math.cube on complex, real part

Percentage Accurate: 82.4% → 99.7%

Time: 12.8s

Alternatives: 9

Speedup: 1.4×

• 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: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.re\_m - \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.im\_m \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(x.im\_m + x.re\_m\right) \cdot x.re\_m\right) \cdot \left(x.re\_m - x.im\_m\right) - \left(\left(x.im\_m + x.im\_m\right) \cdot x.re\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(\left(x.re\_m - x.im\_m\right) \cdot x.re\_m\right) \cdot \left(x.im\_m + x.re\_m\right)}} - \left(x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       5e+246)
    (-
     (* (* (+ x.im_m x.re_m) x.re_m) (- x.re_m x.im_m))
     (* (* (+ x.im_m x.im_m) x.re_m) x.im_m))
    (-
     (/ 1.0 (/ 1.0 (* (* (- x.re_m x.im_m) x.re_m) (+ x.im_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_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+246) {
		tmp = (((x_46_im_m + x_46_re_m) * x_46_re_m) * (x_46_re_m - x_46_im_m)) - (((x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m);
	} else {
		tmp = (1.0 / (1.0 / (((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);
	}
	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+246) then
        tmp = (((x_46im_m + x_46re_m) * x_46re_m) * (x_46re_m - x_46im_m)) - (((x_46im_m + x_46im_m) * x_46re_m) * x_46im_m)
    else
        tmp = (1.0d0 / (1.0d0 / (((x_46re_m - x_46im_m) * x_46re_m) * (x_46im_m + x_46re_m)))) - (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_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+246) {
		tmp = (((x_46_im_m + x_46_re_m) * x_46_re_m) * (x_46_re_m - x_46_im_m)) - (((x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m);
	} else {
		tmp = (1.0 / (1.0 / (((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);
	}
	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+246:
		tmp = (((x_46_im_m + x_46_re_m) * x_46_re_m) * (x_46_re_m - x_46_im_m)) - (((x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m)
	else:
		tmp = (1.0 / (1.0 / (((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)
	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+246)
		tmp = Float64(Float64(Float64(Float64(x_46_im_m + x_46_re_m) * x_46_re_m) * Float64(x_46_re_m - x_46_im_m)) - Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 / 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(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_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+246)
		tmp = (((x_46_im_m + x_46_re_m) * x_46_re_m) * (x_46_re_m - x_46_im_m)) - (((x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m);
	else
		tmp = (1.0 / (1.0 / (((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);
	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+246], N[(N[(N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * N[(x$46$re$95$m - x$46$im$95$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[(1.0 / N[(1.0 / 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]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m + 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)) < 4.99999999999999976e246

   1. Initial program 96.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 96.1

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

   4. Applied rewrites 96.1%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if 4.99999999999999976e246 < (-.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 60.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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 60.9

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

   4. Applied rewrites 60.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. flip-- N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift--.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. associate-*l/ N/A

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

      21. clear-num N/A

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

      22. lower-/.f64 N/A

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

      23. lower-/.f64 N/A

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

   6. Applied rewrites 56.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-- N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   8. Applied rewrites 85.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. associate-*r/ N/A

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

      10. +-inverses N/A

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

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

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

      12. +-inverses N/A

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

      13. flip-+ N/A

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

      14. lift-+.f64 90.9

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

   10. Applied rewrites 90.9%

       \[\leadsto \frac{1}{\frac{1}{\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right)}} - \color{blue}{\left(x.im + x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(x.im + x.re\right) \cdot x.re\right) \cdot \left(x.re - x.im\right) - \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right)}} - \left(x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

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

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.re_m)
        (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.im_m))
       INFINITY)
    (-
     (* (* (+ x.im_m x.re_m) x.re_m) (- x.re_m x.im_m))
     (* (* (+ x.im_m x.im_m) x.re_m) x.im_m))
    (*
     (* (fma (/ x.re_m x.im_m) (/ x.re_m x.im_m) -3.0) 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_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)) <= ((double) INFINITY)) {
		tmp = (((x_46_im_m + x_46_re_m) * x_46_re_m) * (x_46_re_m - x_46_im_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_re_m / x_46_im_m), -3.0) * x_46_re_m) * (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 (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)) <= Inf)
		tmp = Float64(Float64(Float64(Float64(x_46_im_m + x_46_re_m) * x_46_re_m) * Float64(x_46_re_m - x_46_im_m)) - Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re_m) * x_46_im_m));
	else
		tmp = Float64(Float64(fma(Float64(x_46_re_m / x_46_im_m), Float64(x_46_re_m / x_46_im_m), -3.0) * x_46_re_m) * Float64(x_46_im_m * x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

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

   1. Initial program 93.8%

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

      1. lift-+.f64 N/A

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

      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 93.8

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

   4. Applied rewrites 93.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im \]
   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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   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. 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 52.2

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

   5. Applied rewrites 52.2%

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 3: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.8 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im\_m \cdot x.im\_m, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-3 \cdot x.im\_m\right) \cdot \left(x.im\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

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.im_m 1.8e+130)
    (* (fma -3.0 (* x.im_m x.im_m) (* x.re_m x.re_m)) x.re_m)
    (* (* -3.0 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_im_m <= 1.8e+130) {
		tmp = fma(-3.0, (x_46_im_m * x_46_im_m), (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = (-3.0 * x_46_im_m) * (x_46_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 (x_46_im_m <= 1.8e+130)
		tmp = Float64(fma(-3.0, Float64(x_46_im_m * x_46_im_m), Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = Float64(Float64(-3.0 * x_46_im_m) * Float64(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[x$46$im$95$m, 1.8e+130], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(-3.0 * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.8000000000000001e130

   1. Initial program 91.8%

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

   3. Taylor expanded in x.re around 0

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

      1. *-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. cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. cancel-sign-sub-inv N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if 1.8000000000000001e130 < 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. 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 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 91.5%

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

Alternative 4: 55.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(\left(-3 \cdot x.im\_m\right) \cdot \left(x.im\_m \cdot x.re\_m\right)\right) \end{array} \]

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 (* (* -3.0 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 * ((-3.0 * 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 * (((-3.0d0) * 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 * ((-3.0 * 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 * ((-3.0 * 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(-3.0 * x_46_im_m) * Float64(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 * ((-3.0 * 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[(-3.0 * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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 48.8

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

5. Applied rewrites 48.8%

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

7. Applied rewrites 54.3%

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

Alternative 5: 55.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(\left(-3 \cdot \left(x.im\_m \cdot x.re\_m\right)\right) \cdot x.im\_m\right) \end{array} \]

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 (* (* -3.0 (* x.im_m x.re_m)) x.im_m)))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * ((-3.0 * (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 * (((-3.0d0) * (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 * ((-3.0 * (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 * ((-3.0 * (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(-3.0 * 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 * ((-3.0 * (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[(-3.0 * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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 48.8

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

5. Applied rewrites 48.8%

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

7. Applied rewrites 54.2%

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

8. Final simplification 54.2%

   \[\leadsto \left(-3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im \]
9. Add Preprocessing

Alternative 6: 55.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(\left(\left(x.im\_m \cdot x.re\_m\right) \cdot x.im\_m\right) \cdot -3\right) \end{array} \]

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) -3.0)))

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) {
	return x_46_re_s * (((x_46_im_m * x_46_re_m) * x_46_im_m) * -3.0);
}

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) * (-3.0d0))
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) * -3.0);
}

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) * -3.0)

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(Float64(x_46_im_m * x_46_re_m) * x_46_im_m) * -3.0))
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) * -3.0);
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[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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 48.8

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

5. Applied rewrites 48.8%

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

7. Applied rewrites 54.2%

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

8. Final simplification 54.2%

   \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3 \]
9. Add Preprocessing

Alternative 7: 55.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(\left(\left(-3 \cdot x.re\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\right) \end{array} \]

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 (* (* (* -3.0 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) {
	return x_46_re_s * (((-3.0 * x_46_re_m) * x_46_im_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 * ((((-3.0d0) * x_46re_m) * x_46im_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 * (((-3.0 * x_46_re_m) * x_46_im_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 * (((-3.0 * x_46_re_m) * x_46_im_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(Float64(-3.0 * x_46_re_m) * x_46_im_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 * (((-3.0 * x_46_re_m) * x_46_im_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[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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 48.8

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

5. Applied rewrites 48.8%

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

7. Applied rewrites 54.2%

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

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 54.2%

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

Alternative 8: 20.1% accurate, 2.9× speedup

Math

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

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 (* (fma (- 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) {
	return x_46_re_s * (fma(-x_46_re_m, x_46_im_m, 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(fma(Float64(-x_46_re_m), x_46_im_m, 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[(N[((-x$46$re$95$m) * x$46$im$95$m + 2.0), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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 85.4

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

4. Applied rewrites 85.4%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im \]
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. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. lift--.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 95.5

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 95.5

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

6. Applied rewrites 95.5%

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

8. Applied rewrites 27.6%

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

9. Taylor expanded in x.re around 0

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

    1. *-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. *-commutative N/A

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

    7. distribute-lft-in N/A

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

    8. +-commutative N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 N/A

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

    13. associate-*r* N/A

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

    14. lower-fma.f64 N/A

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

    15. mul-1-neg N/A

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

    16. lower-neg.f64 20.6

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

11. Applied rewrites 20.6%

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

Alternative 9: 2.9% 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 85.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
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 85.4

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

4. Applied rewrites 85.4%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im \]
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. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. lift--.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 95.5

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 95.5

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

6. Applied rewrites 95.5%

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

8. Applied rewrites 27.6%

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

9. Taylor expanded in x.re around 0

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

    1. lower-*.f64 3.4

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

11. Applied rewrites 3.4%

    \[\leadsto \color{blue}{2 \cdot x.im} \]
12. 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 2024331 
(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)))