math.cube on complex, imaginary part

Percentage Accurate: 82.4% → 96.4%

Time: 6.1s

Alternatives: 4

Speedup: 0.5×

• 45.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ \mathbf{if}\;x.im \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) + t\_0 \leq \infty:\\ \;\;\;\;t\_0 + \left(x.im \cdot \left(x.re\_m + x.im\right)\right) \cdot \left(x.re\_m - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* x.re_m (+ (* x.re_m x.im) (* x.re_m x.im)))))
   (if (<= (+ (* x.im (- (* x.re_m x.re_m) (* x.im x.im))) t_0) INFINITY)
     (+ t_0 (* (* x.im (+ x.re_m x.im)) (- x.re_m x.im)))
     (* x.im (* x.re_m (* x.re_m -3.0))))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	double tmp;
	if (((x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + t_0) <= ((double) INFINITY)) {
		tmp = t_0 + ((x_46_im * (x_46_re_m + x_46_im)) * (x_46_re_m - x_46_im));
	} else {
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	double tmp;
	if (((x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + t_0) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + ((x_46_im * (x_46_re_m + x_46_im)) * (x_46_re_m - x_46_im));
	} else {
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im))
	tmp = 0
	if ((x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + t_0) <= math.inf:
		tmp = t_0 + ((x_46_im * (x_46_re_m + x_46_im)) * (x_46_re_m - x_46_im))
	else:
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) + t_0) <= Inf)
		tmp = Float64(t_0 + Float64(Float64(x_46_im * Float64(x_46_re_m + x_46_im)) * Float64(x_46_re_m - x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_re_m * -3.0)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im));
	tmp = 0.0;
	if (((x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + t_0) <= Inf)
		tmp = t_0 + ((x_46_im * (x_46_re_m + x_46_im)) * (x_46_re_m - x_46_im));
	else
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], Infinity], N[(t$95$0 + N[(N[(x$46$im * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$re$95$m * -3.0), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 94.8%

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

      1. *-commutative 94.8%

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

      2. difference-of-squares 94.8%

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

      3. associate-*r* 99.8%

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

   4. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. *-commutative 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. associate-*r* 0.0%

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

      4. associate-*r* 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x.re around inf 18.2%

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

      1. associate-*r* 18.2%

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

      2. *-commutative 18.2%

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

      3. associate-*l* 18.2%

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

   9. Simplified 18.2%

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

       1. metadata-eval 18.2%

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

       2. associate-*r* 18.2%

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

       3. neg-mul-1 18.2%

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

       4. add-sqr-sqrt 0.0%

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

       5. associate-*r* 0.0%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 0.0%

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

       8. sqr-neg 0.0%

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

       9. sqrt-unprod 0.0%

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

       10. add-sqr-sqrt 0.0%

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

       11. unpow2 0.0%

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

       12. sqrt-prod 0.0%

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

       13. add-sqr-sqrt 0.0%

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

       14. add-sqr-sqrt 0.0%

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

       15. sqrt-unprod 69.7%

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

       16. sqr-neg 69.7%

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

       17. sqrt-unprod 69.7%

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

       18. add-sqr-sqrt 69.7%

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

       19. unpow2 69.7%

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

       20. sqrt-prod 57.6%

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

       21. add-sqr-sqrt 81.8%

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

   11. Applied egg-rr 81.8%

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

4. Final simplification 97.4%

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

Alternative 2: 57.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 3.3 \cdot 10^{+126}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 3.3e+126)
   (* x.re_m (* x.re_m (* x.im 3.0)))
   (* x.im (* x.re_m (* x.re_m -3.0)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.3e+126) {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 3.0));
	} else {
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 3.3d+126) then
        tmp = x_46re_m * (x_46re_m * (x_46im * 3.0d0))
    else
        tmp = x_46im * (x_46re_m * (x_46re_m * (-3.0d0)))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.3e+126) {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 3.0));
	} else {
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 3.3e+126:
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 3.0))
	else:
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 3.3e+126)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im * 3.0)));
	else
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_re_m * -3.0)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 3.3e+126)
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 3.0));
	else
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 3.3e+126], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.30000000000000013e126

   1. Initial program 87.0%

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

   3. Simplified 91.2%

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

      1. *-commutative 91.2%

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

      2. add-sqr-sqrt 44.2%

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

      3. associate-*r* 44.2%

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

      4. associate-*r* 44.2%

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

   6. Applied egg-rr 44.2%

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

   7. Taylor expanded in x.re around inf 54.9%

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

      1. associate-*r* 54.9%

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

      2. *-commutative 54.9%

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

      3. associate-*l* 54.9%

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

   9. Simplified 54.9%

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

       1. associate-*r* 54.9%

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

       2. unpow2 54.9%

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

       3. associate-*r* 60.0%

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

   11. Applied egg-rr 60.0%

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

   if 3.30000000000000013e126 < x.im

   1. Initial program 56.8%

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

   3. Simplified 56.8%

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

      1. *-commutative 56.8%

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

      2. add-sqr-sqrt 27.0%

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

      3. associate-*r* 27.0%

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

      4. associate-*r* 27.0%

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

   6. Applied egg-rr 27.0%

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

   7. Taylor expanded in x.re around inf 11.3%

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

      1. associate-*r* 11.3%

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

      2. *-commutative 11.3%

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

      3. associate-*l* 11.3%

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

   9. Simplified 11.3%

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

       1. metadata-eval 11.3%

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

       2. associate-*r* 11.3%

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

       3. neg-mul-1 11.3%

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

       4. add-sqr-sqrt 0.3%

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

       5. associate-*r* 0.3%

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

       6. add-sqr-sqrt 0.3%

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

       7. sqrt-unprod 0.3%

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

       8. sqr-neg 0.3%

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

       9. sqrt-unprod 0.3%

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

       10. add-sqr-sqrt 0.3%

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

       11. unpow2 0.3%

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

       12. sqrt-prod 0.2%

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

       13. add-sqr-sqrt 0.3%

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

       14. add-sqr-sqrt 0.3%

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

       15. sqrt-unprod 28.4%

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

       16. sqr-neg 28.4%

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

       17. sqrt-unprod 28.3%

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

       18. add-sqr-sqrt 28.3%

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

       19. unpow2 28.3%

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

       20. sqrt-prod 22.6%

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

       21. add-sqr-sqrt 34.4%

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

   11. Applied egg-rr 34.4%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.3 \cdot 10^{+126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 3 \cdot 10^{+126}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 3e+126)
   (* x.im (* x.re_m (* x.re_m 3.0)))
   (* x.im (* x.re_m (* x.re_m -3.0)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 3e+126) {
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * 3.0));
	} else {
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 3d+126) then
        tmp = x_46im * (x_46re_m * (x_46re_m * 3.0d0))
    else
        tmp = x_46im * (x_46re_m * (x_46re_m * (-3.0d0)))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 3e+126) {
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * 3.0));
	} else {
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 3e+126:
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * 3.0))
	else:
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 3e+126)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_re_m * 3.0)));
	else
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_re_m * -3.0)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 3e+126)
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * 3.0));
	else
		tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 3e+126], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.0000000000000002e126

   1. Initial program 87.0%

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

   3. Simplified 91.2%

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

      1. *-commutative 91.2%

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

      2. add-sqr-sqrt 44.2%

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

      3. associate-*r* 44.2%

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

      4. associate-*r* 44.2%

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

   6. Applied egg-rr 44.2%

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

   7. Taylor expanded in x.re around inf 54.9%

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

      1. associate-*r* 54.9%

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

      2. *-commutative 54.9%

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

      3. associate-*l* 54.9%

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

   9. Simplified 54.9%

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

       1. unpow2 54.9%

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

       2. associate-*r* 54.8%

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

   11. Applied egg-rr 54.8%

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

   if 3.0000000000000002e126 < x.im

   1. Initial program 56.8%

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

   3. Simplified 56.8%

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

      1. *-commutative 56.8%

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

      2. add-sqr-sqrt 27.0%

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

      3. associate-*r* 27.0%

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

      4. associate-*r* 27.0%

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

   6. Applied egg-rr 27.0%

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

   7. Taylor expanded in x.re around inf 11.3%

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

      1. associate-*r* 11.3%

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

      2. *-commutative 11.3%

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

      3. associate-*l* 11.3%

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

   9. Simplified 11.3%

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

       1. metadata-eval 11.3%

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

       2. associate-*r* 11.3%

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

       3. neg-mul-1 11.3%

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

       4. add-sqr-sqrt 0.3%

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

       5. associate-*r* 0.3%

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

       6. add-sqr-sqrt 0.3%

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

       7. sqrt-unprod 0.3%

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

       8. sqr-neg 0.3%

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

       9. sqrt-unprod 0.3%

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

       10. add-sqr-sqrt 0.3%

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

       11. unpow2 0.3%

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

       12. sqrt-prod 0.2%

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

       13. add-sqr-sqrt 0.3%

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

       14. add-sqr-sqrt 0.3%

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

       15. sqrt-unprod 28.4%

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

       16. sqr-neg 28.4%

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

       17. sqrt-unprod 28.3%

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

       18. add-sqr-sqrt 28.3%

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

       19. unpow2 28.3%

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

       20. sqrt-prod 22.6%

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

       21. add-sqr-sqrt 34.4%

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

   11. Applied egg-rr 34.4%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3 \cdot 10^{+126}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 24.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot -3\right)\right) \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im) :precision binary64 (* x.im (* x.re_m (* x.re_m -3.0))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	return x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46im * (x_46re_m * (x_46re_m * (-3.0d0)))
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	return x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	return x_46_im * (x_46_re_m * (x_46_re_m * -3.0))

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	return Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_re_m * -3.0)))
end

MATLAB

x.re_m = abs(x_46_re);
function tmp = code(x_46_re_m, x_46_im)
	tmp = x_46_im * (x_46_re_m * (x_46_re_m * -3.0));
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := N[(x$46$im * N[(x$46$re$95$m * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

3. Simplified 86.2%

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

   1. *-commutative 86.2%

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

   2. add-sqr-sqrt 41.7%

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

   3. associate-*r* 41.7%

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

   4. associate-*r* 41.7%

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

6. Applied egg-rr 41.7%

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

7. Taylor expanded in x.re around inf 48.6%

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

   1. associate-*r* 48.6%

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

   2. *-commutative 48.6%

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

   3. associate-*l* 48.6%

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

9. Simplified 48.6%

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

    1. metadata-eval 48.6%

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

    2. associate-*r* 48.6%

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

    3. neg-mul-1 48.6%

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

    4. add-sqr-sqrt 4.8%

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

    5. associate-*r* 4.8%

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

    6. add-sqr-sqrt 4.8%

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

    7. sqrt-unprod 4.8%

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

    8. sqr-neg 4.8%

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

    9. sqrt-unprod 4.8%

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

    10. add-sqr-sqrt 4.8%

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

    11. unpow2 4.8%

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

    12. sqrt-prod 3.0%

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

    13. add-sqr-sqrt 4.8%

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

    14. add-sqr-sqrt 4.8%

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

    15. sqrt-unprod 34.5%

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

    16. sqr-neg 34.5%

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

    17. sqrt-unprod 38.9%

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

    18. add-sqr-sqrt 38.9%

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

    19. unpow2 38.9%

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

    20. sqrt-prod 15.8%

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

    21. add-sqr-sqrt 25.0%

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

11. Applied egg-rr 25.0%

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

12. Final simplification 25.0%

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

Developer target: 91.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - 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_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - 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_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024096 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))