Asymptote C

Average Accuracy: 53.2% → 100.0%

Time: 12.3s

Precision: binary64

Cost: 14148

Math

\[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

↓

\[\begin{array}{l} t_0 := \frac{1}{x \cdot x}\\ \mathbf{if}\;x \leq -100000:\\ \;\;\;\;\frac{-1}{{x}^{4}} - \left(\left(t_0 + \frac{3}{{x}^{3}}\right) + \frac{3}{x}\right)\\ \mathbf{elif}\;x \leq 150000:\\ \;\;\;\;\frac{x \cdot -3 + -1}{\left(1 - x\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - t_0\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x x))))
   (if (<= x -100000.0)
     (- (/ -1.0 (pow x 4.0)) (+ (+ t_0 (/ 3.0 (pow x 3.0))) (/ 3.0 x)))
     (if (<= x 150000.0)
       (/ (+ (* x -3.0) -1.0) (* (- 1.0 x) (- -1.0 x)))
       (+ (- (/ -3.0 x) t_0) (/ (/ -3.0 x) (* x x)))))))

C

double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}

↓

double code(double x) {
	double t_0 = 1.0 / (x * x);
	double tmp;
	if (x <= -100000.0) {
		tmp = (-1.0 / pow(x, 4.0)) - ((t_0 + (3.0 / pow(x, 3.0))) + (3.0 / x));
	} else if (x <= 150000.0) {
		tmp = ((x * -3.0) + -1.0) / ((1.0 - x) * (-1.0 - x));
	} else {
		tmp = ((-3.0 / x) - t_0) + ((-3.0 / x) / (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / (x + 1.0d0)) - ((x + 1.0d0) / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (x * x)
    if (x <= (-100000.0d0)) then
        tmp = ((-1.0d0) / (x ** 4.0d0)) - ((t_0 + (3.0d0 / (x ** 3.0d0))) + (3.0d0 / x))
    else if (x <= 150000.0d0) then
        tmp = ((x * (-3.0d0)) + (-1.0d0)) / ((1.0d0 - x) * ((-1.0d0) - x))
    else
        tmp = (((-3.0d0) / x) - t_0) + (((-3.0d0) / x) / (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}

↓

public static double code(double x) {
	double t_0 = 1.0 / (x * x);
	double tmp;
	if (x <= -100000.0) {
		tmp = (-1.0 / Math.pow(x, 4.0)) - ((t_0 + (3.0 / Math.pow(x, 3.0))) + (3.0 / x));
	} else if (x <= 150000.0) {
		tmp = ((x * -3.0) + -1.0) / ((1.0 - x) * (-1.0 - x));
	} else {
		tmp = ((-3.0 / x) - t_0) + ((-3.0 / x) / (x * x));
	}
	return tmp;
}

Python

def code(x):
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))

↓

def code(x):
	t_0 = 1.0 / (x * x)
	tmp = 0
	if x <= -100000.0:
		tmp = (-1.0 / math.pow(x, 4.0)) - ((t_0 + (3.0 / math.pow(x, 3.0))) + (3.0 / x))
	elif x <= 150000.0:
		tmp = ((x * -3.0) + -1.0) / ((1.0 - x) * (-1.0 - x))
	else:
		tmp = ((-3.0 / x) - t_0) + ((-3.0 / x) / (x * x))
	return tmp

Julia

function code(x)
	return Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0)))
end

↓

function code(x)
	t_0 = Float64(1.0 / Float64(x * x))
	tmp = 0.0
	if (x <= -100000.0)
		tmp = Float64(Float64(-1.0 / (x ^ 4.0)) - Float64(Float64(t_0 + Float64(3.0 / (x ^ 3.0))) + Float64(3.0 / x)));
	elseif (x <= 150000.0)
		tmp = Float64(Float64(Float64(x * -3.0) + -1.0) / Float64(Float64(1.0 - x) * Float64(-1.0 - x)));
	else
		tmp = Float64(Float64(Float64(-3.0 / x) - t_0) + Float64(Float64(-3.0 / x) / Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
end

↓

function tmp_2 = code(x)
	t_0 = 1.0 / (x * x);
	tmp = 0.0;
	if (x <= -100000.0)
		tmp = (-1.0 / (x ^ 4.0)) - ((t_0 + (3.0 / (x ^ 3.0))) + (3.0 / x));
	elseif (x <= 150000.0)
		tmp = ((x * -3.0) + -1.0) / ((1.0 - x) * (-1.0 - x));
	else
		tmp = ((-3.0 / x) - t_0) + ((-3.0 / x) / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -100000.0], N[(N[(-1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 + N[(3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 150000.0], N[(N[(N[(x * -3.0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(1.0 - x), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-3.0 / x), $MachinePrecision] - t$95$0), $MachinePrecision] + N[(N[(-3.0 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e5

   1. Initial program 7.2%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Simplified 7.2%

      \[\leadsto \color{blue}{\frac{-1 - x}{x + -1} - \frac{x}{-1 - x}} \]
      Proof

      [Start] 7.2	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 7.2	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 7.2	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 7.2	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 7.2	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 7.2	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 7.2	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 7.2	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 7.2	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 7.2	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 7.2	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 7.2	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 7.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 7.2	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 99.5%

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{4}} + \left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)\right)\right)} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{4}} + \left(\left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right) + \frac{3}{x}\right)\right)} \]
      Proof

      [Start] 99.5	\[ -\left(\frac{1}{{x}^{4}} + \left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)\right)\right) \]
      associate-+r+ [=>] 99.5	\[ -\left(\frac{1}{{x}^{4}} + \color{blue}{\left(\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{{x}^{3}}\right) + 3 \cdot \frac{1}{x}\right)}\right) \]
      unpow2 [=>] 99.5	\[ -\left(\frac{1}{{x}^{4}} + \left(\left(\frac{1}{\color{blue}{x \cdot x}} + 3 \cdot \frac{1}{{x}^{3}}\right) + 3 \cdot \frac{1}{x}\right)\right) \]
      associate-*r/ [=>] 99.5	\[ -\left(\frac{1}{{x}^{4}} + \left(\left(\frac{1}{x \cdot x} + \color{blue}{\frac{3 \cdot 1}{{x}^{3}}}\right) + 3 \cdot \frac{1}{x}\right)\right) \]
      metadata-eval [=>] 99.5	\[ -\left(\frac{1}{{x}^{4}} + \left(\left(\frac{1}{x \cdot x} + \frac{\color{blue}{3}}{{x}^{3}}\right) + 3 \cdot \frac{1}{x}\right)\right) \]
      associate-*r/ [=>] 100.0	\[ -\left(\frac{1}{{x}^{4}} + \left(\left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right) + \color{blue}{\frac{3 \cdot 1}{x}}\right)\right) \]
      metadata-eval [=>] 100.0	\[ -\left(\frac{1}{{x}^{4}} + \left(\left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right) + \frac{\color{blue}{3}}{x}\right)\right) \]

   if -1e5 < x < 1.5e5

   1. Initial program 99.8%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-1 - x}{x + -1} - \frac{x}{-1 - x}} \]
      Proof

      [Start] 99.8	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 99.8	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 99.8	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 99.8	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 99.8	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 99.8	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 99.8	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 99.8	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 99.8	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 99.8	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 99.8	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 99.8	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 99.8	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 99.8	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 99.8	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 99.8	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 99.8	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 99.8	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 99.8	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 99.8	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 99.8	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 99.8	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 99.8	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{-3 \cdot x - 1}}{\left(1 - x\right) \cdot \left(-1 - x\right)} \]

   if 1.5e5 < x

   1. Initial program 6.9%

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

   2. Simplified 6.9%

      \[\leadsto \color{blue}{\frac{-1 - x}{x + -1} - \frac{x}{-1 - x}} \]
      Proof

      [Start] 6.9	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 6.9	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 6.9	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 6.9	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 6.9	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 6.9	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 6.9	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 6.9	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 6.9	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 6.9	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 6.9	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 6.9	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 6.9	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 6.9	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 6.9	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 6.9	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 6.9	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 6.9	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 6.9	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 6.9	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 6.9	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 6.9	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 6.9	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 99.5%

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

   4. Simplified 100.0%

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

      [Start] 99.5	\[ -\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right) \]
      distribute-neg-in [=>] 99.5	\[ \color{blue}{\left(-3 \cdot \frac{1}{{x}^{3}}\right) + \left(-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)} \]
      +-commutative [=>] 99.5	\[ \color{blue}{\left(-\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right)} \]
      distribute-neg-in [=>] 99.5	\[ \color{blue}{\left(\left(-\frac{1}{{x}^{2}}\right) + \left(-3 \cdot \frac{1}{x}\right)\right)} + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      unpow2 [=>] 99.5	\[ \left(\left(-\frac{1}{\color{blue}{x \cdot x}}\right) + \left(-3 \cdot \frac{1}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      distribute-neg-frac [=>] 99.5	\[ \left(\color{blue}{\frac{-1}{x \cdot x}} + \left(-3 \cdot \frac{1}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      metadata-eval [=>] 99.5	\[ \left(\frac{\color{blue}{-1}}{x \cdot x} + \left(-3 \cdot \frac{1}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      associate-*r/ [=>] 100.0	\[ \left(\frac{-1}{x \cdot x} + \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      metadata-eval [=>] 100.0	\[ \left(\frac{-1}{x \cdot x} + \left(-\frac{\color{blue}{3}}{x}\right)\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      distribute-neg-frac [=>] 100.0	\[ \left(\frac{-1}{x \cdot x} + \color{blue}{\frac{-3}{x}}\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      metadata-eval [=>] 100.0	\[ \left(\frac{-1}{x \cdot x} + \frac{\color{blue}{-3}}{x}\right) + \left(-3 \cdot \frac{1}{{x}^{3}}\right) \]
      associate-*r/ [=>] 100.0	\[ \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \left(-\color{blue}{\frac{3 \cdot 1}{{x}^{3}}}\right) \]
      metadata-eval [=>] 100.0	\[ \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \left(-\frac{\color{blue}{3}}{{x}^{3}}\right) \]
      distribute-neg-frac [=>] 100.0	\[ \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \color{blue}{\frac{-3}{{x}^{3}}} \]
      metadata-eval [=>] 100.0	\[ \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{\color{blue}{-3}}{{x}^{3}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \color{blue}{\frac{1}{x} \cdot \frac{-3}{x \cdot x}} \]

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -100000:\\ \;\;\;\;\frac{-1}{{x}^{4}} - \left(\left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right) + \frac{3}{x}\right)\\ \mathbf{elif}\;x \leq 150000:\\ \;\;\;\;\frac{x \cdot -3 + -1}{\left(1 - x\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 150000:\\ \;\;\;\;\frac{x \cdot -3 + -1}{\left(1 - x\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -400000 \lor \neg \left(x \leq 450000\right):\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\frac{x \cdot -3 + -1}{\left(1 - x\right) \cdot \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.16\right):\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} + x \cdot 2\\ \end{array} \]

Alternative 5
Accuracy	99.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array} \]

Alternative 6
Accuracy	98.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 7
Accuracy	97.9%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 8
Accuracy	49.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023141 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))