math.cube on complex, real part

Percentage Accurate: 81.8% → 99.6%

Time: 12.9s

Alternatives: 17

Speedup: 0.7×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-209}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot \left(x.im + x.im\right), -x.im, \left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+209}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.re\_m, x.re\_m, \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, x.re\_m \cdot \left(x.re\_m + x.im\right), x.im + x.im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))))
   (*
    x.re_s
    (if (<= t_0 -5e-209)
      (fma
       (* x.re_m (+ x.im x.im))
       (- x.im)
       (* (- x.re_m x.im) (* x.re_m x.im)))
      (if (<= t_0 1e+209)
        (* x.re_m (fma x.re_m x.re_m (* (* x.im x.im) -3.0)))
        (fma (- x.re_m x.im) (* x.re_m (+ x.re_m x.im)) (+ x.im x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_0 <= -5e-209) {
		tmp = fma((x_46_re_m * (x_46_im + x_46_im)), -x_46_im, ((x_46_re_m - x_46_im) * (x_46_re_m * x_46_im)));
	} else if (t_0 <= 1e+209) {
		tmp = x_46_re_m * fma(x_46_re_m, x_46_re_m, ((x_46_im * x_46_im) * -3.0));
	} else {
		tmp = fma((x_46_re_m - x_46_im), (x_46_re_m * (x_46_re_m + x_46_im)), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	tmp = 0.0
	if (t_0 <= -5e-209)
		tmp = fma(Float64(x_46_re_m * Float64(x_46_im + x_46_im)), Float64(-x_46_im), Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * x_46_im)));
	elseif (t_0 <= 1e+209)
		tmp = Float64(x_46_re_m * fma(x_46_re_m, x_46_re_m, Float64(Float64(x_46_im * x_46_im) * -3.0)));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -5e-209], N[(N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision] * (-x$46$im) + N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+209], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m + N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $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.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -5.0000000000000005e-209

   1. Initial program 82.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 83.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -5.0000000000000005e-209 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.0000000000000001e209

   1. Initial program 99.7%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.0000000000000001e209 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 66.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 69.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 78.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-209}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re - x.im\right) \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+209}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+237}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+209}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.im \cdot x.im, -3, x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, x.re\_m \cdot \left(x.re\_m + x.im\right), x.im + x.im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))))
   (*
    x.re_s
    (if (<= t_0 -4e+237)
      (* x.im (* x.re_m (* x.im -3.0)))
      (if (<= t_0 1e+209)
        (* x.re_m (fma (* x.im x.im) -3.0 (* x.re_m x.re_m)))
        (fma (- x.re_m x.im) (* x.re_m (+ x.re_m x.im)) (+ x.im x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_0 <= -4e+237) {
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	} else if (t_0 <= 1e+209) {
		tmp = x_46_re_m * fma((x_46_im * x_46_im), -3.0, (x_46_re_m * x_46_re_m));
	} else {
		tmp = fma((x_46_re_m - x_46_im), (x_46_re_m * (x_46_re_m + x_46_im)), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	tmp = 0.0
	if (t_0 <= -4e+237)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * -3.0)));
	elseif (t_0 <= 1e+209)
		tmp = Float64(x_46_re_m * fma(Float64(x_46_im * x_46_im), -3.0, Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -4e+237], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+209], N[(x$46$re$95$m * N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0 + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $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.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -3.99999999999999976e237

   1. Initial program 69.5%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt-out-- N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 69.4

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

   5. Applied rewrites 69.4%

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

   7. Applied rewrites 99.8%

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

   if -3.99999999999999976e237 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.0000000000000001e209

   1. Initial program 99.7%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   if 1.0000000000000001e209 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 72.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 81.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -4 \cdot 10^{+237}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+209}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im \cdot x.im, -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 86.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024228 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

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