math.cube on complex, real part

Percentage Accurate: 82.8% → 99.8%

Time: 10.0s

Alternatives: 9

Speedup: 1.4×

• 43.6% 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: 82.8% 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: 99.8% accurate, 0.4× speedup

Math

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

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
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       INFINITY)
    (-
     (* (- x.re_m x.im) (* (+ x.im x.re_m) x.re_m))
     (* (* (+ x.im x.im) x.re_m) x.im))
    (* (* (fma (/ (/ x.re_m x.im) x.im) x.re_m -3.0) (* x.im x.im)) x.re_m))))

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 tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= ((double) INFINITY)) {
		tmp = ((x_46_re_m - x_46_im) * ((x_46_im + x_46_re_m) * x_46_re_m)) - (((x_46_im + x_46_im) * x_46_re_m) * x_46_im);
	} else {
		tmp = (fma(((x_46_re_m / x_46_im) / x_46_im), x_46_re_m, -3.0) * (x_46_im * x_46_im)) * x_46_re_m;
	}
	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)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= Inf)
		tmp = Float64(Float64(Float64(x_46_re_m - x_46_im) * Float64(Float64(x_46_im + x_46_re_m) * x_46_re_m)) - Float64(Float64(Float64(x_46_im + x_46_im) * x_46_re_m) * x_46_im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(x_46_re_m / x_46_im) / x_46_im), x_46_re_m, -3.0) * Float64(x_46_im * x_46_im)) * x_46_re_m);
	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_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x$46$re$95$m / x$46$im), $MachinePrecision] / x$46$im), $MachinePrecision] * x$46$re$95$m + -3.0), $MachinePrecision] * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $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.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < +inf.0

   1. Initial program 94.6%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if +inf.0 < (-.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 0.0%

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

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x.im around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-*.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 27.3

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

   7. Applied rewrites 27.3%

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

   8. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate--l+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   10. Applied rewrites 72.7%

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

   11. Taylor expanded in x.im around inf

       \[\leadsto \left({x.im}^{2} \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} - 3\right)\right) \cdot x.re \]
   12. Step-by-step derivation

   13. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 95.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       -2e-281)
    (* (* (* x.im x.re_m) x.im) -3.0)
    (* (* x.re_m x.re_m) x.re_m))))

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 tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281) {
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46im * x_46re_m) + (x_46im * x_46re_m)) * x_46im)) <= (-2d-281)) then
        tmp = ((x_46im * x_46re_m) * x_46im) * (-3.0d0)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281) {
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281:
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	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)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281)
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * x_46_im) * -3.0);
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281)
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0;
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = 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_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e-281], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $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.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2e-281

   1. Initial program 89.8%

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

   3. Taylor expanded in x.im around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 39.8

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

   5. Applied rewrites 39.8%

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

   7. Applied rewrites 49.7%

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

   if -2e-281 < (-.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 84.3%

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

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 83.0

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in x.im around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-*.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 52.1

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

   7. Applied rewrites 52.1%

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

   8. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate--l+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   10. Applied rewrites 94.7%

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

   11. Taylor expanded in x.im around 0

       \[\leadsto {x.re}^{2} \cdot x.re \]
   12. Step-by-step derivation

   13. Applied rewrites 68.0%

       \[\leadsto \left(x.re \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\left(\left(x.im \cdot x.im\right) \cdot x.re\_m\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       -2e-281)
    (* (* (* x.im x.im) x.re_m) -3.0)
    (* (* x.re_m x.re_m) x.re_m))))

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 tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281) {
		tmp = ((x_46_im * x_46_im) * x_46_re_m) * -3.0;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46im * x_46re_m) + (x_46im * x_46re_m)) * x_46im)) <= (-2d-281)) then
        tmp = ((x_46im * x_46im) * x_46re_m) * (-3.0d0)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281) {
		tmp = ((x_46_im * x_46_im) * x_46_re_m) * -3.0;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281:
		tmp = ((x_46_im * x_46_im) * x_46_re_m) * -3.0
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	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)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281)
		tmp = Float64(Float64(Float64(x_46_im * x_46_im) * x_46_re_m) * -3.0);
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281)
		tmp = ((x_46_im * x_46_im) * x_46_re_m) * -3.0;
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = 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_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e-281], N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $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.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2e-281

   1. Initial program 89.8%

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

   3. Taylor expanded in x.im around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 39.8

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

   5. Applied rewrites 39.8%

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

   if -2e-281 < (-.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 84.3%

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

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 83.0

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in x.im around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-*.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 52.1

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

   7. Applied rewrites 52.1%

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

   8. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate--l+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   10. Applied rewrites 94.7%

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

   11. Taylor expanded in x.im around 0

       \[\leadsto {x.re}^{2} \cdot x.re \]
   12. Step-by-step derivation

   13. Applied rewrites 68.0%

       \[\leadsto \left(x.re \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\left(\left(-x.re\_m\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       -2e-281)
    (* (* (- x.re_m) x.im) x.im)
    (* (* x.re_m x.re_m) x.re_m))))

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 tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281) {
		tmp = (-x_46_re_m * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im * x_46im)) * x_46re_m) - (((x_46im * x_46re_m) + (x_46im * x_46re_m)) * x_46im)) <= (-2d-281)) then
        tmp = (-x_46re_m * x_46im) * x_46im
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281) {
		tmp = (-x_46_re_m * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281:
		tmp = (-x_46_re_m * x_46_im) * x_46_im
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	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)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281)
		tmp = Float64(Float64(Float64(-x_46_re_m) * x_46_im) * x_46_im);
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -2e-281)
		tmp = (-x_46_re_m * x_46_im) * x_46_im;
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	end
	tmp_2 = 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_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -2e-281], N[(N[((-x$46$re$95$m) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $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.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2e-281

   1. Initial program 89.8%

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

   3. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 68.2

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

      4. lower-+.f64 N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. distribute-neg-frac N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. distribute-neg-in N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-neg.f64 50.0

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

   7. Applied rewrites 50.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-neg-out N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

   9. Applied rewrites 50.1%

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

   10. Taylor expanded in x.im around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 21.7

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

   12. Applied rewrites 21.7%

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

   if -2e-281 < (-.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 84.3%

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

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 83.0

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in x.im around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-*.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 52.1

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

   7. Applied rewrites 52.1%

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

   8. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate--l+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   10. Applied rewrites 94.7%

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

   11. Taylor expanded in x.im around 0

       \[\leadsto {x.re}^{2} \cdot x.re \]
   12. Step-by-step derivation

   13. Applied rewrites 68.0%

       \[\leadsto \left(x.re \cdot x.re\right) \cdot x.re \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\left(\left(-x.re\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 2.7 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(-3 \cdot x.im, x.im, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right) \cdot -3\\ \end{array} \end{array} \]

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
 (*
  x.re_s
  (if (<= x.im 2.7e+173)
    (* (fma (* -3.0 x.im) x.im (* x.re_m x.re_m)) x.re_m)
    (* (* (* x.im x.re_m) x.im) -3.0))))

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 tmp;
	if (x_46_im <= 2.7e+173) {
		tmp = fma((-3.0 * x_46_im), x_46_im, (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0;
	}
	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)
	tmp = 0.0
	if (x_46_im <= 2.7e+173)
		tmp = Float64(fma(Float64(-3.0 * x_46_im), x_46_im, Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * x_46_im) * -3.0);
	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_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 2.7e+173], N[(N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$im + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.7000000000000001e173

   1. Initial program 87.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 86.0

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

   4. Applied rewrites 86.0%

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

   5. Taylor expanded in x.im around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-*.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate--l+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   10. Applied rewrites 94.1%

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

   if 2.7000000000000001e173 < x.im

   1. Initial program 72.9%

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

   3. Taylor expanded in x.im around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 95.1%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.7 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(-3 \cdot x.im, x.im, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m, -3 \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right) \cdot -3\\ \end{array} \end{array} \]

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
 (*
  x.re_s
  (if (<= x.im 1.05e+160)
    (* (fma x.re_m x.re_m (* -3.0 (* x.im x.im))) x.re_m)
    (* (* (* x.im x.re_m) x.im) -3.0))))

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 tmp;
	if (x_46_im <= 1.05e+160) {
		tmp = fma(x_46_re_m, x_46_re_m, (-3.0 * (x_46_im * x_46_im))) * x_46_re_m;
	} else {
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0;
	}
	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)
	tmp = 0.0
	if (x_46_im <= 1.05e+160)
		tmp = Float64(fma(x_46_re_m, x_46_re_m, Float64(-3.0 * Float64(x_46_im * x_46_im))) * x_46_re_m);
	else
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * x_46_im) * -3.0);
	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_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 1.05e+160], N[(N[(x$46$re$95$m * x$46$re$95$m + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.04999999999999998e160

   1. Initial program 87.6%

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

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 85.9

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

   4. Applied rewrites 85.9%

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

   5. Taylor expanded in x.im around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-*.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 44.0

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

   7. Applied rewrites 44.0%

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

   8. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. associate--l+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   10. Applied rewrites 94.0%

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

   12. Applied rewrites 94.5%

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

   if 1.04999999999999998e160 < x.im

   1. Initial program 75.2%

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

   3. Taylor expanded in x.im around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 95.5%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re, -3 \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 4.2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right) \cdot -3\\ \end{array} \end{array} \]

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
 (*
  x.re_s
  (if (<= x.im 4.2e+150)
    (* (fma -3.0 (* x.im x.im) (* x.re_m x.re_m)) x.re_m)
    (* (* (* x.im x.re_m) x.im) -3.0))))

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 tmp;
	if (x_46_im <= 4.2e+150) {
		tmp = fma(-3.0, (x_46_im * x_46_im), (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = ((x_46_im * x_46_re_m) * x_46_im) * -3.0;
	}
	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)
	tmp = 0.0
	if (x_46_im <= 4.2e+150)
		tmp = Float64(fma(-3.0, Float64(x_46_im * x_46_im), Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * x_46_im) * -3.0);
	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_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 4.2e+150], N[(N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.19999999999999996e150

   1. Initial program 87.9%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 92.7%

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

   if 4.19999999999999996e150 < x.im

   1. Initial program 74.2%

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

   3. Taylor expanded in x.im around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 92.1%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.2 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.8% accurate, 3.6× speedup

Math

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

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
 (* x.re_s (* (* x.re_m x.re_m) x.re_m)))

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) {
	return x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m);
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((x_46re_m * x_46re_m) * x_46re_m)
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m);
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m)

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)
	return Float64(x_46_re_s * Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m);
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_] := N[(x$46$re$95$s * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.5%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
2. Add Preprocessing
3. 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. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 84.5

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

4. Applied rewrites 84.5%

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

5. Taylor expanded in x.im around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

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

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

   7. lower-*.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 47.2

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

7. Applied rewrites 47.2%

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

8. Taylor expanded in x.im around 0

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

   1. +-commutative N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. *-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. distribute-lft-out N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

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

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

   12. associate--l+ N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

10. Applied rewrites 92.7%

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

11. Taylor expanded in x.im around 0

    \[\leadsto {x.re}^{2} \cdot x.re \]
12. Step-by-step derivation

13. Applied rewrites 61.1%

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

Alternative 9: 24.2% accurate, 3.6× speedup

Math

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

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
 (* x.re_s (* (* x.im x.im) x.re_m)))

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) {
	return x_46_re_s * ((x_46_im * x_46_im) * x_46_re_m);
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((x_46im * x_46im) * x_46re_m)
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * ((x_46_im * x_46_im) * x_46_re_m);
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * ((x_46_im * x_46_im) * x_46_re_m)

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)
	return Float64(x_46_re_s * Float64(Float64(x_46_im * x_46_im) * x_46_re_m))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * ((x_46_im * x_46_im) * x_46_re_m);
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_] := N[(x$46$re$95$s * N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.5%

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

3. Taylor expanded in x.im around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 69.7

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

5. Applied rewrites 69.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 69.7

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

   4. lower-+.f64 N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. +-inverses N/A

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

   9. distribute-neg-frac N/A

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

   10. +-inverses N/A

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

   11. +-inverses N/A

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

   12. flip-+ N/A

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

   13. distribute-neg-in N/A

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

   14. lower-+.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-neg.f64 60.7

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

7. Applied rewrites 60.7%

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

8. Taylor expanded in x.im around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-rgt-identity N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 20.9

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

10. Applied rewrites 20.9%

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

Developer Target 1: 87.8% 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 2024264 
(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)))