math.cube on complex, real part

Percentage Accurate: 82.0% → 99.7%

Time: 18.4s

Alternatives: 10

Speedup: 1.4×

• 45.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.0% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\left(\left(x.re\_m - x.im\_m\right) \cdot x.re\_m\right) \cdot \left(x.im\_m + x.re\_m\right) - \left(\left(x.im\_m + x.im\_m\right) \cdot x.re\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m + x.re\_m, \mathsf{fma}\left(\frac{x.re\_m}{x.im\_m}, x.re\_m, -x.re\_m\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       2e+303)
    (-
     (* (* (- x.re_m x.im_m) x.re_m) (+ x.im_m x.re_m))
     (* (* (+ x.im_m x.im_m) x.re_m) x.im_m))
    (fma
     (+ x.im_m x.re_m)
     (* (fma (/ x.re_m x.im_m) x.re_m (- x.re_m)) x.im_m)
     (* 2.0 x.im_m)))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 2e+303) {
		tmp = (((x_46_re_m - x_46_im_m) * x_46_re_m) * (x_46_im_m + x_46_re_m)) - (((x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m);
	} else {
		tmp = fma((x_46_im_m + x_46_re_m), (fma((x_46_re_m / x_46_im_m), x_46_re_m, -x_46_re_m) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 2e+303)
		tmp = Float64(Float64(Float64(Float64(x_46_re_m - x_46_im_m) * x_46_re_m) * Float64(x_46_im_m + x_46_re_m)) - Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m));
	else
		tmp = fma(Float64(x_46_im_m + x_46_re_m), Float64(fma(Float64(x_46_re_m / x_46_im_m), x_46_re_m, Float64(-x_46_re_m)) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := 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$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], 2e+303], N[(N[(N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * N[(N[(N[(x$46$re$95$m / x$46$im$95$m), $MachinePrecision] * x$46$re$95$m + (-x$46$re$95$m)), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 2e303

   1. Initial program 96.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if 2e303 < (-.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 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(\color{blue}{\left(0 - x.re\right)} + \frac{{x.re}^{2}}{x.im}\right)\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      3. associate-+l- N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(x.im \cdot \color{blue}{\left(0 - \left(x.re - \frac{{x.re}^{2}}{x.im}\right)\right)}\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      4. unsub-neg N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(0 - \color{blue}{\left(x.re + \left(\mathsf{neg}\left(\frac{{x.re}^{2}}{x.im}\right)\right)\right)}\right)\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      5. mul-1-neg N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(0 - \left(x.re + \color{blue}{-1 \cdot \frac{{x.re}^{2}}{x.im}}\right)\right)\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      6. neg-sub0 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \left(\color{blue}{\left(0 - \left(x.re + -1 \cdot \frac{{x.re}^{2}}{x.im}\right)\right)} \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      10. mul-1-neg N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(0 - \left(x.re + \color{blue}{\left(\mathsf{neg}\left(\frac{{x.re}^{2}}{x.im}\right)\right)}\right)\right) \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      11. unsub-neg N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(0 - \color{blue}{\left(x.re - \frac{{x.re}^{2}}{x.im}\right)}\right) \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      12. associate-+l- N/A

          \[\leadsto \left(x.im + x.re\right) \cdot \left(\color{blue}{\left(\left(0 - x.re\right) + \frac{{x.re}^{2}}{x.im}\right)} \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      13. neg-sub0 N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. mul-1-neg N/A

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

      22. lower-neg.f64 73.8

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

   7. Applied rewrites 73.8%

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

   9. Applied rewrites 88.1%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re, \mathsf{fma}\left(\frac{x.re}{x.im}, x.re, -x.re\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\left(-3 \cdot x.im\_m\right) \cdot \left(x.im\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       -1e-322)
    (* (* -3.0 x.im_m) (* x.im_m x.re_m))
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322) {
		tmp = (-3.0 * x_46_im_m) * (x_46_im_m * x_46_re_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-1d-322)) then
        tmp = ((-3.0d0) * x_46im_m) * (x_46im_m * x_46re_m)
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322) {
		tmp = (-3.0 * x_46_im_m) * (x_46_im_m * x_46_re_m);
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322:
		tmp = (-3.0 * x_46_im_m) * (x_46_im_m * x_46_re_m)
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322)
		tmp = Float64(Float64(-3.0 * x_46_im_m) * Float64(x_46_im_m * x_46_re_m));
	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.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322)
		tmp = (-3.0 * x_46_im_m) * (x_46_im_m * x_46_re_m);
	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.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := 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$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -1e-322], N[(N[(-3.0 * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $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)) < -9.88131e-323

   1. Initial program 93.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 55.4

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

   5. Applied rewrites 55.4%

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

   7. Applied rewrites 62.2%

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

   if -9.88131e-323 < (-.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 74.4%

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

   3. 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 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.6

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

   10. Applied rewrites 69.6%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\left(-3 \cdot x.im\right) \cdot \left(x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\_m\right) \cdot x.im\_m\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       -1e-322)
    (* (* (* x.im_m x.re_m) x.im_m) -3.0)
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322) {
		tmp = ((x_46_im_m * x_46_re_m) * x_46_im_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.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-1d-322)) then
        tmp = ((x_46im_m * x_46re_m) * x_46im_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.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322) {
		tmp = ((x_46_im_m * x_46_re_m) * x_46_im_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.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322:
		tmp = ((x_46_im_m * x_46_re_m) * x_46_im_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.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re_m) * x_46_im_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.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322)
		tmp = ((x_46_im_m * x_46_re_m) * x_46_im_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.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := 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$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -1e-322], N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im$95$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)) < -9.88131e-323

   1. Initial program 93.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 55.4

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

   5. Applied rewrites 55.4%

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

   7. Applied rewrites 62.1%

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

   if -9.88131e-323 < (-.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 74.4%

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

   3. 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 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.6

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

   10. Applied rewrites 69.6%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.im\_m\right) \cdot x.re\_m\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       -1e-322)
    (* (* (* x.im_m x.im_m) x.re_m) -3.0)
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322) {
		tmp = ((x_46_im_m * x_46_im_m) * 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.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-1d-322)) then
        tmp = ((x_46im_m * x_46im_m) * 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.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322) {
		tmp = ((x_46_im_m * x_46_im_m) * 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.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322:
		tmp = ((x_46_im_m * x_46_im_m) * 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.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_im_m) * 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.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322)
		tmp = ((x_46_im_m * x_46_im_m) * 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.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := 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$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -1e-322], N[(N[(N[(x$46$im$95$m * x$46$im$95$m), $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)) < -9.88131e-323

   1. Initial program 93.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 55.4

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

   5. Applied rewrites 55.4%

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

   if -9.88131e-323 < (-.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 74.4%

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

   3. 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 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.6

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

   10. Applied rewrites 69.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.re\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       -1e-322)
    (* (* (- x.im_m) x.re_m) x.im_m)
    (* (* x.re_m x.re_m) x.re_m))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322) {
		tmp = (-x_46_im_m * x_46_re_m) * x_46_im_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-1d-322)) then
        tmp = (-x_46im_m * x_46re_m) * x_46im_m
    else
        tmp = (x_46re_m * x_46re_m) * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322) {
		tmp = (-x_46_im_m * x_46_re_m) * x_46_im_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322:
		tmp = (-x_46_im_m * x_46_re_m) * x_46_im_m
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322)
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_re_m) * x_46_im_m);
	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.im_m = abs(x_46_im);
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_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-322)
		tmp = (-x_46_im_m * x_46_re_m) * x_46_im_m;
	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.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := 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$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -1e-322], N[(N[((-x$46$im$95$m) * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $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)) < -9.88131e-323

   1. Initial program 93.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 39.2%

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

   8. Taylor expanded in x.re around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-*r* N/A

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

      17. distribute-rgt-out N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 31.0

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

   10. Applied rewrites 31.0%

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

   11. Taylor expanded in x.im around inf

       \[\leadsto \left(\left(-1 \cdot x.im\right) \cdot x.re\right) \cdot x.im \]
   12. Step-by-step derivation

   13. Applied rewrites 32.3%

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

   if -9.88131e-323 < (-.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 74.4%

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

   3. 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 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.6

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

   10. Applied rewrites 69.6%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(x.re\_m - x.im\_m\right) \cdot x.re\_m\right) \cdot \left(x.im\_m + x.re\_m\right) - \left(\left(x.im\_m + x.im\_m\right) \cdot x.re\_m\right) \cdot x.im\_m\\ \mathbf{elif}\;x.re\_m \leq 9.8 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m, -3 \cdot \left(x.im\_m \cdot x.im\_m\right)\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2e-24)
    (-
     (* (* (- x.re_m x.im_m) x.re_m) (+ x.im_m x.re_m))
     (* (* (+ x.im_m x.im_m) x.re_m) x.im_m))
    (if (<= x.re_m 9.8e+231)
      (* (fma x.re_m x.re_m (* -3.0 (* x.im_m x.im_m))) x.re_m)
      (* (* x.re_m x.re_m) x.re_m)))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 2e-24) {
		tmp = (((x_46_re_m - x_46_im_m) * x_46_re_m) * (x_46_im_m + x_46_re_m)) - (((x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m);
	} else if (x_46_re_m <= 9.8e+231) {
		tmp = fma(x_46_re_m, x_46_re_m, (-3.0 * (x_46_im_m * x_46_im_m))) * x_46_re_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (x_46_re_m <= 2e-24)
		tmp = Float64(Float64(Float64(Float64(x_46_re_m - x_46_im_m) * x_46_re_m) * Float64(x_46_im_m + x_46_re_m)) - Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m));
	elseif (x_46_re_m <= 9.8e+231)
		tmp = Float64(fma(x_46_re_m, x_46_re_m, Float64(-3.0 * Float64(x_46_im_m * x_46_im_m))) * x_46_re_m);
	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

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2e-24], N[(N[(N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 9.8e+231], N[(N[(x$46$re$95$m * x$46$re$95$m + N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $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 3 regimes

2. if x.re < 1.99999999999999985e-24

   1. Initial program 82.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 82.4

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

   4. Applied rewrites 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 92.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 92.7

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

   6. Applied rewrites 92.7%

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

   if 1.99999999999999985e-24 < x.re < 9.80000000000000042e231

   1. Initial program 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft1-in N/A

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

      10. unpow3 N/A

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

      11. unpow2 N/A

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

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

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

      13. +-commutative N/A

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

      14. associate--l- N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 83.0%

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

   9. Applied rewrites 95.3%

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

   if 9.80000000000000042e231 < x.re

   1. Initial program 70.0%

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

   3. 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 70.0

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{elif}\;x.re \leq 9.8 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re, -3 \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.9 \cdot 10^{-124}:\\ \;\;\;\;\left(-3 \cdot x.im\_m\right) \cdot \left(x.im\_m \cdot x.re\_m\right)\\ \mathbf{elif}\;x.re\_m \leq 9.8 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m, -3 \cdot \left(x.im\_m \cdot x.im\_m\right)\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.9e-124)
    (* (* -3.0 x.im_m) (* x.im_m x.re_m))
    (if (<= x.re_m 9.8e+231)
      (* (fma x.re_m x.re_m (* -3.0 (* x.im_m x.im_m))) x.re_m)
      (* (* x.re_m x.re_m) x.re_m)))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (x_46_re_m <= 2.9e-124) {
		tmp = (-3.0 * x_46_im_m) * (x_46_im_m * x_46_re_m);
	} else if (x_46_re_m <= 9.8e+231) {
		tmp = fma(x_46_re_m, x_46_re_m, (-3.0 * (x_46_im_m * x_46_im_m))) * x_46_re_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (x_46_re_m <= 2.9e-124)
		tmp = Float64(Float64(-3.0 * x_46_im_m) * Float64(x_46_im_m * x_46_re_m));
	elseif (x_46_re_m <= 9.8e+231)
		tmp = Float64(fma(x_46_re_m, x_46_re_m, Float64(-3.0 * Float64(x_46_im_m * x_46_im_m))) * x_46_re_m);
	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

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.9e-124], N[(N[(-3.0 * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 9.8e+231], N[(N[(x$46$re$95$m * x$46$re$95$m + N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $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 3 regimes

2. if x.re < 2.9000000000000002e-124

   1. Initial program 81.0%

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

   3. 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 61.8

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

   5. Applied rewrites 61.8%

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

   7. Applied rewrites 71.2%

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

   if 2.9000000000000002e-124 < x.re < 9.80000000000000042e231

   1. Initial program 81.8%

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

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.re}{x.re \cdot x.re + x.im \cdot x.im} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. Applied rewrites 37.9%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.re}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft1-in N/A

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

      10. unpow3 N/A

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

      11. unpow2 N/A

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

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

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

      13. +-commutative N/A

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

      14. associate--l- N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

   7. Applied rewrites 85.2%

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

   9. Applied rewrites 94.3%

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

   if 9.80000000000000042e231 < x.re

   1. Initial program 70.0%

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

   3. 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 70.0

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.9 \cdot 10^{-124}:\\ \;\;\;\;\left(-3 \cdot x.im\right) \cdot \left(x.im \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq 9.8 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re, -3 \cdot \left(x.im \cdot x.im\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.35 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im\_m \cdot x.im\_m, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-3 \cdot x.im\_m\right) \cdot x.re\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 1.35e+145)
    (* (fma -3.0 (* x.im_m x.im_m) (* x.re_m x.re_m)) x.re_m)
    (* (* (* -3.0 x.im_m) x.re_m) x.im_m))))

C

x.im_m = fabs(x_46_im);
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_m) {
	double tmp;
	if (x_46_im_m <= 1.35e+145) {
		tmp = fma(-3.0, (x_46_im_m * x_46_im_m), (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = ((-3.0 * x_46_im_m) * x_46_re_m) * x_46_im_m;
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
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_m)
	tmp = 0.0
	if (x_46_im_m <= 1.35e+145)
		tmp = Float64(fma(-3.0, Float64(x_46_im_m * x_46_im_m), Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = Float64(Float64(Float64(-3.0 * x_46_im_m) * x_46_re_m) * x_46_im_m);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 1.35e+145], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(N[(-3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.35000000000000011e145

   1. Initial program 85.5%

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

   3. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(-1 \cdot x.re - 2 \cdot x.re\right) + {x.re}^{3}} \]
   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 91.7%

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

   if 1.35000000000000011e145 < x.im

   1. Initial program 45.0%

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

   3. 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 70.0

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 87.3%

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

   9. Applied rewrites 87.4%

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

   11. Applied rewrites 87.5%

       \[\leadsto \left(\left(-3 \cdot x.im\right) \cdot x.re\right) \cdot x.im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.0% accurate, 3.6× speedup

Math

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

FPCore

x.im_m = (fabs.f64 x.im)
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_m)
 :precision binary64
 (* x.re_s (* (* x.re_m x.re_m) x.re_m)))

C

x.im_m = fabs(x_46_im);
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_m) {
	return x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m);
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * ((x_46re_m * x_46re_m) * x_46re_m)
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	return x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m);
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	return x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m)

Julia

x.im_m = abs(x_46_im)
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_m)
	return Float64(x_46_re_s * Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m))
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = x_46_re_s * ((x_46_re_m * x_46_re_m) * x_46_re_m);
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
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$95$m_] := 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 80.4%

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

3. 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 65.5

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

5. Applied rewrites 65.5%

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

7. Applied rewrites 56.7%

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

8. Taylor expanded in x.im around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 60.0

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

10. Applied rewrites 60.0%

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

11. Final simplification 60.0%

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

Alternative 10: 2.8% accurate, 6.7× speedup

Math

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

FPCore

x.im_m = (fabs.f64 x.im)
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_m) :precision binary64 (* x.re_s (* 2.0 x.im_m)))

C

x.im_m = fabs(x_46_im);
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_m) {
	return x_46_re_s * (2.0 * x_46_im_m);
}

Fortran

x.im_m = abs(x_46im)
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_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * (2.0d0 * x_46im_m)
end function

Java

x.im_m = Math.abs(x_46_im);
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_m) {
	return x_46_re_s * (2.0 * x_46_im_m);
}

Python

x.im_m = math.fabs(x_46_im)
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_m):
	return x_46_re_s * (2.0 * x_46_im_m)

Julia

x.im_m = abs(x_46_im)
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_m)
	return Float64(x_46_re_s * Float64(2.0 * x_46_im_m))
end

MATLAB

x.im_m = abs(x_46_im);
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_m)
	tmp = x_46_re_s * (2.0 * x_46_im_m);
end

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. difference-of-squares N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 90.4

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in x.im around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(\color{blue}{\left(0 - x.re\right)} + \frac{{x.re}^{2}}{x.im}\right)\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   3. associate-+l- N/A

      \[\leadsto \left(x.im + x.re\right) \cdot \left(x.im \cdot \color{blue}{\left(0 - \left(x.re - \frac{{x.re}^{2}}{x.im}\right)\right)}\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. unsub-neg N/A

      \[\leadsto \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(0 - \color{blue}{\left(x.re + \left(\mathsf{neg}\left(\frac{{x.re}^{2}}{x.im}\right)\right)\right)}\right)\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. mul-1-neg N/A

      \[\leadsto \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(0 - \left(x.re + \color{blue}{-1 \cdot \frac{{x.re}^{2}}{x.im}}\right)\right)\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. neg-sub0 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. neg-sub0 N/A

      \[\leadsto \left(x.im + x.re\right) \cdot \left(\color{blue}{\left(0 - \left(x.re + -1 \cdot \frac{{x.re}^{2}}{x.im}\right)\right)} \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   10. mul-1-neg N/A

       \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(0 - \left(x.re + \color{blue}{\left(\mathsf{neg}\left(\frac{{x.re}^{2}}{x.im}\right)\right)}\right)\right) \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   11. unsub-neg N/A

       \[\leadsto \left(x.im + x.re\right) \cdot \left(\left(0 - \color{blue}{\left(x.re - \frac{{x.re}^{2}}{x.im}\right)}\right) \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   12. associate-+l- N/A

       \[\leadsto \left(x.im + x.re\right) \cdot \left(\color{blue}{\left(\left(0 - x.re\right) + \frac{{x.re}^{2}}{x.im}\right)} \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   13. neg-sub0 N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

   16. unpow2 N/A

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

   17. associate-/l* N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 N/A

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

   20. lower-/.f64 N/A

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

   21. mul-1-neg N/A

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

   22. lower-neg.f64 88.3

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

7. Applied rewrites 88.3%

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

9. Applied rewrites 56.4%

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

10. Taylor expanded in x.re around 0

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

    1. lower-*.f64 3.4

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

12. Applied rewrites 3.4%

    \[\leadsto \color{blue}{2 \cdot x.im} \]
13. Add Preprocessing

Developer Target 1: 99.8% 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 2024276 
(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)))