math.cube on complex, imaginary part

Percentage Accurate: 82.2% → 96.5%

Time: 5.8s

Alternatives: 6

Speedup: 1.3×

• 46.0% 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: 82.2% 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: 96.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.05 \cdot 10^{+158}:\\ \;\;\;\;\left(-x.im\right) \cdot \mathsf{fma}\left(-3 \cdot x.re\_m, x.re\_m, x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot 3\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 1.05e+158)
   (* (- x.im) (fma (* -3.0 x.re_m) x.re_m (* x.im x.im)))
   (* (* (* x.im x.re_m) 3.0) x.re_m)))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.05e+158) {
		tmp = -x_46_im * fma((-3.0 * x_46_re_m), x_46_re_m, (x_46_im * x_46_im));
	} else {
		tmp = ((x_46_im * x_46_re_m) * 3.0) * x_46_re_m;
	}
	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_re_m <= 1.05e+158)
		tmp = Float64(Float64(-x_46_im) * fma(Float64(-3.0 * x_46_re_m), x_46_re_m, Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * 3.0) * x_46_re_m);
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 1.05e+158], N[((-x$46$im) * N[(N[(-3.0 * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.0499999999999999e158

   1. Initial program 87.5%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. distribute-lft1-in N/A

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

   5. Applied rewrites 96.9%

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

   7. Applied rewrites 94.3%

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

   if 1.0499999999999999e158 < x.re

   1. Initial program 44.6%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 91.5%

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

Alternative 2: 59.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-279} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot 3\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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.im)) x.im)
          (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.re_m))))
   (if (or (<= t_0 -1e-279) (not (<= t_0 INFINITY)))
     (* (- x.im) (* x.im x.im))
     (* (* (* x.im x.re_m) 3.0) x.re_m))))

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_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -1e-279) || !(t_0 <= ((double) INFINITY))) {
		tmp = -x_46_im * (x_46_im * x_46_im);
	} else {
		tmp = ((x_46_im * x_46_re_m) * 3.0) * x_46_re_m;
	}
	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_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -1e-279) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = -x_46_im * (x_46_im * x_46_im);
	} else {
		tmp = ((x_46_im * x_46_re_m) * 3.0) * x_46_re_m;
	}
	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_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if (t_0 <= -1e-279) or not (t_0 <= math.inf):
		tmp = -x_46_im * (x_46_im * x_46_im)
	else:
		tmp = ((x_46_im * x_46_re_m) * 3.0) * x_46_re_m
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if ((t_0 <= -1e-279) || !(t_0 <= Inf))
		tmp = Float64(Float64(-x_46_im) * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * 3.0) * x_46_re_m);
	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_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if ((t_0 <= -1e-279) || ~((t_0 <= Inf)))
		tmp = -x_46_im * (x_46_im * x_46_im);
	else
		tmp = ((x_46_im * x_46_re_m) * 3.0) * x_46_re_m;
	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[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-279], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[((-x$46$im) * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re$95$m), $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)) < -1.00000000000000006e-279 or +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 71.8%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. distribute-lft1-in N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 54.6%

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

   if -1.00000000000000006e-279 < (+.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 95.4%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 58.9

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 58.9%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-279} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot 3\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-279} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot x.re\_m\right)\\ \end{array} \end{array} \]

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.im)) x.im)
          (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.re_m))))
   (if (or (<= t_0 -1e-279) (not (<= t_0 INFINITY)))
     (* (- x.im) (* x.im x.im))
     (* 3.0 (* (* x.im x.re_m) x.re_m)))))

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_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -1e-279) || !(t_0 <= ((double) INFINITY))) {
		tmp = -x_46_im * (x_46_im * x_46_im);
	} else {
		tmp = 3.0 * ((x_46_im * x_46_re_m) * x_46_re_m);
	}
	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_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -1e-279) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = -x_46_im * (x_46_im * x_46_im);
	} else {
		tmp = 3.0 * ((x_46_im * x_46_re_m) * x_46_re_m);
	}
	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_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if (t_0 <= -1e-279) or not (t_0 <= math.inf):
		tmp = -x_46_im * (x_46_im * x_46_im)
	else:
		tmp = 3.0 * ((x_46_im * x_46_re_m) * x_46_re_m)
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if ((t_0 <= -1e-279) || !(t_0 <= Inf))
		tmp = Float64(Float64(-x_46_im) * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_im * x_46_re_m) * x_46_re_m));
	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_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if ((t_0 <= -1e-279) || ~((t_0 <= Inf)))
		tmp = -x_46_im * (x_46_im * x_46_im);
	else
		tmp = 3.0 * ((x_46_im * x_46_re_m) * x_46_re_m);
	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[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-279], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[((-x$46$im) * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.00000000000000006e-279 or +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 71.8%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. distribute-lft1-in N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 54.6%

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

   if -1.00000000000000006e-279 < (+.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 95.4%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 58.9

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 58.8%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-279} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.75 \cdot 10^{+151}:\\ \;\;\;\;\left(-x.im\right) \cdot \mathsf{fma}\left(x.im, x.im, -3 \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot 3\right) \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 1.75e+151)
   (* (- x.im) (fma x.im x.im (* -3.0 (* x.re_m x.re_m))))
   (* (* (* x.im x.re_m) 3.0) x.re_m)))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.75e+151) {
		tmp = -x_46_im * fma(x_46_im, x_46_im, (-3.0 * (x_46_re_m * x_46_re_m)));
	} else {
		tmp = ((x_46_im * x_46_re_m) * 3.0) * x_46_re_m;
	}
	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_re_m <= 1.75e+151)
		tmp = Float64(Float64(-x_46_im) * fma(x_46_im, x_46_im, Float64(-3.0 * Float64(x_46_re_m * x_46_re_m))));
	else
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * 3.0) * x_46_re_m);
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 1.75e+151], N[((-x$46$im) * N[(x$46$im * x$46$im + N[(-3.0 * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.7500000000000001e151

   1. Initial program 87.5%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. distribute-lft1-in N/A

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

   5. Applied rewrites 96.9%

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

   if 1.7500000000000001e151 < x.re

   1. Initial program 44.6%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 91.5%

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

Alternative 5: 62.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 9.5 \cdot 10^{+150}:\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 9.5e+150) (* (- x.im) (* x.im x.im)) (* (* x.im x.im) x.im)))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 9.5e+150) {
		tmp = -x_46_im * (x_46_im * x_46_im);
	} else {
		tmp = (x_46_im * x_46_im) * x_46_im;
	}
	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_46re_m <= 9.5d+150) then
        tmp = -x_46im * (x_46im * x_46im)
    else
        tmp = (x_46im * x_46im) * x_46im
    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_re_m <= 9.5e+150) {
		tmp = -x_46_im * (x_46_im * x_46_im);
	} else {
		tmp = (x_46_im * x_46_im) * x_46_im;
	}
	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_re_m <= 9.5e+150:
		tmp = -x_46_im * (x_46_im * x_46_im)
	else:
		tmp = (x_46_im * x_46_im) * x_46_im
	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_re_m <= 9.5e+150)
		tmp = Float64(Float64(-x_46_im) * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(Float64(x_46_im * x_46_im) * x_46_im);
	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_re_m <= 9.5e+150)
		tmp = -x_46_im * (x_46_im * x_46_im);
	else
		tmp = (x_46_im * x_46_im) * x_46_im;
	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$46$re$95$m, 9.5e+150], N[((-x$46$im) * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 9.5000000000000001e150

   1. Initial program 87.5%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. distribute-lft1-in N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 66.9%

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

   if 9.5000000000000001e150 < x.re

   1. Initial program 44.6%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. cube-neg-rev N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-neg.f64 9.3

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

   5. Applied rewrites 9.3%

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

   7. Applied rewrites 22.6%

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

Alternative 6: 20.5% accurate, 3.6× speedup

Math

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

FPCore

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

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_im) * x_46_im;
}

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_46im) * x_46im
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_im) * x_46_im;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. cube-neg-rev N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-neg.f64 61.6

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

5. Applied rewrites 61.6%

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

7. Applied rewrites 21.8%

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

Developer Target 1: 91.1% 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 2024326 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 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)))