math.cube on complex, real part

Percentage Accurate: 82.5% → 99.8%

Time: 7.9s

Alternatives: 8

Speedup: 0.6×

• 44.4% 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: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.im\_m + x.re\_m\right) \cdot x.re\_m\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq 4 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m \cdot x.re\_m, -2 \cdot x.im\_m, \left(x.re\_m - x.im\_m\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, t\_0, 0\right)\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_0 (* (+ x.im_m x.re_m) x.re_m)))
   (*
    x.re_s
    (if (<=
         (-
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
          (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
         4e+205)
      (fma (* x.im_m x.re_m) (* -2.0 x.im_m) (* (- x.re_m x.im_m) t_0))
      (fma (- x.re_m x.im_m) t_0 0.0)))))

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 t_0 = (x_46_im_m + x_46_re_m) * x_46_re_m;
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 4e+205) {
		tmp = fma((x_46_im_m * x_46_re_m), (-2.0 * x_46_im_m), ((x_46_re_m - x_46_im_m) * t_0));
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), t_0, 0.0);
	}
	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)
	t_0 = Float64(Float64(x_46_im_m + x_46_re_m) * x_46_re_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 4e+205)
		tmp = fma(Float64(x_46_im_m * x_46_re_m), Float64(-2.0 * x_46_im_m), Float64(Float64(x_46_re_m - x_46_im_m) * t_0));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), t_0, 0.0);
	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_] := Block[{t$95$0 = N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], 4e+205], N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * N[(-2.0 * x$46$im$95$m), $MachinePrecision] + N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * t$95$0 + 0.0), $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)) < 4.00000000000000007e205

   1. Initial program 95.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 95.5

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

   4. Applied rewrites 95.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.7%

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

   if 4.00000000000000007e205 < (-.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 53.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 53.4

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

   4. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 77.4

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

   6. Applied rewrites 77.4%

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

   8. Applied rewrites 83.6%

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

   10. Applied rewrites 86.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 4 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot x.re, -2 \cdot x.im, \left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq 4 \cdot 10^{+205}:\\ \;\;\;\;\left(\left(x.re\_m - x.im\_m\right) \cdot x.re\_m\right) \cdot \left(x.im\_m + x.re\_m\right) - \left(\left(x.im\_m + x.im\_m\right) \cdot x.re\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, \left(x.im\_m + x.re\_m\right) \cdot x.re\_m, 0\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       4e+205)
    (-
     (* (* (- x.re_m x.im_m) x.re_m) (+ x.im_m x.re_m))
     (* (* (+ x.im_m x.im_m) x.re_m) x.im_m))
    (fma (- x.re_m x.im_m) (* (+ x.im_m x.re_m) x.re_m) 0.0))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 4e+205) {
		tmp = (((x_46_re_m - x_46_im_m) * x_46_re_m) * (x_46_im_m + x_46_re_m)) - (((x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m);
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), ((x_46_im_m + x_46_re_m) * x_46_re_m), 0.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 4e+205)
		tmp = Float64(Float64(Float64(Float64(x_46_re_m - x_46_im_m) * x_46_re_m) * Float64(x_46_im_m + x_46_re_m)) - Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(Float64(x_46_im_m + x_46_re_m) * x_46_re_m), 0.0);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], 4e+205], N[(N[(N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + 0.0), $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)) < 4.00000000000000007e205

   1. Initial program 95.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 95.5

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

   4. Applied rewrites 95.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if 4.00000000000000007e205 < (-.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 53.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 53.4

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

   4. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 77.4

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

   6. Applied rewrites 77.4%

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

   8. Applied rewrites 83.6%

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

   10. Applied rewrites 86.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 4 \cdot 10^{+205}:\\ \;\;\;\;\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -4 \cdot 10^{-320}:\\ \;\;\;\;\left(-3 \cdot \left(x.im\_m \cdot x.re\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, \left(x.im\_m + x.re\_m\right) \cdot x.re\_m, 0\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       -4e-320)
    (* (* -3.0 (* x.im_m x.re_m)) x.im_m)
    (fma (- x.re_m x.im_m) (* (+ x.im_m x.re_m) x.re_m) 0.0))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -4e-320) {
		tmp = (-3.0 * (x_46_im_m * x_46_re_m)) * x_46_im_m;
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), ((x_46_im_m + x_46_re_m) * x_46_re_m), 0.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -4e-320)
		tmp = Float64(Float64(-3.0 * Float64(x_46_im_m * x_46_re_m)) * x_46_im_m);
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(Float64(x_46_im_m + x_46_re_m) * x_46_re_m), 0.0);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-320], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + 0.0), $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)) < -3.99996e-320

   1. Initial program 92.4%

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

   3. Taylor expanded in x.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. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

   7. Applied rewrites 52.1%

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

   if -3.99996e-320 < (-.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 75.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 75.7

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

   4. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 88.1

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

   6. Applied rewrites 88.1%

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

   8. Applied rewrites 91.4%

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

   10. Applied rewrites 76.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -4 \cdot 10^{-320}:\\ \;\;\;\;\left(-3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq 5 \cdot 10^{-289}:\\ \;\;\;\;\left(-3 \cdot \left(x.im\_m \cdot x.re\_m\right)\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       5e-289)
    (* (* -3.0 (* x.im_m x.re_m)) x.im_m)
    (* x.re_m x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-289) {
		tmp = (-3.0 * (x_46_im_m * x_46_re_m)) * x_46_im_m;
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= 5d-289) then
        tmp = ((-3.0d0) * (x_46im_m * x_46re_m)) * x_46im_m
    else
        tmp = x_46re_m * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-289) {
		tmp = (-3.0 * (x_46_im_m * x_46_re_m)) * x_46_im_m;
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-289:
		tmp = (-3.0 * (x_46_im_m * x_46_re_m)) * x_46_im_m
	else:
		tmp = x_46_re_m * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-289)
		tmp = Float64(Float64(-3.0 * Float64(x_46_im_m * x_46_re_m)) * x_46_im_m);
	else
		tmp = Float64(x_46_re_m * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-289)
		tmp = (-3.0 * (x_46_im_m * x_46_re_m)) * x_46_im_m;
	else
		tmp = x_46_re_m * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], 5e-289], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(x$46$re$95$m * 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)) < 5.00000000000000029e-289

   1. Initial program 94.2%

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

   3. Taylor expanded in x.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. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 61.9%

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

   if 5.00000000000000029e-289 < (-.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 69.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 69.6

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

   4. Applied rewrites 69.6%

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

   6. Applied rewrites 30.3%

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

   7. Taylor expanded in x.re around inf

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

         \[\leadsto {x.re}^{2} \cdot \color{blue}{1} \]

      7. *-rgt-identity N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 31.7

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

   9. Applied rewrites 31.7%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 5 \cdot 10^{-289}:\\ \;\;\;\;\left(-3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq 5 \cdot 10^{-289}:\\ \;\;\;\;\left(\left(-3 \cdot x.re\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       5e-289)
    (* (* (* -3.0 x.re_m) x.im_m) x.im_m)
    (* x.re_m x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-289) {
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m;
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= 5d-289) then
        tmp = (((-3.0d0) * x_46re_m) * x_46im_m) * x_46im_m
    else
        tmp = x_46re_m * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-289) {
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m;
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-289:
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m
	else:
		tmp = x_46_re_m * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-289)
		tmp = Float64(Float64(Float64(-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(x_46_re_m * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= 5e-289)
		tmp = ((-3.0 * x_46_re_m) * x_46_im_m) * x_46_im_m;
	else
		tmp = x_46_re_m * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], 5e-289], N[(N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(x$46$re$95$m * 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)) < 5.00000000000000029e-289

   1. Initial program 94.2%

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

   3. Taylor expanded in x.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. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 61.8%

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

   8. Taylor expanded in x.re around 0

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

   10. Applied rewrites 61.9%

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

   if 5.00000000000000029e-289 < (-.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 69.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 69.6

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

   4. Applied rewrites 69.6%

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

   6. Applied rewrites 30.3%

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

   7. Taylor expanded in x.re around inf

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

         \[\leadsto {x.re}^{2} \cdot \color{blue}{1} \]

      7. *-rgt-identity N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 31.7

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

   9. Applied rewrites 31.7%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 5 \cdot 10^{-289}:\\ \;\;\;\;\left(\left(-3 \cdot x.re\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq -1 \cdot 10^{-180}:\\ \;\;\;\;-2 \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       -1e-180)
    (* -2.0 x.im_m)
    (* x.re_m x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-180) {
		tmp = -2.0 * x_46_im_m;
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)) * x_46re_m) - (((x_46im_m * x_46re_m) + (x_46im_m * x_46re_m)) * x_46im_m)) <= (-1d-180)) then
        tmp = (-2.0d0) * x_46im_m
    else
        tmp = x_46re_m * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-180) {
		tmp = -2.0 * x_46_im_m;
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-180:
		tmp = -2.0 * x_46_im_m
	else:
		tmp = x_46_re_m * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_re_m) - Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-180)
		tmp = Float64(-2.0 * x_46_im_m);
	else
		tmp = Float64(x_46_re_m * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_re_m) - (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_im_m)) <= -1e-180)
		tmp = -2.0 * x_46_im_m;
	else
		tmp = x_46_re_m * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -1e-180], N[(-2.0 * x$46$im$95$m), $MachinePrecision], N[(x$46$re$95$m * 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)) < -1e-180

   1. Initial program 92.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 92.0

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

   4. Applied rewrites 92.0%

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

   6. Applied rewrites 20.9%

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

   7. Taylor expanded in x.im around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 1.9

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

   9. Applied rewrites 1.9%

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

   10. Taylor expanded in x.re around 0

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

   12. Applied rewrites 3.0%

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

   if -1e-180 < (-.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 76.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 76.6

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

   4. Applied rewrites 76.6%

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

   6. Applied rewrites 24.5%

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

   7. Taylor expanded in x.re around inf

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

         \[\leadsto {x.re}^{2} \cdot \color{blue}{1} \]

      7. *-rgt-identity N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 40.8

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

   9. Applied rewrites 40.8%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -1 \cdot 10^{-180}:\\ \;\;\;\;-2 \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.2 \cdot 10^{+165}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (* x.re_s (if (<= x.im_m 1.2e+165) (* x.re_m x.re_m) (* (- x.im_m) x.im_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.2e+165) {
		tmp = x_46_re_m * x_46_re_m;
	} else {
		tmp = -x_46_im_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_46im_m <= 1.2d+165) then
        tmp = x_46re_m * x_46re_m
    else
        tmp = -x_46im_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_im_m <= 1.2e+165) {
		tmp = x_46_re_m * x_46_re_m;
	} else {
		tmp = -x_46_im_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_im_m <= 1.2e+165:
		tmp = x_46_re_m * x_46_re_m
	else:
		tmp = -x_46_im_m * x_46_im_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.2e+165)
		tmp = Float64(x_46_re_m * x_46_re_m);
	else
		tmp = Float64(Float64(-x_46_im_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_im_m <= 1.2e+165)
		tmp = x_46_re_m * x_46_re_m;
	else
		tmp = -x_46_im_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[x$46$im$95$m, 1.2e+165], 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]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.2e165

   1. Initial program 86.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 86.1

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

   4. Applied rewrites 86.1%

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

   6. Applied rewrites 21.0%

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

   7. Taylor expanded in x.re around inf

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

         \[\leadsto {x.re}^{2} \cdot \color{blue}{1} \]

      7. *-rgt-identity N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 27.6

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

   9. Applied rewrites 27.6%

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

   if 1.2e165 < x.im

   1. Initial program 51.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 51.9

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 40.1%

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

   7. Taylor expanded in x.re around inf

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. metadata-eval N/A

         \[\leadsto {x.re}^{2} \cdot \color{blue}{1} \]

      7. *-rgt-identity N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 12.0

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

   9. Applied rewrites 12.0%

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

   10. Taylor expanded in x.im around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. lower-neg.f64 43.7

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

   12. Applied rewrites 43.7%

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

Alternative 8: 4.4% accurate, 6.7× speedup

Math

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

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m) :precision binary64 (* x.re_s (* -2.0 x.im_m)))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (-2.0 * x_46_im_m);
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * ((-2.0d0) * x_46im_m)
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (-2.0 * x_46_im_m);
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * (-2.0 * x_46_im_m)

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(-2.0 * x_46_im_m))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * (-2.0 * x_46_im_m);
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * N[(-2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 82.3

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

4. Applied rewrites 82.3%

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

6. Applied rewrites 23.1%

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

7. Taylor expanded in x.im around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 16.7

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

9. Applied rewrites 16.7%

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

10. Taylor expanded in x.re around 0

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

12. Applied rewrites 3.5%

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

Developer Target 1: 99.7% 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 2024296 
(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)))