math.cube on complex, real part

Percentage Accurate: 82.2% → 99.5%

Time: 7.7s

Alternatives: 7

Speedup: 0.6×

• 44.8% 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: 82.2% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left(x.re\_m \cdot \left(x.im\_m + x.re\_m\right)\right) \cdot \left(x.re\_m - x.im\_m\right) - \left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, \mathsf{fma}\left(\frac{x.re\_m}{x.im\_m}, x.re\_m, x.re\_m\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \]

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.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       2e+301)
    (-
     (* (* x.re_m (+ x.im_m x.re_m)) (- x.re_m x.im_m))
     (* (* x.re_m (+ x.im_m x.im_m)) x.im_m))
    (fma
     (- x.re_m x.im_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_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 2e+301) {
		tmp = ((x_46_re_m * (x_46_im_m + x_46_re_m)) * (x_46_re_m - x_46_im_m)) - ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_im_m);
	} else {
		tmp = fma((x_46_re_m - x_46_im_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_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 2e+301)
		tmp = Float64(Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_re_m)) * Float64(x_46_re_m - x_46_im_m)) - Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)) * x_46_im_m));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(fma(Float64(x_46_re_m / x_46_im_m), x_46_re_m, 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$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], 2e+301], N[(N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$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)) < 2.00000000000000011e301

   1. Initial program 95.4%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if 2.00000000000000011e301 < (-.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 46.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 70.0

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

   4. Applied rewrites 70.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 70.0

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

   6. Applied rewrites 70.0%

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

   7. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 70.0

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

   9. Applied rewrites 70.0%

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

       1. lift--.f64 N/A

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

       2. sub-neg N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

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

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

   11. Applied rewrites 82.2%

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

Alternative 2: 96.7% accurate, 0.5× speedup

Math

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

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.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       INFINITY)
    (-
     (* (* x.re_m (+ x.im_m x.re_m)) (- x.re_m x.im_m))
     (* (* x.re_m (+ x.im_m x.im_m)) x.im_m))
    (* (* x.im_m x.im_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_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= ((double) INFINITY)) {
		tmp = ((x_46_re_m * (x_46_im_m + x_46_re_m)) * (x_46_re_m - x_46_im_m)) - ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_im_m);
	} else {
		tmp = (x_46_im_m * x_46_im_m) * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

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_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_re_m * (x_46_im_m + x_46_re_m)) * (x_46_re_m - x_46_im_m)) - ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_im_m);
	} else {
		tmp = (x_46_im_m * x_46_im_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_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= math.inf:
		tmp = ((x_46_re_m * (x_46_im_m + x_46_re_m)) * (x_46_re_m - x_46_im_m)) - ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_im_m)
	else:
		tmp = (x_46_im_m * x_46_im_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_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= Inf)
		tmp = Float64(Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_re_m)) * Float64(x_46_re_m - x_46_im_m)) - Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)) * x_46_im_m));
	else
		tmp = Float64(Float64(x_46_im_m * x_46_im_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_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= Inf)
		tmp = ((x_46_re_m * (x_46_im_m + x_46_re_m)) * (x_46_re_m - x_46_im_m)) - ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_im_m);
	else
		tmp = (x_46_im_m * x_46_im_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$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$im$95$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)) < +inf.0

   1. Initial program 91.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-neg-frac2 N/A

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

   4. Applied rewrites 33.3%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 83.3

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

   7. Applied rewrites 83.3%

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

Alternative 3: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -2 \cdot 10^{-321}:\\ \;\;\;\;\left(\left(-3 \cdot x.re\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m, x.im\_m \cdot x.im\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       -2e-321)
    (* (* (* -3.0 x.re_m) x.im_m) x.im_m)
    (* (fma x.re_m x.re_m (* x.im_m x.im_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_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321) {
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m;
	} else {
		tmp = fma(x_46_re_m, x_46_re_m, (x_46_im_m * x_46_im_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_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321)
		tmp = Float64(Float64(Float64(-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(fma(x_46_re_m, x_46_re_m, Float64(x_46_im_m * x_46_im_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[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$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -2e-321], N[(N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m + N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2.00097e-321

   1. Initial program 91.6%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 50.4%

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

   if -2.00097e-321 < (-.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 76.3%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-neg-frac2 N/A

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.7

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

   7. Applied rewrites 61.7%

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

   9. Applied rewrites 61.7%

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

Alternative 4: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -2 \cdot 10^{-321}:\\ \;\;\;\;\left(\left(-3 \cdot x.re\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m, x.im\_m, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       -2e-321)
    (* (* (* -3.0 x.re_m) x.im_m) x.im_m)
    (* (fma x.im_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_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321) {
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m;
	} else {
		tmp = fma(x_46_im_m, x_46_im_m, (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_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321)
		tmp = Float64(Float64(Float64(-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(fma(x_46_im_m, x_46_im_m, 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[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$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -2e-321], N[(N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$im$95$m + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2.00097e-321

   1. Initial program 91.6%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 50.4%

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

   if -2.00097e-321 < (-.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 76.3%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-neg-frac2 N/A

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.7

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

   7. Applied rewrites 61.7%

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

Alternative 5: 63.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -2 \cdot 10^{-321}:\\ \;\;\;\;\left(\left(-3 \cdot x.re\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.re\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

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.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.im_m))
       -2e-321)
    (* (* (* -3.0 x.re_m) x.im_m) x.im_m)
    (* (* 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_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321) {
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_im_m * x_46_re_m) * x_46_im_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_46re_m * x_46im_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-2d-321)) then
        tmp = (((-3.0d0) * x_46re_m) * x_46im_m) * x_46im_m
    else
        tmp = (x_46im_m * x_46re_m) * x_46im_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_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321) {
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_im_m * x_46_re_m) * x_46_im_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_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321:
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m
	else:
		tmp = (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 (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_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321)
		tmp = Float64(Float64(Float64(-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_im_m * x_46_re_m) * x_46_im_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_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -2e-321)
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m;
	else
		tmp = (x_46_im_m * x_46_re_m) * x_46_im_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$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -2e-321], N[(N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im$95$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)) < -2.00097e-321

   1. Initial program 91.6%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 50.4%

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

   if -2.00097e-321 < (-.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 76.3%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-neg-frac2 N/A

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 39.9

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

   7. Applied rewrites 39.9%

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

   9. Applied rewrites 39.9%

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

Alternative 6: 23.8% accurate, 3.6× speedup

Math

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

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 (* (* 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) {
	return x_46_re_s * ((x_46_im_m * x_46_re_m) * 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 * ((x_46im_m * x_46re_m) * 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 * ((x_46_im_m * x_46_re_m) * 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 * ((x_46_im_m * x_46_re_m) * 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(Float64(x_46_im_m * x_46_re_m) * 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 * ((x_46_im_m * x_46_re_m) * 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[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.5%

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

   1. lift-+.f64 N/A

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

   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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. flip-+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. +-inverses N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-neg-frac2 N/A

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

4. Applied rewrites 52.7%

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

5. Taylor expanded in x.re around 0

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-rgt-identity N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 24.9

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

7. Applied rewrites 24.9%

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

9. Applied rewrites 25.0%

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

Alternative 7: 23.7% accurate, 3.6× speedup

Math

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

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.im_m x.im_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_im_m * x_46_im_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_46im_m * x_46im_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_im_m * x_46_im_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_im_m * x_46_im_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_im_m * x_46_im_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_im_m * x_46_im_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$46$im$95$m * x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.5%

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

   1. lift-+.f64 N/A

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

   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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. flip-+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. +-inverses N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-neg-frac2 N/A

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

4. Applied rewrites 52.7%

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

5. Taylor expanded in x.re around 0

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-rgt-identity N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 24.9

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

7. Applied rewrites 24.9%

   \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot x.re} \]
8. 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 2024326 
(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)))