math.cube on complex, real part

Percentage Accurate: 82.7% → 99.7%

Time: 9.8s

Alternatives: 7

Speedup: 1.2×

• 45.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 4.3 \cdot 10^{+86}:\\ \;\;\;\;{x.re\_m}^{3} - x.im \cdot \left(x.re\_m \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m + x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\right)\right) + x.im \cdot 0\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 4.3e+86)
    (- (pow x.re_m 3.0) (* x.im (* x.re_m (* 3.0 x.im))))
    (+ (* (+ x.re_m x.im) (* x.re_m (- x.re_m x.im))) (* x.im 0.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 4.3e+86) {
		tmp = pow(x_46_re_m, 3.0) - (x_46_im * (x_46_re_m * (3.0 * x_46_im)));
	} else {
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 4.3d+86) then
        tmp = (x_46re_m ** 3.0d0) - (x_46im * (x_46re_m * (3.0d0 * x_46im)))
    else
        tmp = ((x_46re_m + x_46im) * (x_46re_m * (x_46re_m - x_46im))) + (x_46im * 0.0d0)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 4.3e+86) {
		tmp = Math.pow(x_46_re_m, 3.0) - (x_46_im * (x_46_re_m * (3.0 * x_46_im)));
	} else {
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 4.3e+86:
		tmp = math.pow(x_46_re_m, 3.0) - (x_46_im * (x_46_re_m * (3.0 * x_46_im)))
	else:
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 4.3e+86)
		tmp = Float64((x_46_re_m ^ 3.0) - Float64(x_46_im * Float64(x_46_re_m * Float64(3.0 * x_46_im))));
	else
		tmp = Float64(Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im))) + Float64(x_46_im * 0.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 4.3e+86)
		tmp = (x_46_re_m ^ 3.0) - (x_46_im * (x_46_re_m * (3.0 * x_46_im)));
	else
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 4.3e+86], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] - N[(x$46$im * N[(x$46$re$95$m * N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * 0.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.3000000000000002e86

   1. Initial program 81.5%

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

   3. Simplified 79.6%

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

      1. associate-*r* 79.6%

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

      2. associate-*r* 89.4%

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

      3. *-commutative 89.4%

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

   6. Applied egg-rr 89.4%

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

   if 4.3000000000000002e86 < x.re

   1. Initial program 58.7%

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

      1. difference-of-squares 69.6%

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

      2. associate-*l* 69.6%

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

   4. Applied egg-rr 69.6%

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

      1. flip-+ 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. div-sub 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. +-inverses 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4.3 \cdot 10^{+86}:\\ \;\;\;\;{x.re}^{3} - x.im \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + x.im \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 1.6 \cdot 10^{+27} \lor \neg \left(x.im \leq 2.8 \cdot 10^{+83}\right) \land x.im \leq 1.12 \cdot 10^{+113}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (or (<= x.im 1.6e+27) (and (not (<= x.im 2.8e+83)) (<= x.im 1.12e+113)))
    (* x.re_m (* x.re_m x.re_m))
    (* -3.0 (* x.im (* x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if ((x_46_im <= 1.6e+27) || (!(x_46_im <= 2.8e+83) && (x_46_im <= 1.12e+113))) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= 1.6d+27) .or. (.not. (x_46im <= 2.8d+83)) .and. (x_46im <= 1.12d+113)) then
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    else
        tmp = (-3.0d0) * (x_46im * (x_46re_m * x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if ((x_46_im <= 1.6e+27) || (!(x_46_im <= 2.8e+83) && (x_46_im <= 1.12e+113))) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if (x_46_im <= 1.6e+27) or (not (x_46_im <= 2.8e+83) and (x_46_im <= 1.12e+113)):
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	else:
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if ((x_46_im <= 1.6e+27) || (!(x_46_im <= 2.8e+83) && (x_46_im <= 1.12e+113)))
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= 1.6e+27) || (~((x_46_im <= 2.8e+83)) && (x_46_im <= 1.12e+113)))
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	else
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[Or[LessEqual[x$46$im, 1.6e+27], And[N[Not[LessEqual[x$46$im, 2.8e+83]], $MachinePrecision], LessEqual[x$46$im, 1.12e+113]]], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.60000000000000008e27 or 2.8e83 < x.im < 1.1200000000000001e113

   1. Initial program 82.8%

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

      1. difference-of-squares 86.3%

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

      2. associate-*l* 91.4%

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

   4. Applied egg-rr 91.4%

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

   5. Taylor expanded in x.re around inf 63.6%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   6. Step-by-step derivation

      1. cube-unmult 63.6%

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

   7. Simplified 63.6%

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

   if 1.60000000000000008e27 < x.im < 2.8e83 or 1.1200000000000001e113 < x.im

   1. Initial program 57.4%

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

   3. Simplified 48.3%

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

      1. associate-*r* 48.3%

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

      2. associate-*r* 67.1%

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

      3. *-commutative 67.1%

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

   6. Applied egg-rr 67.1%

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

      1. cube-unmult 67.1%

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

      2. cancel-sign-sub-inv 67.1%

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

      3. associate-*r* 67.1%

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

   8. Applied egg-rr 67.1%

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

      1. associate-*l* 67.1%

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

   10. Applied egg-rr 67.1%

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

   11. Taylor expanded in x.re around 0 61.1%

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

       1. *-commutative 61.1%

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

       2. unpow2 61.1%

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

       3. associate-*l* 79.8%

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

       4. *-commutative 79.8%

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

       5. *-commutative 79.8%

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

   13. Simplified 79.8%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.6 \cdot 10^{+27} \lor \neg \left(x.im \leq 2.8 \cdot 10^{+83}\right) \land x.im \leq 1.12 \cdot 10^{+113}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x.im \leq 4.4 \cdot 10^{+83}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.2 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* x.re_m (* x.re_m x.re_m))))
   (*
    x.re_s
    (if (<= x.im 1.5e+27)
      t_0
      (if (<= x.im 4.4e+83)
        (* -3.0 (* x.im (* x.re_m x.im)))
        (if (<= x.im 1.2e+113) t_0 (* (* x.re_m x.im) (* x.im -3.0))))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * (x_46_re_m * x_46_re_m);
	double tmp;
	if (x_46_im <= 1.5e+27) {
		tmp = t_0;
	} else if (x_46_im <= 4.4e+83) {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	} else if (x_46_im <= 1.2e+113) {
		tmp = t_0;
	} else {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re_m * (x_46re_m * x_46re_m)
    if (x_46im <= 1.5d+27) then
        tmp = t_0
    else if (x_46im <= 4.4d+83) then
        tmp = (-3.0d0) * (x_46im * (x_46re_m * x_46im))
    else if (x_46im <= 1.2d+113) then
        tmp = t_0
    else
        tmp = (x_46re_m * x_46im) * (x_46im * (-3.0d0))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = x_46_re_m * (x_46_re_m * x_46_re_m);
	double tmp;
	if (x_46_im <= 1.5e+27) {
		tmp = t_0;
	} else if (x_46_im <= 4.4e+83) {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	} else if (x_46_im <= 1.2e+113) {
		tmp = t_0;
	} else {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = x_46_re_m * (x_46_re_m * x_46_re_m)
	tmp = 0
	if x_46_im <= 1.5e+27:
		tmp = t_0
	elif x_46_im <= 4.4e+83:
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im))
	elif x_46_im <= 1.2e+113:
		tmp = t_0
	else:
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m))
	tmp = 0.0
	if (x_46_im <= 1.5e+27)
		tmp = t_0;
	elseif (x_46_im <= 4.4e+83)
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im)));
	elseif (x_46_im <= 1.2e+113)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = x_46_re_m * (x_46_re_m * x_46_re_m);
	tmp = 0.0;
	if (x_46_im <= 1.5e+27)
		tmp = t_0;
	elseif (x_46_im <= 4.4e+83)
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	elseif (x_46_im <= 1.2e+113)
		tmp = t_0;
	else
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[x$46$im, 1.5e+27], t$95$0, If[LessEqual[x$46$im, 4.4e+83], N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.2e+113], t$95$0, N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.im < 1.49999999999999988e27 or 4.39999999999999997e83 < x.im < 1.19999999999999992e113

   1. Initial program 82.8%

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

      1. difference-of-squares 86.3%

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

      2. associate-*l* 91.4%

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

   4. Applied egg-rr 91.4%

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

   5. Taylor expanded in x.re around inf 63.6%

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   6. Step-by-step derivation

      1. cube-unmult 63.6%

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

   7. Simplified 63.6%

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

   if 1.49999999999999988e27 < x.im < 4.39999999999999997e83

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r* 99.6%

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

      2. associate-*r* 99.4%

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

      3. *-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. cube-unmult 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. associate-*r* 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-*l* 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in x.re around 0 72.3%

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

       1. *-commutative 72.3%

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

       2. unpow2 72.3%

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

       3. associate-*l* 72.5%

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

       4. *-commutative 72.5%

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

       5. *-commutative 72.5%

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

   13. Simplified 72.5%

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

   if 1.19999999999999992e113 < x.im

   1. Initial program 46.9%

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

      1. difference-of-squares 58.2%

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

      2. associate-*l* 81.6%

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

   4. Applied egg-rr 81.6%

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

   5. Taylor expanded in x.re around 0 58.2%

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

      1. distribute-rgt-out-- 58.2%

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

      2. metadata-eval 58.2%

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

      3. *-commutative 58.2%

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

      4. *-commutative 58.2%

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

      5. associate-*r* 58.2%

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

      6. metadata-eval 58.2%

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

      7. distribute-lft-neg-in 58.2%

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

      8. associate-*r* 58.2%

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

      9. *-commutative 58.2%

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

      10. unpow2 58.2%

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

      11. associate-*r* 58.2%

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

      12. *-commutative 58.2%

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

      13. distribute-rgt-neg-in 58.2%

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

      14. distribute-rgt-neg-in 58.2%

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

      15. distribute-rgt-neg-in 58.2%

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

      16. metadata-eval 58.2%

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

   7. Simplified 58.2%

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

      1. associate-*r* 81.7%

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

   9. Applied egg-rr 81.7%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{+27}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 4.4 \cdot 10^{+83}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.2 \cdot 10^{+113}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 10^{+62}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right) - x.im \cdot \left(x.re\_m \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m + x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\right)\right) + x.im \cdot 0\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1e+62)
    (- (* x.re_m (* x.re_m x.re_m)) (* x.im (* x.re_m (* 3.0 x.im))))
    (+ (* (+ x.re_m x.im) (* x.re_m (- x.re_m x.im))) (* x.im 0.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+62) {
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) - (x_46_im * (x_46_re_m * (3.0 * x_46_im)));
	} else {
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 1d+62) then
        tmp = (x_46re_m * (x_46re_m * x_46re_m)) - (x_46im * (x_46re_m * (3.0d0 * x_46im)))
    else
        tmp = ((x_46re_m + x_46im) * (x_46re_m * (x_46re_m - x_46im))) + (x_46im * 0.0d0)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+62) {
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) - (x_46_im * (x_46_re_m * (3.0 * x_46_im)));
	} else {
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 1e+62:
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) - (x_46_im * (x_46_re_m * (3.0 * x_46_im)))
	else:
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1e+62)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)) - Float64(x_46_im * Float64(x_46_re_m * Float64(3.0 * x_46_im))));
	else
		tmp = Float64(Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im))) + Float64(x_46_im * 0.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 1e+62)
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) - (x_46_im * (x_46_re_m * (3.0 * x_46_im)));
	else
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1e+62], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re$95$m * N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * 0.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.00000000000000004e62

   1. Initial program 80.8%

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

   3. Simplified 78.9%

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

      1. associate-*r* 78.9%

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

      2. associate-*r* 89.0%

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

      3. *-commutative 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. cube-unmult 89.0%

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

      2. cancel-sign-sub-inv 89.0%

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

      3. associate-*r* 88.9%

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

   8. Applied egg-rr 88.9%

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

      1. associate-*l* 89.0%

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

   10. Applied egg-rr 89.0%

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

   if 1.00000000000000004e62 < x.re

   1. Initial program 64.1%

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

      1. difference-of-squares 73.6%

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

      2. associate-*l* 73.6%

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

   4. Applied egg-rr 73.6%

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

      1. flip-+ 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. div-sub 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. +-inverses 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 10^{+62}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) - x.im \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + x.im \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 10^{+62}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right) + -3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m + x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\right)\right) + x.im \cdot 0\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1e+62)
    (+ (* x.re_m (* x.re_m x.re_m)) (* -3.0 (* x.im (* x.re_m x.im))))
    (+ (* (+ x.re_m x.im) (* x.re_m (- x.re_m x.im))) (* x.im 0.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+62) {
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	} else {
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 1d+62) then
        tmp = (x_46re_m * (x_46re_m * x_46re_m)) + ((-3.0d0) * (x_46im * (x_46re_m * x_46im)))
    else
        tmp = ((x_46re_m + x_46im) * (x_46re_m * (x_46re_m - x_46im))) + (x_46im * 0.0d0)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+62) {
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	} else {
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 1e+62:
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)))
	else:
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1e+62)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)) + Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im))) + Float64(x_46_im * 0.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 1e+62)
		tmp = (x_46_re_m * (x_46_re_m * x_46_re_m)) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	else
		tmp = ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))) + (x_46_im * 0.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1e+62], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] + N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * 0.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.00000000000000004e62

   1. Initial program 80.8%

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

   3. Simplified 78.9%

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

      1. associate-*r* 78.9%

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

      2. associate-*r* 89.0%

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

      3. *-commutative 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. cube-unmult 89.0%

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

      2. cancel-sign-sub-inv 89.0%

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

      3. associate-*r* 88.9%

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

   8. Applied egg-rr 88.9%

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

      1. *-commutative 88.9%

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

      2. distribute-rgt-neg-in 88.9%

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

      3. metadata-eval 88.9%

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

   10. Applied egg-rr 88.9%

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

       1. associate-*r* 89.0%

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

       2. *-commutative 89.0%

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

   12. Simplified 89.0%

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

   if 1.00000000000000004e62 < x.re

   1. Initial program 64.1%

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

      1. difference-of-squares 73.6%

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

      2. associate-*l* 73.6%

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

   4. Applied egg-rr 73.6%

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

      1. flip-+ 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. div-sub 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. +-inverses 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 10^{+62}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) + x.im \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 3.7e+151)
    (* x.re_m (- (* x.re_m x.re_m) (* x.im (* 3.0 x.im))))
    (* (* x.re_m x.im) (* x.im -3.0)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.7e+151) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * (3.0 * x_46_im)));
	} else {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 3.7d+151) then
        tmp = x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * (3.0d0 * x_46im)))
    else
        tmp = (x_46re_m * x_46im) * (x_46im * (-3.0d0))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.7e+151) {
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * (3.0 * x_46_im)));
	} else {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 3.7e+151:
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * (3.0 * x_46_im)))
	else:
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 3.7e+151)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * Float64(3.0 * x_46_im))));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 3.7e+151)
		tmp = x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * (3.0 * x_46_im)));
	else
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 3.7e+151], N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.6999999999999997e151

   1. Initial program 83.7%

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

   3. Simplified 81.4%

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

      1. cube-mult 81.4%

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

      2. distribute-lft-out-- 88.7%

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

      3. *-commutative 88.7%

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

      4. associate-*l* 88.7%

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

   6. Applied egg-rr 88.7%

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

   if 3.6999999999999997e151 < x.im

   1. Initial program 41.1%

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

      1. difference-of-squares 54.3%

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

      2. associate-*l* 81.4%

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

   4. Applied egg-rr 81.4%

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

   5. Taylor expanded in x.re around 0 54.3%

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

      1. distribute-rgt-out-- 54.3%

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

      2. metadata-eval 54.3%

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

      3. *-commutative 54.3%

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

      4. *-commutative 54.3%

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

      5. associate-*r* 54.3%

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

      6. metadata-eval 54.3%

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

      7. distribute-lft-neg-in 54.3%

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

      8. associate-*r* 54.3%

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

      9. *-commutative 54.3%

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

      10. unpow2 54.3%

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

      11. associate-*r* 54.3%

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

      12. *-commutative 54.3%

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

      13. distribute-rgt-neg-in 54.3%

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

      14. distribute-rgt-neg-in 54.3%

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

      15. distribute-rgt-neg-in 54.3%

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

      16. metadata-eval 54.3%

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

   7. Simplified 54.3%

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

      1. associate-*r* 81.5%

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

   9. Applied egg-rr 81.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.7 \cdot 10^{+151}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. difference-of-squares 82.1%

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

   2. associate-*l* 90.1%

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

4. Applied egg-rr 90.1%

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

5. Taylor expanded in x.re around inf 54.5%

   \[\leadsto \color{blue}{{x.re}^{3}} \]
6. Step-by-step derivation

   1. cube-unmult 54.5%

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

7. Simplified 54.5%

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

Developer target: 87.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024107 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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