math.cube on complex, imaginary part

Percentage Accurate: 83.1% → 99.7%

Time: 12.9s

Alternatives: 13

Speedup: 0.5×

• 44.3% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.re \cdot x.re, x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\right) \cdot x.re\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, \left(x.im\_m + x.re\right) \cdot x.im\_m, x.re + x.re\right)\\ \end{array} \end{array} \end{array} \]

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+120)
      (* (fma -3.0 (* x.re x.re) (* x.im_m x.im_m)) (- x.im_m))
      (if (<= t_0 INFINITY)
        (* (* (* x.im_m x.re) x.re) 3.0)
        (fma (- x.re x.im_m) (* (+ x.im_m x.re) x.im_m) (+ x.re x.re)))))))

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+120) {
		tmp = fma(-3.0, (x_46_re * x_46_re), (x_46_im_m * x_46_im_m)) * -x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((x_46_im_m * x_46_re) * x_46_re) * 3.0;
	} else {
		tmp = fma((x_46_re - x_46_im_m), ((x_46_im_m + x_46_re) * x_46_im_m), (x_46_re + x_46_re));
	}
	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+120)
		tmp = Float64(fma(-3.0, Float64(x_46_re * x_46_re), Float64(x_46_im_m * x_46_im_m)) * Float64(-x_46_im_m));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re) * x_46_re) * 3.0);
	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(x_46_re + x_46_re));
	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+120], N[(N[(-3.0 * 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], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] * 3.0), $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[(x$46$re + x$46$re), $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)) < 5.00000000000000019e120

   1. Initial program 94.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 94.0%

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

   if 5.00000000000000019e120 < (+.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 84.6%

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

   3. 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 52.7%

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

   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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 35.7

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. lower-+.f64 35.7

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

   4. Applied rewrites 35.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 35.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 35.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 35.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. flip-+ N/A

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

      21. +-inverses N/A

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

      22. +-inverses N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq 5 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.re \cdot x.re, x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.im, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, \left(x.im\_m + x.re\right) \cdot x.im\_m, x.re + x.re\right)\\ \end{array} \end{array} \end{array} \]

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 -2e-272)
      (* (* (- x.im_m) x.im_m) x.im_m)
      (if (<= t_0 INFINITY)
        (* (* 3.0 x.re) (* x.im_m x.re))
        (fma (- x.re x.im_m) (* (+ x.im_m x.re) x.im_m) (+ x.re x.re)))))))

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 <= -2e-272) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (3.0 * x_46_re) * (x_46_im_m * x_46_re);
	} else {
		tmp = fma((x_46_re - x_46_im_m), ((x_46_im_m + x_46_re) * x_46_im_m), (x_46_re + x_46_re));
	}
	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 <= -2e-272)
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(3.0 * x_46_re) * Float64(x_46_im_m * 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(x_46_re + x_46_re));
	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, -2e-272], 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 * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $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[(x$46$re + x$46$re), $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)) < -1.99999999999999986e-272

   1. Initial program 90.2%

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

   3. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 46.0

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

   5. Applied rewrites 46.0%

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

   if -1.99999999999999986e-272 < (+.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 92.1%

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

   3. Taylor expanded in x.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 66.4%

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

   7. Applied rewrites 66.5%

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

   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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 35.7

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. lower-+.f64 35.7

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

   4. Applied rewrites 35.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 35.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 35.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 35.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. flip-+ N/A

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

      21. +-inverses N/A

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

      22. +-inverses N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.im, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 -2e-272)
      t_0
      (if (<= t_1 INFINITY) (* (* 3.0 x.re) (* x.im_m 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 <= -2e-272) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (3.0 * x_46_re) * (x_46_im_m * 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 <= -2e-272) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (3.0 * x_46_re) * (x_46_im_m * 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 <= -2e-272:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (3.0 * x_46_re) * (x_46_im_m * 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(Float64(-x_46_im_m) * x_46_im_m) * 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 <= -2e-272)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(3.0 * x_46_re) * Float64(x_46_im_m * 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 <= -2e-272)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (3.0 * x_46_re) * (x_46_im_m * 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, -2e-272], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(3.0 * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $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)) < -1.99999999999999986e-272 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 69.8%

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

   3. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   if -1.99999999999999986e-272 < (+.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 92.1%

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

   3. Taylor expanded in x.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 66.4%

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

   7. Applied rewrites 66.5%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im\_m \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 -2e-272)
      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 <= -2e-272) {
		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 <= -2e-272) {
		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 <= -2e-272:
		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(Float64(-x_46_im_m) * x_46_im_m) * 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 <= -2e-272)
		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 <= -2e-272)
		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, -2e-272], 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)) < -1.99999999999999986e-272 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 69.8%

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

   3. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   if -1.99999999999999986e-272 < (+.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 92.1%

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

   3. Taylor expanded in x.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 66.4%

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

   7. Applied rewrites 66.4%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_im_m * x_46_re) + (x_46_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) INFINITY)) {
		tmp = fma((x_46_im_m + x_46_re), ((x_46_re - x_46_im_m) * x_46_im_m), (((x_46_re + x_46_re) * x_46_im_m) * x_46_re));
	} else {
		tmp = fma((x_46_re - x_46_im_m), ((x_46_im_m + x_46_re) * x_46_im_m), (x_46_re + x_46_re));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(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)) <= Inf)
		tmp = fma(Float64(x_46_im_m + x_46_re), Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m), Float64(Float64(Float64(x_46_re + x_46_re) * x_46_im_m) * 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(x_46_re + x_46_re));
	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[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], Infinity], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$re), $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[(x$46$re + x$46$re), $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 91.3%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 99.8

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 35.7

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. lower-+.f64 35.7

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

   4. Applied rewrites 35.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 35.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 35.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 35.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. flip-+ N/A

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

      21. +-inverses N/A

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

      22. +-inverses N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_im_m * x_46_re) + (x_46_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) INFINITY)) {
		tmp = fma((-x_46_im_m * x_46_im_m), x_46_im_m, ((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), (x_46_re + x_46_re));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(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)) <= Inf)
		tmp = fma(Float64(Float64(-x_46_im_m) * x_46_im_m), x_46_im_m, 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(x_46_re + x_46_re));
	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[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], Infinity], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m + N[(N[(3.0 * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $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[(x$46$re + x$46$re), $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 91.3%

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

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 35.1%

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

   5. Taylor expanded in x.re around 0

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

   7. Applied rewrites 99.8%

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

   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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 35.7

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. lower-+.f64 35.7

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

   4. Applied rewrites 35.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 35.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 35.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 35.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. flip-+ N/A

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

      21. +-inverses N/A

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

      22. +-inverses N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 7: 28.9% accurate, 0.7× speedup

Math

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

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<=
       (+
        (* (+ (* x.im_m x.re) (* x.im_m x.re)) x.re)
        (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))
       1e+66)
    (* (+ x.re x.re) x.im_m)
    (* (+ x.re x.re) x.re))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_im_m * x_46_re) + (x_46_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)) <= 1e+66) {
		tmp = (x_46_re + x_46_re) * x_46_im_m;
	} else {
		tmp = (x_46_re + x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46im_m * x_46re) + (x_46im_m * x_46re)) * x_46re) + (((x_46re * x_46re) - (x_46im_m * x_46im_m)) * x_46im_m)) <= 1d+66) then
        tmp = (x_46re + x_46re) * x_46im_m
    else
        tmp = (x_46re + x_46re) * x_46re
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_im_m * x_46_re) + (x_46_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)) <= 1e+66) {
		tmp = (x_46_re + x_46_re) * x_46_im_m;
	} else {
		tmp = (x_46_re + x_46_re) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if ((((x_46_im_m * x_46_re) + (x_46_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)) <= 1e+66:
		tmp = (x_46_re + x_46_re) * x_46_im_m
	else:
		tmp = (x_46_re + x_46_re) * x_46_re
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(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)) <= 1e+66)
		tmp = Float64(Float64(x_46_re + x_46_re) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_re + x_46_re) * x_46_re);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_im_m * x_46_re) + (x_46_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)) <= 1e+66)
		tmp = (x_46_re + x_46_re) * x_46_im_m;
	else
		tmp = (x_46_re + x_46_re) * x_46_re;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[N[(N[(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], 1e+66], N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 9.99999999999999945e65

   1. Initial program 93.9%

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

      1. 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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 99.8

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. flip-+ N/A

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

      21. +-inverses N/A

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

      22. +-inverses N/A

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

   6. Applied rewrites 50.9%

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

   7. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{2 \cdot x.re} \]
   8. Step-by-step derivation

      1. lower-*.f64 3.9

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

   9. Applied rewrites 3.9%

      \[\leadsto \color{blue}{2 \cdot x.re} \]
   10. Step-by-step derivation

   11. Applied rewrites 19.7%

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

   if 9.99999999999999945e65 < (+.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 61.7%

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

      1. 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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 81.8

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. lower-+.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 81.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 81.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. flip-+ N/A

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

      21. +-inverses N/A

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

      22. +-inverses N/A

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

   6. Applied rewrites 84.5%

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

   7. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{2 \cdot x.re} \]
   8. Step-by-step derivation

      1. lower-*.f64 3.2

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

   9. Applied rewrites 3.2%

      \[\leadsto \color{blue}{2 \cdot x.re} \]
   10. Step-by-step derivation

   11. Applied rewrites 30.4%

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

4. Final simplification 23.9%

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

Alternative 8: 66.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 2.2 \cdot 10^{+131}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.re\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 2.2e+131)
    (* (* (- x.im_m) x.im_m) x.im_m)
    (* (+ x.re x.re) (* x.im_m x.re)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 2.2e+131) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_re + x_46_re) * (x_46_im_m * x_46_re);
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 2.2d+131) then
        tmp = (-x_46im_m * x_46im_m) * x_46im_m
    else
        tmp = (x_46re + x_46re) * (x_46im_m * x_46re)
    end if
    code = x_46im_s * tmp
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 2.2e+131) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_re + x_46_re) * (x_46_im_m * x_46_re);
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 2.2e+131:
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = (x_46_re + x_46_re) * (x_46_im_m * x_46_re)
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 2.2e+131)
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_re + x_46_re) * Float64(x_46_im_m * x_46_re));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 2.2e+131)
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = (x_46_re + x_46_re) * (x_46_im_m * x_46_re);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 2.2e+131], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re + x$46$re), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.1999999999999999e131

   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. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if 2.1999999999999999e131 < x.re

   1. Initial program 61.3%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 88.8

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. lower-+.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 88.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. flip-+ N/A

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

      21. +-inverses N/A

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

      22. +-inverses N/A

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

   6. Applied rewrites 80.4%

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

   7. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{2 \cdot x.re} \]
   8. Step-by-step derivation

      1. lower-*.f64 4.6

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

   9. Applied rewrites 4.6%

      \[\leadsto \color{blue}{2 \cdot x.re} \]
   10. Step-by-step derivation

   11. Applied rewrites 71.1%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.2 \cdot 10^{+131}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.re\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 8 \cdot 10^{+131}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m, x.re, 2\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 8e+131)
    (* (* (- x.im_m) x.im_m) x.im_m)
    (* (fma x.im_m x.re 2.0) x.re))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 8e+131) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = fma(x_46_im_m, x_46_re, 2.0) * x_46_re;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 8e+131)
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(fma(x_46_im_m, x_46_re, 2.0) * x_46_re);
	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$46$re, 8e+131], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$re + 2.0), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.9999999999999993e131

   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. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if 7.9999999999999993e131 < x.re

   1. Initial program 61.3%

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

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 88.8

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f64 N/A

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

      19. lower-+.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 88.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft-in N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. flip-+ N/A

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

      21. +-inverses N/A

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

      22. +-inverses N/A

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

   6. Applied rewrites 80.4%

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

   7. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 69.3

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

   9. Applied rewrites 69.3%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 8 \cdot 10^{+131}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re, 2\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.1% accurate, 3.6× speedup

Math

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

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.re x.re))))

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_re * x_46_re));
}

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_46re * x_46re))
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_re * x_46_re));
}

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_re * x_46_re))

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(x_46_im_m * Float64(x_46_re * x_46_re)))
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_re * x_46_re));
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[(x$46$im$95$m * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 92.8

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

   13. lift-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-out N/A

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

   18. lower-*.f64 N/A

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

   19. lower-+.f64 92.8

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

4. Applied rewrites 92.8%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 92.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 92.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 92.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. distribute-lft-in N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. flip-+ N/A

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

   21. +-inverses N/A

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

   22. +-inverses N/A

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

6. Applied rewrites 64.0%

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

7. Taylor expanded in x.re around inf

   \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 38.0

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

9. Applied rewrites 38.0%

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

10. Final simplification 38.0%

    \[\leadsto x.im \cdot \left(x.re \cdot x.re\right) \]
11. Add Preprocessing

Alternative 11: 19.3% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(x_46_re + x_46_re) * 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_re + x_46_re) * 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$46$re + x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 92.8

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

   13. lift-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-out N/A

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

   18. lower-*.f64 N/A

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

   19. lower-+.f64 92.8

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

4. Applied rewrites 92.8%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 92.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 92.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 92.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. distribute-lft-in N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. flip-+ N/A

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

   21. +-inverses N/A

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

   22. +-inverses N/A

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

6. Applied rewrites 64.0%

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

7. Taylor expanded in x.im around 0

   \[\leadsto \color{blue}{2 \cdot x.re} \]
8. Step-by-step derivation

   1. lower-*.f64 3.6

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

9. Applied rewrites 3.6%

   \[\leadsto \color{blue}{2 \cdot x.re} \]
10. Step-by-step derivation

11. Applied rewrites 17.1%

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

Alternative 12: 3.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \left(8 \cdot \left(x.re + x.re\right)\right) \end{array} \]

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 (* 8.0 (+ x.re x.re))))

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 * (8.0 * (x_46_re + x_46_re));
}

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 * (8.0d0 * (x_46re + x_46re))
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 * (8.0 * (x_46_re + x_46_re));
}

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 * (8.0 * (x_46_re + x_46_re))

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(8.0 * Float64(x_46_re + x_46_re)))
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 * (8.0 * (x_46_re + x_46_re));
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[(8.0 * N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 92.8

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

   13. lift-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-out N/A

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

   18. lower-*.f64 N/A

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

   19. lower-+.f64 92.8

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

4. Applied rewrites 92.8%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 92.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 92.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 92.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. distribute-lft-in N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. flip-+ N/A

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

   21. +-inverses N/A

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

   22. +-inverses N/A

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

6. Applied rewrites 64.0%

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

7. Taylor expanded in x.im around 0

   \[\leadsto \color{blue}{2 \cdot x.re} \]
8. Step-by-step derivation

   1. lower-*.f64 3.6

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

9. Applied rewrites 3.6%

   \[\leadsto \color{blue}{2 \cdot x.re} \]
10. Step-by-step derivation

11. Applied rewrites 3.6%

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

12. Final simplification 3.6%

    \[\leadsto 8 \cdot \left(x.re + x.re\right) \]
13. Add Preprocessing

Alternative 13: 3.7% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

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(x_46_re + x_46_re))
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * (x_46_re + x_46_re);
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[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 92.8

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

   13. lift-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-out N/A

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

   18. lower-*.f64 N/A

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

   19. lower-+.f64 92.8

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

4. Applied rewrites 92.8%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 92.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 92.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 92.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. distribute-lft-in N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. flip-+ N/A

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

   21. +-inverses N/A

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

   22. +-inverses N/A

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

6. Applied rewrites 64.0%

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

7. Taylor expanded in x.im around 0

   \[\leadsto \color{blue}{2 \cdot x.re} \]
8. Step-by-step derivation

   1. lower-*.f64 3.6

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

9. Applied rewrites 3.6%

   \[\leadsto \color{blue}{2 \cdot x.re} \]
10. Step-by-step derivation

11. Applied rewrites 3.6%

    \[\leadsto \color{blue}{x.re + x.re} \]
12. Add Preprocessing

Developer Target 1: 92.0% 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 2024235 
(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)))