math.cube on complex, imaginary part

Percentage Accurate: 83.1% → 98.9%

Time: 7.1s

Alternatives: 6

Speedup: 1.1×

• 43.3% 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: 83.1% 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: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4 \cdot 10^{-113}:\\ \;\;\;\;\left(\left(3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.re\\ \mathbf{elif}\;x.im\_m \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\left(-x.im\_m\right) \cdot \mathsf{fma}\left(-3, x.re \cdot x.re, x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, \left(x.im\_m + x.re\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 4e-113)
    (* (* (* 3.0 x.im_m) x.re) x.re)
    (if (<= x.im_m 5e+81)
      (* (- x.im_m) (fma -3.0 (* x.re x.re) (* x.im_m x.im_m)))
      (fma (- x.re x.im_m) (* (+ x.im_m x.re) x.im_m) (* 2.0 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 <= 4e-113) {
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	} else if (x_46_im_m <= 5e+81) {
		tmp = -x_46_im_m * fma(-3.0, (x_46_re * x_46_re), (x_46_im_m * x_46_im_m));
	} else {
		tmp = fma((x_46_re - x_46_im_m), ((x_46_im_m + x_46_re) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 4e-113)
		tmp = Float64(Float64(Float64(3.0 * x_46_im_m) * x_46_re) * x_46_re);
	elseif (x_46_im_m <= 5e+81)
		tmp = Float64(Float64(-x_46_im_m) * fma(-3.0, Float64(x_46_re * x_46_re), Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(Float64(x_46_im_m + x_46_re) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(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, 4e-113], N[(N[(N[(3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[x$46$im$95$m, 5e+81], N[((-x$46$im$95$m) * N[(-3.0 * N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 3.99999999999999991e-113

   1. Initial program 85.7%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 68.5%

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

   if 3.99999999999999991e-113 < x.im < 4.9999999999999998e81

   1. Initial program 99.7%

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

   3. Taylor expanded in x.im around 0

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

   5. Applied rewrites 99.7%

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

   if 4.9999999999999998e81 < x.im

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 84.1

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

   4. Applied rewrites 84.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 77.9

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

   6. Applied rewrites 77.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. difference-of-squares N/A

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

      12. lift--.f64 N/A

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

      13. flip-+ N/A

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

      14. distribute-rgt-out N/A

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

   8. Applied rewrites 100.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4 \cdot 10^{-113}:\\ \;\;\;\;\left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\left(-x.im\right) \cdot \mathsf{fma}\left(-3, x.re \cdot x.re, x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im\_m \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, \left(x.im\_m + x.re\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (+ (* x.im_m x.re) (* x.im_m x.re)) x.re)
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_0 -5e-308)
      (* (* x.im_m x.im_m) (- x.im_m))
      (if (<= t_0 INFINITY)
        (* (* 3.0 (* x.im_m x.re)) x.re)
        (fma (- x.re x.im_m) (* (+ x.im_m x.re) x.im_m) (* 2.0 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_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= -5e-308) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re;
	} else {
		tmp = fma((x_46_re - x_46_im_m), ((x_46_im_m + x_46_re) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= -5e-308)
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(3.0 * Float64(x_46_im_m * x_46_re)) * x_46_re);
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(Float64(x_46_im_m + x_46_re) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -5e-308], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(3.0 * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.99999999999999955e-308

   1. Initial program 93.7%

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

   3. Taylor expanded in x.im around 0

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 49.1%

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

   if -4.99999999999999955e-308 < (+.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 95.4%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 65.5%

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

   if +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 0.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 25.0

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

   4. Applied rewrites 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 25.0

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

   6. Applied rewrites 25.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. difference-of-squares N/A

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

      12. lift--.f64 N/A

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

      13. flip-+ N/A

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

      14. distribute-rgt-out N/A

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

   8. Applied rewrites 100.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ t_1 := \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im\_m \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* (* x.im_m x.im_m) (- x.im_m)))
        (t_1
         (+
          (* (+ (* x.im_m x.re) (* x.im_m x.re)) x.re)
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_1 -5e-308)
      t_0
      (if (<= t_1 INFINITY) (* (* 3.0 (* x.im_m x.re)) x.re) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -5e-308) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re;
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -5e-308) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re;
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (x_46_im_m * x_46_im_m) * -x_46_im_m
	t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m)
	tmp = 0
	if t_1 <= -5e-308:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m))
	t_1 = Float64(Float64(Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_1 <= -5e-308)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(3.0 * Float64(x_46_im_m * x_46_re)) * x_46_re);
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	tmp = 0.0;
	if (t_1 <= -5e-308)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -5e-308], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(3.0 * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], t$95$0]]), $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)) < -4.99999999999999955e-308 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 79.0%

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

   3. Taylor expanded in x.im around 0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 53.1%

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

   if -4.99999999999999955e-308 < (+.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 95.4%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 65.5%

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

4. Final simplification 59.3%

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

Alternative 4: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ t_1 := \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* (* x.im_m x.im_m) (- x.im_m)))
        (t_1
         (+
          (* (+ (* x.im_m x.re) (* x.im_m x.re)) x.re)
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_1 -5e-308)
      t_0
      (if (<= t_1 INFINITY) (* (* (* 3.0 x.im_m) x.re) x.re) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -5e-308) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -5e-308) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (x_46_im_m * x_46_im_m) * -x_46_im_m
	t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m)
	tmp = 0
	if t_1 <= -5e-308:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m))
	t_1 = Float64(Float64(Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_1 <= -5e-308)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(3.0 * x_46_im_m) * x_46_re) * x_46_re);
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	tmp = 0.0;
	if (t_1 <= -5e-308)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = ((3.0 * x_46_im_m) * x_46_re) * x_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -5e-308], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision], t$95$0]]), $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)) < -4.99999999999999955e-308 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 79.0%

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

   3. Taylor expanded in x.im around 0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in x.im around inf

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

   8. Applied rewrites 53.1%

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

   if -4.99999999999999955e-308 < (+.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 95.4%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 65.6%

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

4. Final simplification 59.3%

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

Alternative 5: 99.8% accurate, 0.5× speedup

Math

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

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* (+ (* x.im_m x.re) (* x.im_m x.re)) x.re)))
   (*
    x.im_s
    (if (<= (+ t_0 (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)) INFINITY)
      (+ (* (* (- x.re x.im_m) x.im_m) (+ x.im_m x.re)) t_0)
      (fma (- x.re x.im_m) (* (+ x.im_m x.re) x.im_m) (* 2.0 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_im_m * x_46_re)) * x_46_re;
	double tmp;
	if ((t_0 + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m)) <= ((double) INFINITY)) {
		tmp = (((x_46_re - x_46_im_m) * x_46_im_m) * (x_46_im_m + x_46_re)) + t_0;
	} else {
		tmp = fma((x_46_re - x_46_im_m), ((x_46_im_m + x_46_re) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if +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 0.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 25.0

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

   4. Applied rewrites 25.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 25.0

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

   6. Applied rewrites 25.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. difference-of-squares N/A

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

      12. lift--.f64 N/A

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

      13. flip-+ N/A

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

      14. distribute-rgt-out N/A

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.6% accurate, 3.1× speedup

Math

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

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.im_m x.im_m) (- x.im_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

3. Taylor expanded in x.im around 0

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

5. Applied rewrites 90.7%

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

6. Taylor expanded in x.im around inf

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

8. Applied rewrites 57.1%

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

Developer Target 1: 91.7% 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 2024249 
(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)))