math.cube on complex, imaginary part

Percentage Accurate: 82.3% → 99.7%

Time: 7.3s

Alternatives: 6

Speedup: 1.1×

• 44.1% 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: 82.3% 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: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.55 \cdot 10^{+102}:\\ \;\;\;\;\left(x.im\_m \cdot \left(x.im\_m + x.re\_m\right)\right) \cdot \left(x.re\_m - x.im\_m\right) + \left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, \mathsf{fma}\left(\frac{x.im\_m}{x.re\_m}, x.im\_m, x.im\_m\right) \cdot x.re\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.55e+102)
    (+
     (* (* x.im_m (+ x.im_m x.re_m)) (- x.re_m x.im_m))
     (* (* x.re_m (+ x.im_m x.im_m)) x.re_m))
    (fma
     (- x.re_m x.im_m)
     (* (fma (/ x.im_m x.re_m) x.im_m x.im_m) x.re_m)
     (* 2.0 x.im_m)))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.55e+102) {
		tmp = ((x_46_im_m * (x_46_im_m + x_46_re_m)) * (x_46_re_m - x_46_im_m)) + ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_re_m);
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (fma((x_46_im_m / x_46_re_m), x_46_im_m, x_46_im_m) * x_46_re_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.55e+102)
		tmp = Float64(Float64(Float64(x_46_im_m * Float64(x_46_im_m + x_46_re_m)) * Float64(x_46_re_m - x_46_im_m)) + Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)) * x_46_re_m));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(fma(Float64(x_46_im_m / x_46_re_m), x_46_im_m, x_46_im_m) * x_46_re_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.55e+102], N[(N[(N[(x$46$im$95$m * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(N[(N[(x$46$im$95$m / x$46$re$95$m), $MachinePrecision] * x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.54999999999999993e102

   1. Initial program 85.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 94.5

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

   4. Applied rewrites 94.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 94.5

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

   6. Applied rewrites 94.5%

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

   if 1.54999999999999993e102 < x.im

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 74.4

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 74.5

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

   7. Applied rewrites 74.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 74.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. flip-+ N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. distribute-lft-in N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

   9. Applied rewrites 100.0%

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

Alternative 2: 95.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-293} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(3 \cdot x.im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (or (<= t_0 -1e-293) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* x.re_m (* x.re_m (* 3.0 x.im_m)))))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -1e-293) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = x_46_re_m * (x_46_re_m * (3.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Java

x.re_m = Math.abs(x_46_re);
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_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -1e-293) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = x_46_re_m * (x_46_re_m * (3.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
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_m, x_46_im_m):
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if (t_0 <= -1e-293) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = x_46_re_m * (x_46_re_m * (3.0 * x_46_im_m))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if ((t_0 <= -1e-293) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(3.0 * x_46_im_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
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_m, x_46_im_m)
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if ((t_0 <= -1e-293) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = x_46_re_m * (x_46_re_m * (3.0 * x_46_im_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -1e-293], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(3.0 * x$46$im$95$m), $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)) < -1.0000000000000001e-293 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 62.8%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-1 \cdot {x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

   8. Applied rewrites 58.7%

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

   if -1.0000000000000001e-293 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 96.3%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 69.2%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-293} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.1 \cdot 10^{-58}:\\ \;\;\;\;\left(x.im\_m \cdot \left(x.im\_m + x.re\_m\right)\right) \cdot \left(x.re\_m - x.im\_m\right) + \left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.re\_m\\ \mathbf{elif}\;x.im\_m \leq 1.65 \cdot 10^{+207}:\\ \;\;\;\;\left(-\mathsf{fma}\left(x.im\_m, x.im\_m, -3 \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.1e-58)
    (+
     (* (* x.im_m (+ x.im_m x.re_m)) (- x.re_m x.im_m))
     (* (* x.re_m (+ x.im_m x.im_m)) x.re_m))
    (if (<= x.im_m 1.65e+207)
      (* (- (fma x.im_m x.im_m (* -3.0 (* x.re_m x.re_m)))) x.im_m)
      (* (* (- x.im_m) x.im_m) x.im_m)))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.1e-58) {
		tmp = ((x_46_im_m * (x_46_im_m + x_46_re_m)) * (x_46_re_m - x_46_im_m)) + ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_re_m);
	} else if (x_46_im_m <= 1.65e+207) {
		tmp = -fma(x_46_im_m, x_46_im_m, (-3.0 * (x_46_re_m * x_46_re_m))) * x_46_im_m;
	} else {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.1e-58)
		tmp = Float64(Float64(Float64(x_46_im_m * Float64(x_46_im_m + x_46_re_m)) * Float64(x_46_re_m - x_46_im_m)) + Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)) * x_46_re_m));
	elseif (x_46_im_m <= 1.65e+207)
		tmp = Float64(Float64(-fma(x_46_im_m, x_46_im_m, Float64(-3.0 * Float64(x_46_re_m * x_46_re_m)))) * x_46_im_m);
	else
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.1e-58], N[(N[(N[(x$46$im$95$m * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 1.65e+207], N[((-N[(x$46$im$95$m * x$46$im$95$m + N[(-3.0 * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * x$46$im$95$m), $MachinePrecision], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 1.10000000000000003e-58

   1. Initial program 82.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.2

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

   4. Applied rewrites 93.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 93.2

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

   6. Applied rewrites 93.2%

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

   if 1.10000000000000003e-58 < x.im < 1.65e207

   1. Initial program 85.1%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

   if 1.65e207 < x.im

   1. Initial program 50.0%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-1 \cdot {x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

   8. Applied rewrites 95.0%

      \[\leadsto \left(\left(-x.im\right) \cdot x.im\right) \cdot x.im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3 \cdot 10^{-117}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(3 \cdot x.im\_m\right)\right)\\ \mathbf{elif}\;x.im\_m \leq 1.65 \cdot 10^{+207}:\\ \;\;\;\;\left(-\mathsf{fma}\left(x.im\_m, x.im\_m, -3 \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 3e-117)
    (* x.re_m (* x.re_m (* 3.0 x.im_m)))
    (if (<= x.im_m 1.65e+207)
      (* (- (fma x.im_m x.im_m (* -3.0 (* x.re_m x.re_m)))) x.im_m)
      (* (* (- x.im_m) x.im_m) x.im_m)))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3e-117) {
		tmp = x_46_re_m * (x_46_re_m * (3.0 * x_46_im_m));
	} else if (x_46_im_m <= 1.65e+207) {
		tmp = -fma(x_46_im_m, x_46_im_m, (-3.0 * (x_46_re_m * x_46_re_m))) * x_46_im_m;
	} else {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 3e-117)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(3.0 * x_46_im_m)));
	elseif (x_46_im_m <= 1.65e+207)
		tmp = Float64(Float64(-fma(x_46_im_m, x_46_im_m, Float64(-3.0 * Float64(x_46_re_m * x_46_re_m)))) * x_46_im_m);
	else
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 3e-117], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 1.65e+207], N[((-N[(x$46$im$95$m * x$46$im$95$m + N[(-3.0 * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * x$46$im$95$m), $MachinePrecision], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 2.99999999999999991e-117

   1. Initial program 81.5%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 66.5%

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

   if 2.99999999999999991e-117 < x.im < 1.65e207

   1. Initial program 85.8%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 98.6%

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

   if 1.65e207 < x.im

   1. Initial program 50.0%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-1 \cdot {x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

   8. Applied rewrites 95.0%

      \[\leadsto \left(\left(-x.im\right) \cdot x.im\right) \cdot x.im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 71.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.12 \cdot 10^{+139}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m, 2\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re_m 1.12e+139)
    (* (* (- x.im_m) x.im_m) x.im_m)
    (* (fma x.re_m x.re_m 2.0) x.im_m))))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1.12e+139) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = fma(x_46_re_m, x_46_re_m, 2.0) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 1.12e+139)
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(fma(x_46_re_m, x_46_re_m, 2.0) * x_46_im_m);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re$95$m, 1.12e+139], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m + 2.0), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.12e139

   1. Initial program 85.1%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-1 \cdot {x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

   8. Applied rewrites 68.1%

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

   if 1.12e139 < x.re

   1. Initial program 51.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 83.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. flip-+ N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. +-inverses N/A

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

      16. flip-+ N/A

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

      17. distribute-lft-in N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

   9. Applied rewrites 84.0%

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

   10. Taylor expanded in x.im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. unpow2 N/A

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

       5. lower-fma.f64 65.0

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

   12. Applied rewrites 65.0%

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

Alternative 6: 21.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \left(\mathsf{fma}\left(x.re\_m, x.re\_m, 2\right) \cdot x.im\_m\right) \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (* x.im_s (* (fma x.re_m x.re_m 2.0) x.im_m)))

C

x.re_m = fabs(x_46_re);
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_m, double x_46_im_m) {
	return x_46_im_s * (fma(x_46_re_m, x_46_re_m, 2.0) * x_46_im_m);
}

Julia

x.re_m = abs(x_46_re)
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_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(fma(x_46_re_m, x_46_re_m, 2.0) * x_46_im_m))
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
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$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(N[(x$46$re$95$m * x$46$re$95$m + 2.0), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower--.f64 90.8

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

4. Applied rewrites 90.8%

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

5. Taylor expanded in x.re around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 88.6

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

7. Applied rewrites 88.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 88.6

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. flip-+ N/A

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

   12. +-inverses N/A

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

   13. +-inverses N/A

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

   14. +-inverses N/A

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

   15. +-inverses N/A

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

   16. flip-+ N/A

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

   17. distribute-lft-in N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

9. Applied rewrites 59.9%

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

10. Taylor expanded in x.im around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. unpow2 N/A

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

    5. lower-fma.f64 23.6

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

12. Applied rewrites 23.6%

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

Developer Target 1: 91.5% 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 2024321 
(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)))