math.cube on complex, real part

Percentage Accurate: 83.1% → 99.4%

Time: 14.1s

Alternatives: 6

Speedup: 1.2×

• 44.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 83.1% 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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.26 \cdot 10^{+86}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right) + x.im \cdot \left(x.im \cdot \left(x.re\_m \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re\_m + x.im\right) \cdot \left(x.re\_m \cdot x.im\right)\right) \cdot \left(-1 + \frac{x.re\_m}{x.im}\right) - \left(x.im + x.im\right)\\ \end{array} \end{array} \]

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 2.26e+86)
    (+ (* x.re_m (* x.re_m x.re_m)) (* x.im (* x.im (* x.re_m -3.0))))
    (-
     (* (* (+ x.re_m x.im) (* x.re_m x.im)) (+ -1.0 (/ x.re_m x.im)))
     (+ x.im 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_re_m <= 2.26e+86) {
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) + (x_46_im * (x_46_im * (x_46_re_m * -3.0)));
	} else {
		tmp = (((x_46_re_m + x_46_im) * (x_46_re_m * x_46_im)) * (-1.0 + (x_46_re_m / x_46_im))) - (x_46_im + 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_46re_m <= 2.26d+86) then
        tmp = (x_46re_m * (x_46re_m * x_46re_m)) + (x_46im * (x_46im * (x_46re_m * (-3.0d0))))
    else
        tmp = (((x_46re_m + x_46im) * (x_46re_m * x_46im)) * ((-1.0d0) + (x_46re_m / x_46im))) - (x_46im + 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_re_m <= 2.26e+86) {
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) + (x_46_im * (x_46_im * (x_46_re_m * -3.0)));
	} else {
		tmp = (((x_46_re_m + x_46_im) * (x_46_re_m * x_46_im)) * (-1.0 + (x_46_re_m / x_46_im))) - (x_46_im + 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_re_m <= 2.26e+86:
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) + (x_46_im * (x_46_im * (x_46_re_m * -3.0)))
	else:
		tmp = (((x_46_re_m + x_46_im) * (x_46_re_m * x_46_im)) * (-1.0 + (x_46_re_m / x_46_im))) - (x_46_im + 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_re_m <= 2.26e+86)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)) + Float64(x_46_im * Float64(x_46_im * Float64(x_46_re_m * -3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m * x_46_im)) * Float64(-1.0 + Float64(x_46_re_m / x_46_im))) - Float64(x_46_im + 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_re_m <= 2.26e+86)
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) + (x_46_im * (x_46_im * (x_46_re_m * -3.0)));
	else
		tmp = (((x_46_re_m + x_46_im) * (x_46_re_m * x_46_im)) * (-1.0 + (x_46_re_m / x_46_im))) - (x_46_im + 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$46$re$95$m, 2.26e+86], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(x$46$im * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x$46$re$95$m / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.26e86

   1. Initial program 84.2%

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

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

      5. distribute-rgt-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 85.5%

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

      1. distribute-lft-in N/A

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

      2. cube-unmult N/A

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

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

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

      4. cube-unmult N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. *-lowering-*.f64 87.6%

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

   6. Applied egg-rr 87.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 87.6%

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

   8. Applied egg-rr 87.6%

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

   if 2.26e86 < x.re

   1. Initial program 72.1%

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

      1. difference-of-squares N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 79.1%

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

   4. Applied egg-rr 79.1%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \color{blue}{\left(x.im \cdot \left(-1 \cdot x.re + \frac{{x.re}^{2}}{x.im}\right)\right)}\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right), x.im\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \left(-1 \cdot x.re + \frac{{x.re}^{2}}{x.im}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \color{blue}{\mathsf{*.f64}\left(x.im, x.re\right)}\right), x.im\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \left(x.re \cdot -1 + \frac{{x.re}^{2}}{x.im}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right), x.im\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \left(x.re \cdot -1 + \frac{x.re \cdot x.re}{x.im}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right), x.im\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \left(x.re \cdot -1 + x.re \cdot \frac{x.re}{x.im}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right), x.im\right)\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \left(x.re \cdot \left(-1 + \frac{x.re}{x.im}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right), x.im\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(-1 + \frac{x.re}{x.im}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right), x.im\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(-1, \left(\frac{x.re}{x.im}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right), x.im\right)\right) \]

      8. /-lowering-/.f64 77.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right), x.im\right)\right) \]

   7. Simplified 77.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x.re + x.im\right) \cdot \left(\left(x.im \cdot x.re\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \color{blue}{\mathsf{*.f64}\left(x.im, x.re\right)}\right), x.im\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x.re + x.im\right) \cdot \left(\left(x.re \cdot x.im\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(\color{blue}{x.im}, x.re\right)\right), x.im\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(x.re + x.im\right) \cdot \left(x.re \cdot x.im\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.im\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\left(x.re + x.im\right) \cdot \left(x.re \cdot x.im\right)\right), \left(-1 + \frac{x.re}{x.im}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.im\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x.re + x.im\right), \left(x.re \cdot x.im\right)\right), \left(-1 + \frac{x.re}{x.im}\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.im\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \left(x.re \cdot x.im\right)\right), \left(-1 + \frac{x.re}{x.im}\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{x.re}, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right), x.im\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \left(-1 + \frac{x.re}{x.im}\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \color{blue}{x.im}\right), \mathsf{*.f64}\left(x.im, x.re\right)\right), x.im\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \left(\frac{x.re}{x.im}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \color{blue}{\mathsf{*.f64}\left(x.im, x.re\right)}\right), x.im\right)\right) \]

      9. /-lowering-/.f64 79.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right), x.im\right)\right) \]

   9. Applied egg-rr 79.1%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(x.im \cdot \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(x.im \cdot \left(x.re \cdot x.im + x.re \cdot \color{blue}{x.im}\right)\right)\right) \]

       3. flip-+ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(x.im \cdot \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\color{blue}{x.re \cdot x.im - x.re \cdot x.im}}\right)\right) \]

       4. fmm-def N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(x.im \cdot \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\mathsf{fma}\left(x.re, \color{blue}{x.im}, \mathsf{neg}\left(x.re \cdot x.im\right)\right)}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(x.im \cdot \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}\right)\right) \]

       6. +-inverses N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(x.im \cdot \frac{0}{\mathsf{fma}\left(\color{blue}{x.re}, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}\right)\right) \]

       7. +-inverses N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(x.im \cdot \frac{x.re \cdot x.im - x.re \cdot x.im}{\mathsf{fma}\left(\color{blue}{x.re}, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}\right)\right) \]

       8. fmm-def N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(x.im \cdot \frac{\mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.re \cdot x.im\right)\right)}{\mathsf{fma}\left(\color{blue}{x.re}, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(x.im \cdot \frac{\mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}{\mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(\frac{x.im \cdot \mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}{\color{blue}{\mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(\frac{x.im \cdot \mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.re \cdot x.im\right)\right)}{\mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}\right)\right) \]

       12. fmm-def N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(\frac{x.im \cdot \left(x.re \cdot x.im - x.re \cdot x.im\right)}{\mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}\right)\right) \]

       13. +-inverses N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(\frac{x.im \cdot 0}{\mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}\right)\right) \]

       14. +-inverses N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(\frac{x.im \cdot \left(x.im - x.im\right)}{\mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}\right)\right) \]

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

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(\frac{x.im \cdot x.im - x.im \cdot x.im}{\mathsf{fma}\left(\color{blue}{x.re}, x.im, \mathsf{neg}\left(x.im \cdot x.re\right)\right)}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(\frac{x.im \cdot x.im - x.im \cdot x.im}{\mathsf{fma}\left(x.re, x.im, \mathsf{neg}\left(x.re \cdot x.im\right)\right)}\right)\right) \]

       17. fmm-def N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(\frac{x.im \cdot x.im - x.im \cdot x.im}{x.re \cdot x.im - \color{blue}{x.re \cdot x.im}}\right)\right) \]

       18. +-inverses N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(\frac{x.im \cdot x.im - x.im \cdot x.im}{0}\right)\right) \]

       19. +-inverses N/A

           \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.re, x.im\right)\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(x.re, x.im\right)\right)\right), \left(\frac{x.im \cdot x.im - x.im \cdot x.im}{x.im - \color{blue}{x.im}}\right)\right) \]

   11. Applied egg-rr 100.0%

       \[\leadsto \left(\left(x.re + x.im\right) \cdot \left(x.re \cdot x.im\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right) - \color{blue}{\left(x.im + x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.26 \cdot 10^{+86}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re + x.im\right) \cdot \left(x.re \cdot x.im\right)\right) \cdot \left(-1 + \frac{x.re}{x.im}\right) - \left(x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 1.15 \cdot 10^{+135}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re\_m \cdot -3\right)\right)\\ \end{array} \end{array} \]

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.im 1.15e+135)
    (* x.re_m (+ (* x.re_m x.re_m) (* x.im (* x.im -3.0))))
    (* x.im (* x.im (* x.re_m -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_im <= 1.15e+135) {
		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_im * (x_46_im * (x_46_re_m * -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_46im <= 1.15d+135) then
        tmp = x_46re_m * ((x_46re_m * x_46re_m) + (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = x_46im * (x_46im * (x_46re_m * (-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_im <= 1.15e+135) {
		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_im * (x_46_im * (x_46_re_m * -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_im <= 1.15e+135:
		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_im * (x_46_im * (x_46_re_m * -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_im <= 1.15e+135)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re_m * -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_im <= 1.15e+135)
		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_im * (x_46_im * (x_46_re_m * -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$46$im, 1.15e+135], N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.1500000000000001e135

   1. Initial program 85.2%

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

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

      5. distribute-rgt-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 89.0%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 89.0%

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

   6. Applied egg-rr 89.0%

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

   if 1.1500000000000001e135 < x.im

   1. Initial program 56.8%

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

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

      5. distribute-rgt-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 56.9%

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

   5. Taylor expanded in x.re around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 71.7%

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

   7. Simplified 71.7%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-lowering-*.f64 88.7%

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

   9. Applied egg-rr 88.7%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

       4. *-lowering-*.f64 88.9%

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

   11. Applied egg-rr 88.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.15 \cdot 10^{+135}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 2.3 \cdot 10^{+123}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m + -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re\_m \cdot -3\right)\right)\\ \end{array} \end{array} \]

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.im 2.3e+123)
    (* x.re_m (+ (* x.re_m x.re_m) (* -3.0 (* x.im x.im))))
    (* x.im (* x.im (* x.re_m -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_im <= 2.3e+123) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re_m * -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_46im <= 2.3d+123) then
        tmp = x_46re_m * ((x_46re_m * x_46re_m) + ((-3.0d0) * (x_46im * x_46im)))
    else
        tmp = x_46im * (x_46im * (x_46re_m * (-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_im <= 2.3e+123) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re_m * -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_im <= 2.3e+123:
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + (-3.0 * (x_46_im * x_46_im)))
	else:
		tmp = x_46_im * (x_46_im * (x_46_re_m * -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_im <= 2.3e+123)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) + Float64(-3.0 * Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re_m * -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_im <= 2.3e+123)
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) + (-3.0 * (x_46_im * x_46_im)));
	else
		tmp = x_46_im * (x_46_im * (x_46_re_m * -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$46$im, 2.3e+123], N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.2999999999999999e123

   1. Initial program 84.9%

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

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

      5. distribute-rgt-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 88.8%

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

   if 2.2999999999999999e123 < x.im

   1. Initial program 62.4%

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

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

      5. distribute-rgt-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 62.4%

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

   5. Taylor expanded in x.re around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 75.3%

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

   7. Simplified 75.3%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-lowering-*.f64 90.1%

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

   9. Applied egg-rr 90.1%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

       4. *-lowering-*.f64 90.3%

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

   11. Applied egg-rr 90.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.3 \cdot 10^{+123}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 1.45 \cdot 10^{+68}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \end{array} \]

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 1.45e+68)
    (* x.re_m (* x.re_m x.re_m))
    (* (* x.re_m x.im) (* 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_im <= 1.45e+68) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = (x_46_re_m * x_46_im) * (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_46im <= 1.45d+68) then
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    else
        tmp = (x_46re_m * x_46im) * (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_im <= 1.45e+68) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = (x_46_re_m * x_46_im) * (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_im <= 1.45e+68:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	else:
		tmp = (x_46_re_m * x_46_im) * (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_im <= 1.45e+68)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_im) * 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_im <= 1.45e+68)
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	else
		tmp = (x_46_re_m * x_46_im) * (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$46$im, 1.45e+68], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.45000000000000006e68

   1. Initial program 85.2%

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

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

      5. distribute-rgt-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 88.4%

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

   5. Taylor expanded in x.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 65.1%

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

   7. Simplified 65.1%

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

   if 1.45000000000000006e68 < x.im

   1. Initial program 65.9%

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

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

      5. distribute-rgt-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 70.8%

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

   5. Taylor expanded in x.re around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 73.3%

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

   7. Simplified 73.3%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-lowering-*.f64 84.9%

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

   9. Applied egg-rr 84.9%

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

Alternative 5: 68.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 1.75 \cdot 10^{+66}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re\_m \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \end{array} \]

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 1.75e+66)
    (* x.re_m (* x.re_m x.re_m))
    (* -3.0 (* x.re_m (* x.im 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 <= 1.75e+66) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = -3.0 * (x_46_re_m * (x_46_im * 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 <= 1.75d+66) then
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    else
        tmp = (-3.0d0) * (x_46re_m * (x_46im * 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 <= 1.75e+66) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = -3.0 * (x_46_re_m * (x_46_im * 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 <= 1.75e+66:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	else:
		tmp = -3.0 * (x_46_re_m * (x_46_im * 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 <= 1.75e+66)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = Float64(-3.0 * Float64(x_46_re_m * Float64(x_46_im * 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 <= 1.75e+66)
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	else
		tmp = -3.0 * (x_46_re_m * (x_46_im * 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, 1.75e+66], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$re$95$m * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.7499999999999999e66

   1. Initial program 85.2%

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

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

      5. distribute-rgt-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 88.4%

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

   5. Taylor expanded in x.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 65.1%

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

   7. Simplified 65.1%

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

   if 1.7499999999999999e66 < x.im

   1. Initial program 65.9%

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

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

      5. distribute-rgt-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 70.8%

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 73.4%

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

   7. Simplified 73.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.75 \cdot 10^{+66}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.6% 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 82.2%

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

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

   5. distribute-rgt-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 85.6%

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

5. Taylor expanded in x.re around 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 57.4%

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

7. Simplified 57.4%

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

Developer Target 1: 87.6% 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 2024154 
(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)))