math.cube on complex, imaginary part

Percentage Accurate: 82.2% → 99.8%

Time: 8.8s

Alternatives: 12

Speedup: 0.5×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \leq 10^{+106}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\_m\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(x.re \cdot 2 + \left(1 + \frac{x.im\_m}{x.re}\right) \cdot \left(x.re - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<=
       (+
        (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
        (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))
       1e+106)
    (- (* 3.0 (* x.re (* x.re x.im_m))) (pow x.im_m 3.0))
    (*
     (* x.re x.im_m)
     (+ (* x.re 2.0) (* (+ 1.0 (/ x.im_m x.re)) (- x.re x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 1e+106) {
		tmp = (3.0 * (x_46_re * (x_46_re * x_46_im_m))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46im_m * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) + (x_46re * ((x_46re * x_46im_m) + (x_46re * x_46im_m)))) <= 1d+106) then
        tmp = (3.0d0 * (x_46re * (x_46re * x_46im_m))) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46re * x_46im_m) * ((x_46re * 2.0d0) + ((1.0d0 + (x_46im_m / x_46re)) * (x_46re - x_46im_m)))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 1e+106) {
		tmp = (3.0 * (x_46_re * (x_46_re * x_46_im_m))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if ((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 1e+106:
		tmp = (3.0 * (x_46_re * (x_46_re * x_46_im_m))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)))) <= 1e+106)
		tmp = Float64(Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im_m))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(Float64(x_46_re * 2.0) + Float64(Float64(1.0 + Float64(x_46_im_m / x_46_re)) * Float64(x_46_re - x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 1e+106)
		tmp = (3.0 * (x_46_re * (x_46_re * x_46_im_m))) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+106], N[(N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re * 2.0), $MachinePrecision] + N[(N[(1.0 + N[(x$46$im$95$m / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 1.00000000000000009e106

   1. Initial program 93.1%

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

   3. Simplified 98.7%

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

   if 1.00000000000000009e106 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 57.4%

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

      1. difference-of-squares 66.4%

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

      2. *-commutative 66.4%

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

   4. Applied egg-rr 66.4%

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

   5. Taylor expanded in x.re around inf 65.4%

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

      1. *-commutative 65.4%

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

      2. count-2 65.4%

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

      3. *-commutative 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. *-un-lft-identity 65.4%

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

      2. *-commutative 65.4%

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

      3. fma-define 65.4%

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

      4. *-commutative 65.4%

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

      5. associate-*l* 61.2%

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

      6. *-commutative 61.2%

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

      7. associate-*l* 61.2%

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

      8. associate-*r* 61.2%

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

      9. pow2 61.2%

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

   9. Applied egg-rr 61.2%

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

       1. *-lft-identity 61.2%

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

       2. fma-undefine 61.2%

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

       3. +-commutative 61.2%

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

       4. unpow2 61.2%

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

       5. *-commutative 61.2%

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

       6. associate-*l* 61.2%

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

       7. *-commutative 61.2%

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

       8. associate-*r* 61.2%

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

       9. *-commutative 61.2%

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

       10. associate-*r* 61.2%

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

       11. *-commutative 61.2%

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

       12. associate-*r* 61.2%

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

       13. *-commutative 61.2%

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

       14. associate-*r* 70.7%

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

       15. distribute-lft-out 94.7%

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

   11. Simplified 94.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+106}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 2 + \left(1 + \frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \leq 4 \cdot 10^{+107}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re \cdot 3\right)\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(x.re \cdot 2 + \left(1 + \frac{x.im\_m}{x.re}\right) \cdot \left(x.re - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<=
       (+
        (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
        (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))
       4e+107)
    (- (* x.re (* x.im_m (* x.re 3.0))) (pow x.im_m 3.0))
    (*
     (* x.re x.im_m)
     (+ (* x.re 2.0) (* (+ 1.0 (/ x.im_m x.re)) (- x.re x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 4e+107) {
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46im_m * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) + (x_46re * ((x_46re * x_46im_m) + (x_46re * x_46im_m)))) <= 4d+107) then
        tmp = (x_46re * (x_46im_m * (x_46re * 3.0d0))) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46re * x_46im_m) * ((x_46re * 2.0d0) + ((1.0d0 + (x_46im_m / x_46re)) * (x_46re - x_46im_m)))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 4e+107) {
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if ((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 4e+107:
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)))) <= 4e+107)
		tmp = Float64(Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re * 3.0))) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(Float64(x_46_re * 2.0) + Float64(Float64(1.0 + Float64(x_46_im_m / x_46_re)) * Float64(x_46_re - x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 4e+107)
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+107], N[(N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re * 2.0), $MachinePrecision] + N[(N[(1.0 + N[(x$46$im$95$m / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 3.9999999999999999e107

   1. Initial program 93.1%

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

   3. Simplified 98.6%

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

   if 3.9999999999999999e107 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 57.4%

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

      1. difference-of-squares 66.4%

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

      2. *-commutative 66.4%

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

   4. Applied egg-rr 66.4%

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

   5. Taylor expanded in x.re around inf 65.4%

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

      1. *-commutative 65.4%

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

      2. count-2 65.4%

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

      3. *-commutative 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. *-un-lft-identity 65.4%

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

      2. *-commutative 65.4%

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

      3. fma-define 65.4%

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

      4. *-commutative 65.4%

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

      5. associate-*l* 61.2%

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

      6. *-commutative 61.2%

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

      7. associate-*l* 61.2%

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

      8. associate-*r* 61.2%

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

      9. pow2 61.2%

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

   9. Applied egg-rr 61.2%

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

       1. *-lft-identity 61.2%

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

       2. fma-undefine 61.2%

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

       3. +-commutative 61.2%

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

       4. unpow2 61.2%

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

       5. *-commutative 61.2%

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

       6. associate-*l* 61.2%

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

       7. *-commutative 61.2%

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

       8. associate-*r* 61.2%

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

       9. *-commutative 61.2%

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

       10. associate-*r* 61.2%

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

       11. *-commutative 61.2%

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

       12. associate-*r* 61.2%

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

       13. *-commutative 61.2%

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

       14. associate-*r* 70.7%

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

       15. distribute-lft-out 94.7%

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

   11. Simplified 94.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 4 \cdot 10^{+107}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 2 + \left(1 + \frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \leq -1 \cdot 10^{-258}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 3\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(x.re \cdot 2 + \left(1 + \frac{x.im\_m}{x.re}\right) \cdot \left(x.re - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<=
       (+
        (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
        (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))
       -1e-258)
    (- (* x.re (* (* x.re x.im_m) 3.0)) (pow x.im_m 3.0))
    (*
     (* x.re x.im_m)
     (+ (* x.re 2.0) (* (+ 1.0 (/ x.im_m x.re)) (- x.re x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= -1e-258) {
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 3.0)) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46im_m * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) + (x_46re * ((x_46re * x_46im_m) + (x_46re * x_46im_m)))) <= (-1d-258)) then
        tmp = (x_46re * ((x_46re * x_46im_m) * 3.0d0)) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46re * x_46im_m) * ((x_46re * 2.0d0) + ((1.0d0 + (x_46im_m / x_46re)) * (x_46re - x_46im_m)))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= -1e-258) {
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 3.0)) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if ((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= -1e-258:
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 3.0)) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)))) <= -1e-258)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 3.0)) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(Float64(x_46_re * 2.0) + Float64(Float64(1.0 + Float64(x_46_im_m / x_46_re)) * Float64(x_46_re - x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= -1e-258)
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 3.0)) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-258], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re * 2.0), $MachinePrecision] + N[(N[(1.0 + N[(x$46$im$95$m / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -9.99999999999999954e-259

   1. Initial program 88.2%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x.im around 0 97.6%

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

   if -9.99999999999999954e-259 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 76.3%

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

      1. difference-of-squares 81.3%

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

      2. *-commutative 81.3%

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in x.re around inf 80.2%

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

      1. *-commutative 80.2%

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

      2. count-2 80.2%

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

      3. *-commutative 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. *-un-lft-identity 80.2%

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

      2. *-commutative 80.2%

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

      3. fma-define 80.2%

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

      4. *-commutative 80.2%

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

      5. associate-*l* 77.8%

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

      6. *-commutative 77.8%

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

      7. associate-*l* 77.8%

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

      8. associate-*r* 77.8%

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

      9. pow2 77.8%

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

   9. Applied egg-rr 77.8%

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

       1. *-lft-identity 77.8%

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

       2. fma-undefine 77.8%

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

       3. +-commutative 77.8%

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

       4. unpow2 77.8%

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

       5. *-commutative 77.8%

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

       6. associate-*l* 77.8%

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

       7. *-commutative 77.8%

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

       8. associate-*r* 77.8%

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

       9. *-commutative 77.8%

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

       10. associate-*r* 77.8%

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

       11. *-commutative 77.8%

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

       12. associate-*r* 77.8%

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

       13. *-commutative 77.8%

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

       14. associate-*r* 81.3%

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

       15. distribute-lft-out 94.5%

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

   11. Simplified 94.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-258}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 2 + \left(1 + \frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;t\_0 + x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(x.re \cdot 2 + \left(1 + \frac{x.im\_m}{x.re}\right) \cdot \left(x.re - x.im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))))
   (*
    x.im_s
    (if (<= (+ t_0 (* x.re (+ (* x.re x.im_m) (* x.re x.im_m)))) 5e+304)
      (+ t_0 (* x.re (* (* x.re x.im_m) 2.0)))
      (*
       (* x.re x.im_m)
       (+ (* x.re 2.0) (* (+ 1.0 (/ x.im_m x.re)) (- x.re x.im_m))))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m));
	double tmp;
	if ((t_0 + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 5e+304) {
		tmp = t_0 + (x_46_re * ((x_46_re * x_46_im_m) * 2.0));
	} else {
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im_m * ((x_46re * x_46re) - (x_46im_m * x_46im_m))
    if ((t_0 + (x_46re * ((x_46re * x_46im_m) + (x_46re * x_46im_m)))) <= 5d+304) then
        tmp = t_0 + (x_46re * ((x_46re * x_46im_m) * 2.0d0))
    else
        tmp = (x_46re * x_46im_m) * ((x_46re * 2.0d0) + ((1.0d0 + (x_46im_m / x_46re)) * (x_46re - x_46im_m)))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m));
	double tmp;
	if ((t_0 + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 5e+304) {
		tmp = t_0 + (x_46_re * ((x_46_re * x_46_im_m) * 2.0));
	} else {
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))
	tmp = 0
	if (t_0 + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 5e+304:
		tmp = t_0 + (x_46_re * ((x_46_re * x_46_im_m) * 2.0))
	else:
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)))
	tmp = 0.0
	if (Float64(t_0 + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)))) <= 5e+304)
		tmp = Float64(t_0 + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 2.0)));
	else
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(Float64(x_46_re * 2.0) + Float64(Float64(1.0 + Float64(x_46_im_m / x_46_re)) * Float64(x_46_re - x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m));
	tmp = 0.0;
	if ((t_0 + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= 5e+304)
		tmp = t_0 + (x_46_re * ((x_46_re * x_46_im_m) * 2.0));
	else
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[N[(t$95$0 + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+304], N[(t$95$0 + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re * 2.0), $MachinePrecision] + N[(N[(1.0 + N[(x$46$im$95$m / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 4.9999999999999997e304

   1. Initial program 93.9%

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

      1. *-commutative 92.8%

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

      2. count-2 92.8%

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

      3. *-commutative 92.8%

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

   4. Applied egg-rr 93.9%

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

   if 4.9999999999999997e304 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 44.1%

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

      1. difference-of-squares 56.0%

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

      2. *-commutative 56.0%

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

   4. Applied egg-rr 56.0%

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

   5. Taylor expanded in x.re around inf 56.0%

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

      1. *-commutative 56.0%

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

      2. count-2 56.0%

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

      3. *-commutative 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. *-un-lft-identity 56.0%

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

      2. *-commutative 56.0%

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

      3. fma-define 56.0%

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

      4. *-commutative 56.0%

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

      5. associate-*l* 56.0%

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

      6. *-commutative 56.0%

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

      7. associate-*l* 56.0%

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

      8. associate-*r* 56.0%

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

      9. pow2 56.0%

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

   9. Applied egg-rr 56.0%

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

       1. *-lft-identity 56.0%

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

       2. fma-undefine 56.0%

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

       3. +-commutative 56.0%

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

       4. unpow2 56.0%

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

       5. *-commutative 56.0%

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

       6. associate-*l* 56.0%

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

       7. *-commutative 56.0%

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

       8. associate-*r* 56.0%

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

       9. *-commutative 56.0%

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

       10. associate-*r* 56.0%

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

       11. *-commutative 56.0%

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

       12. associate-*r* 56.0%

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

       13. *-commutative 56.0%

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

       14. associate-*r* 68.6%

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

       15. distribute-lft-out 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 2 + \left(1 + \frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 4.4 \cdot 10^{-110}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 2\right) + x.im\_m \cdot \left(x.im\_m \cdot \left(x.re - x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(x.re \cdot 2 + \left(1 + \frac{x.im\_m}{x.re}\right) \cdot \left(x.re - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 4.4e-110)
    (+ (* x.re (* (* x.re x.im_m) 2.0)) (* x.im_m (* x.im_m (- x.re x.im_m))))
    (*
     (* x.re x.im_m)
     (+ (* x.re 2.0) (* (+ 1.0 (/ x.im_m x.re)) (- x.re x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 4.4e-110) {
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)));
	} else {
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 4.4d-110) then
        tmp = (x_46re * ((x_46re * x_46im_m) * 2.0d0)) + (x_46im_m * (x_46im_m * (x_46re - x_46im_m)))
    else
        tmp = (x_46re * x_46im_m) * ((x_46re * 2.0d0) + ((1.0d0 + (x_46im_m / x_46re)) * (x_46re - x_46im_m)))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 4.4e-110) {
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)));
	} else {
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 4.4e-110:
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)))
	else:
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 4.4e-110)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 2.0)) + Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re - x_46_im_m))));
	else
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(Float64(x_46_re * 2.0) + Float64(Float64(1.0 + Float64(x_46_im_m / x_46_re)) * Float64(x_46_re - x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 4.4e-110)
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)));
	else
		tmp = (x_46_re * x_46_im_m) * ((x_46_re * 2.0) + ((1.0 + (x_46_im_m / x_46_re)) * (x_46_re - x_46_im_m)));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 4.4e-110], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re * 2.0), $MachinePrecision] + N[(N[(1.0 + N[(x$46$im$95$m / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.3999999999999999e-110

   1. Initial program 84.0%

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

      1. difference-of-squares 86.4%

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

      2. *-commutative 86.4%

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

   4. Applied egg-rr 86.4%

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

   5. Taylor expanded in x.re around inf 85.3%

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

      1. *-commutative 85.3%

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

      2. count-2 85.3%

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

      3. *-commutative 85.3%

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

   7. Applied egg-rr 85.3%

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

   8. Taylor expanded in x.re around 0 66.6%

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

   if 4.3999999999999999e-110 < x.re

   1. Initial program 74.3%

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

      1. difference-of-squares 79.0%

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

      2. *-commutative 79.0%

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in x.re around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. count-2 79.0%

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

      3. *-commutative 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. *-commutative 79.0%

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

      3. fma-define 79.1%

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

      4. *-commutative 79.1%

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

      5. associate-*l* 79.0%

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

      6. *-commutative 79.0%

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

      7. associate-*l* 79.0%

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

      8. associate-*r* 79.0%

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

      9. pow2 79.0%

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

   9. Applied egg-rr 79.0%

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

       1. *-lft-identity 79.0%

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

       2. fma-undefine 79.0%

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

       3. +-commutative 79.0%

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

       4. unpow2 79.0%

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

       5. *-commutative 79.0%

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

       6. associate-*l* 79.0%

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

       7. *-commutative 79.0%

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

       8. associate-*r* 79.0%

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

       9. *-commutative 79.0%

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

       10. associate-*r* 79.0%

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

       11. *-commutative 79.0%

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

       12. associate-*r* 79.0%

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

       13. *-commutative 79.0%

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

       14. associate-*r* 88.9%

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

       15. distribute-lft-out 99.6%

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

   11. Simplified 99.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4.4 \cdot 10^{-110}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right) + x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 2 + \left(1 + \frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 2\right) + x.im\_m \cdot \left(x.re \cdot \left(x.re - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 1.7e-12)
    (+ (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m))) (* x.re (* x.re x.im_m)))
    (+ (* x.re (* (* x.re x.im_m) 2.0)) (* x.im_m (* x.re (- x.re x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1.7e-12) {
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * (x_46_re * x_46_im_m));
	} else {
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_re * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 1.7d-12) then
        tmp = (x_46im_m * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) + (x_46re * (x_46re * x_46im_m))
    else
        tmp = (x_46re * ((x_46re * x_46im_m) * 2.0d0)) + (x_46im_m * (x_46re * (x_46re - x_46im_m)))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1.7e-12) {
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * (x_46_re * x_46_im_m));
	} else {
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_re * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 1.7e-12:
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * (x_46_re * x_46_im_m))
	else:
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_re * (x_46_re - x_46_im_m)))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 1.7e-12)
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(x_46_re * x_46_im_m)));
	else
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 2.0)) + Float64(x_46_im_m * Float64(x_46_re * Float64(x_46_re - x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 1.7e-12)
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * (x_46_re * x_46_im_m));
	else
		tmp = (x_46_re * ((x_46_re * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_re * (x_46_re - x_46_im_m)));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 1.7e-12], N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m * N[(x$46$re * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.7e-12

   1. Initial program 85.9%

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

      1. *-commutative 87.0%

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

      2. count-2 87.0%

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

      3. *-commutative 87.0%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in x.re around 0 85.9%

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

   7. Simplified 71.7%

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

   if 1.7e-12 < x.re

   1. Initial program 64.7%

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

      1. difference-of-squares 71.2%

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

      2. *-commutative 71.2%

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

   4. Applied egg-rr 71.2%

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

   5. Taylor expanded in x.re around inf 71.2%

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

      1. *-commutative 71.2%

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

      2. count-2 71.2%

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

      3. *-commutative 71.2%

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

   7. Applied egg-rr 71.2%

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

   8. Taylor expanded in x.re around inf 60.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right) + x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 2\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;t\_0 + x.im\_m \cdot \left(x.im\_m \cdot \left(x.re - x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + x.im\_m \cdot \left(x.re \cdot \left(x.re - x.im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re (* (* x.re x.im_m) 2.0))))
   (*
    x.im_s
    (if (<= x.re 1.7e-12)
      (+ t_0 (* x.im_m (* x.im_m (- x.re x.im_m))))
      (+ t_0 (* x.im_m (* x.re (- x.re x.im_m))))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_re * ((x_46_re * x_46_im_m) * 2.0);
	double tmp;
	if (x_46_re <= 1.7e-12) {
		tmp = t_0 + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)));
	} else {
		tmp = t_0 + (x_46_im_m * (x_46_re * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re * ((x_46re * x_46im_m) * 2.0d0)
    if (x_46re <= 1.7d-12) then
        tmp = t_0 + (x_46im_m * (x_46im_m * (x_46re - x_46im_m)))
    else
        tmp = t_0 + (x_46im_m * (x_46re * (x_46re - x_46im_m)))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = x_46_re * ((x_46_re * x_46_im_m) * 2.0);
	double tmp;
	if (x_46_re <= 1.7e-12) {
		tmp = t_0 + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)));
	} else {
		tmp = t_0 + (x_46_im_m * (x_46_re * (x_46_re - x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = x_46_re * ((x_46_re * x_46_im_m) * 2.0)
	tmp = 0
	if x_46_re <= 1.7e-12:
		tmp = t_0 + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)))
	else:
		tmp = t_0 + (x_46_im_m * (x_46_re * (x_46_re - x_46_im_m)))
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 2.0))
	tmp = 0.0
	if (x_46_re <= 1.7e-12)
		tmp = Float64(t_0 + Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re - x_46_im_m))));
	else
		tmp = Float64(t_0 + Float64(x_46_im_m * Float64(x_46_re * Float64(x_46_re - x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = x_46_re * ((x_46_re * x_46_im_m) * 2.0);
	tmp = 0.0;
	if (x_46_re <= 1.7e-12)
		tmp = t_0 + (x_46_im_m * (x_46_im_m * (x_46_re - x_46_im_m)));
	else
		tmp = t_0 + (x_46_im_m * (x_46_re * (x_46_re - x_46_im_m)));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$re, 1.7e-12], N[(t$95$0 + N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(x$46$im$95$m * N[(x$46$re * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.7e-12

   1. Initial program 85.9%

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

      1. difference-of-squares 88.0%

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

      2. *-commutative 88.0%

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

   4. Applied egg-rr 88.0%

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

   5. Taylor expanded in x.re around inf 87.0%

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

      1. *-commutative 87.0%

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

      2. count-2 87.0%

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

      3. *-commutative 87.0%

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

   7. Applied egg-rr 87.0%

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

   8. Taylor expanded in x.re around 0 69.4%

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

   if 1.7e-12 < x.re

   1. Initial program 64.7%

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

      1. difference-of-squares 71.2%

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

      2. *-commutative 71.2%

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

   4. Applied egg-rr 71.2%

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

   5. Taylor expanded in x.re around inf 71.2%

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

      1. *-commutative 71.2%

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

      2. count-2 71.2%

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

      3. *-commutative 71.2%

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

   7. Applied egg-rr 71.2%

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

   8. Taylor expanded in x.re around inf 60.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right) + x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right) + x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5.8 \cdot 10^{-45}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot -3\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5.8e-45)
    (* (* x.re x.re) (* x.im_m 3.0))
    (+ (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m))) (* x.re -3.0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5.8e-45) {
		tmp = (x_46_re * x_46_re) * (x_46_im_m * 3.0);
	} else {
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * -3.0);
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 5.8d-45) then
        tmp = (x_46re * x_46re) * (x_46im_m * 3.0d0)
    else
        tmp = (x_46im_m * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) + (x_46re * (-3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5.8e-45) {
		tmp = (x_46_re * x_46_re) * (x_46_im_m * 3.0);
	} else {
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * -3.0);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 5.8e-45:
		tmp = (x_46_re * x_46_re) * (x_46_im_m * 3.0)
	else:
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * -3.0)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5.8e-45)
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_im_m * 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * -3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 5.8e-45)
		tmp = (x_46_re * x_46_re) * (x_46_im_m * 3.0);
	else
		tmp = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * -3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5.8e-45], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.8e-45

   1. Initial program 81.9%

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

      1. +-commutative 81.9%

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

      2. *-commutative 81.9%

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

      3. sqr-neg 81.9%

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

      4. fma-define 82.4%

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

      5. *-commutative 82.4%

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

      6. distribute-rgt-out 82.4%

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

      7. count-2 82.4%

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

      8. *-commutative 82.4%

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

      9. *-commutative 82.4%

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

      10. sqr-neg 82.4%

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

   3. Simplified 82.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im \cdot 2\right), x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around inf 60.2%

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

      1. unpow2 60.2%

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

   7. Applied egg-rr 60.2%

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

      1. *-un-lft-identity 60.2%

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

      2. distribute-rgt-out 60.2%

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

      3. metadata-eval 60.2%

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

   9. Applied egg-rr 60.2%

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

   if 5.8e-45 < x.im

   1. Initial program 78.3%

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

      1. *-commutative 83.6%

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

      2. count-2 83.6%

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

      3. *-commutative 83.6%

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

   4. Applied egg-rr 78.3%

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

      1. pow1 78.3%

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

      2. *-commutative 78.3%

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

      3. *-commutative 78.3%

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

      4. count-2 78.3%

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

      5. *-commutative 78.3%

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

      6. *-commutative 78.3%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + {\left(x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)}^{1} \]

      9. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + {\left(x.re \cdot \frac{0}{\color{blue}{0}}\right)}^{1} \]

   6. Applied egg-rr 0.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{{\left(x.re \cdot \frac{0}{0}\right)}^{1}} \]

   8. Simplified 75.7%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5.8 \cdot 10^{-45}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.4 \cdot 10^{+182}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im\_m \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 3.4e+182)
    (* (* x.re x.re) (* x.im_m 3.0))
    (* (* x.re x.re) (* x.im_m -3.0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3.4e+182) {
		tmp = (x_46_re * x_46_re) * (x_46_im_m * 3.0);
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_im_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 3.4d+182) then
        tmp = (x_46re * x_46re) * (x_46im_m * 3.0d0)
    else
        tmp = (x_46re * x_46re) * (x_46im_m * (-3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3.4e+182) {
		tmp = (x_46_re * x_46_re) * (x_46_im_m * 3.0);
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_im_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 3.4e+182:
		tmp = (x_46_re * x_46_re) * (x_46_im_m * 3.0)
	else:
		tmp = (x_46_re * x_46_re) * (x_46_im_m * -3.0)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 3.4e+182)
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_im_m * 3.0));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_im_m * -3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 3.4e+182)
		tmp = (x_46_re * x_46_re) * (x_46_im_m * 3.0);
	else
		tmp = (x_46_re * x_46_re) * (x_46_im_m * -3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 3.4e+182], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.39999999999999987e182

   1. Initial program 85.2%

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

      1. +-commutative 85.2%

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

      2. *-commutative 85.2%

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

      3. sqr-neg 85.2%

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

      4. fma-define 85.6%

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

      5. *-commutative 85.6%

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

      6. distribute-rgt-out 85.6%

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

      7. count-2 85.6%

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

      8. *-commutative 85.6%

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

      9. *-commutative 85.6%

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

      10. sqr-neg 85.6%

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

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im \cdot 2\right), x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around inf 56.2%

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

      1. unpow2 56.2%

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

   7. Applied egg-rr 56.2%

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

      1. *-un-lft-identity 56.2%

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

      2. distribute-rgt-out 56.2%

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

      3. metadata-eval 56.2%

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

   9. Applied egg-rr 56.2%

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

   if 3.39999999999999987e182 < x.im

   1. Initial program 42.3%

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

      1. +-commutative 42.3%

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

      2. *-commutative 42.3%

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

      3. sqr-neg 42.3%

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

      4. fma-define 46.2%

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

      5. *-commutative 46.2%

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

      6. distribute-rgt-out 46.2%

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

      7. count-2 46.2%

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

      8. *-commutative 46.2%

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

      9. *-commutative 46.2%

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

      10. sqr-neg 46.2%

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

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im \cdot 2\right), x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x.re around inf 11.8%

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

      1. unpow2 11.8%

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

   7. Applied egg-rr 11.8%

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

   8. Taylor expanded in x.im around 0 11.8%

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

   10. Simplified 48.4%

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

Alternative 10: 23.5% accurate, 2.7× speedup

Math

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

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (* x.im_s (* (* x.re x.re) (* x.im_m -3.0))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * ((x_46_re * x_46_re) * (x_46_im_m * -3.0));
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * ((x_46re * x_46re) * (x_46im_m * (-3.0d0)))
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * ((x_46_re * x_46_re) * (x_46_im_m * -3.0));
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	return x_46_im_s * ((x_46_re * x_46_re) * (x_46_im_m * -3.0))

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(x_46_re * x_46_re) * Float64(x_46_im_m * -3.0)))
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * ((x_46_re * x_46_re) * (x_46_im_m * -3.0));
end

Wolfram

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

Derivation

1. Initial program 80.8%

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

   1. +-commutative 80.8%

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

   2. *-commutative 80.8%

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

   3. sqr-neg 80.8%

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

   4. fma-define 81.6%

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

   5. *-commutative 81.6%

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

   6. distribute-rgt-out 81.6%

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

   7. count-2 81.6%

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

   8. *-commutative 81.6%

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

   9. *-commutative 81.6%

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

   10. sqr-neg 81.6%

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

3. Simplified 81.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im \cdot 2\right), x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x.re around inf 51.7%

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

   1. unpow2 51.7%

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

7. Applied egg-rr 51.7%

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

8. Taylor expanded in x.im around 0 51.7%

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

10. Simplified 25.3%

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

Alternative 11: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot 3 \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m) :precision binary64 (* x.im_s 3.0))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * 3.0;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * 3.0d0
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * 3.0;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	return x_46_im_s * 3.0

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * 3.0)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * 3.0;
end

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Simplified 86.2%

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

6. Applied egg-rr 11.8%

   \[\leadsto \color{blue}{\left({\left({x.re}^{2} \cdot \left(x.im \cdot 3\right)\right)}^{3} - {x.im}^{9}\right) \cdot \frac{1}{\left({x.re}^{4} \cdot {x.im}^{2}\right) \cdot 9 + {x.im}^{3} \cdot \mathsf{fma}\left(x.im \cdot {x.re}^{2}, 3, {x.im}^{3}\right)}} \]

8. Simplified 11.3%

   \[\leadsto \color{blue}{-\left(-3 - {x.im}^{9}\right)} \]

9. Taylor expanded in x.im around 0 2.6%

   \[\leadsto \color{blue}{3} \]
10. Add Preprocessing

Alternative 12: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot -3 \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m) :precision binary64 (* x.im_s -3.0))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (-3.0d0)
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	return x_46_im_s * -3.0

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * -3.0)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * -3.0;
end

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Simplified 86.2%

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

5. Taylor expanded in x.re around 0 56.2%

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]

7. Simplified 2.9%

   \[\leadsto \color{blue}{-3} \]
8. Add Preprocessing

Developer Target 1: 91.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

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

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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