 94.3%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   6. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

         \[\leadsto x.re \cdot {x.re}^{\color{blue}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left({x.re}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot \color{blue}{x.re}\right)\right) \]

      5. *-lowering-*.f64 73.2%

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right) \]

   7. Simplified 73.2%

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

   if 8e11 < x.im

   1. Initial program 66.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re + \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right) + \color{blue}{\left(x.im + x.im\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot x.re + \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)} + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(\mathsf{neg}\left(\left(x.im + x.im\right) \cdot x.im\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \color{blue}{\left(x.im + x.im\right) \cdot x.im}\right)\right)\right) \]

      17. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \left(2 \cdot x.im\right) \cdot x.im\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - 2 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right)\right) \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \color{blue}{\left({x.im}^{2} \cdot x.re\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \left(\left(x.im \cdot x.im\right) \cdot x.re\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(x.im, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      5. *-lowering-*.f64 68.6%

         \[\leadsto \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 68.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 800000000000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_re_m * (x_46_re_m * x_46_re_m));
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * (x_46re_m * (x_46re_m * x_46re_m))
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_re_m * (x_46_re_m * x_46_re_m));
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * (x_46_re_m * (x_46_re_m * x_46_re_m))

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * (x_46_re_m * (x_46_re_m * x_46_re_m));
end

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. *-commutative N/A

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

   5. distribute-lft-out N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-out N/A

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

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)}\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x.re, \left(\left(x.re \cdot x.re + \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right) + \color{blue}{\left(x.im + x.im\right)} \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

   12. associate-+l+ N/A

       \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot x.re + \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \color{blue}{\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\color{blue}{\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)} + \left(x.im + x.im\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

   15. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) + \left(\mathsf{neg}\left(\left(x.im + x.im\right) \cdot x.im\right)\right)\right)\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \color{blue}{\left(x.im + x.im\right) \cdot x.im}\right)\right)\right) \]

   17. count-2 N/A

       \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - \left(2 \cdot x.im\right) \cdot x.im\right)\right)\right) \]

   18. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right) - 2 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right)\right) \]

3. Simplified 91.0%

   \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im\right) \cdot -3\right)} \]
4. Add Preprocessing

5. Taylor expanded in x.re around inf

   \[\leadsto \color{blue}{{x.re}^{3}} \]
6. Step-by-step derivation

   1. cube-mult N/A

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

   2. unpow2 N/A

      \[\leadsto x.re \cdot {x.re}^{\color{blue}{2}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left({x.re}^{2}\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x.re, \left(x.re \cdot \color{blue}{x.re}\right)\right) \]

   5. *-lowering-*.f64 62.7%

      \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right) \]

7. Simplified 62.7%

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

Developer Target 1: 87.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * 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_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

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

Reproduce

herbie shell --seed 2024141 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

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