Asymptote C

Average Accuracy: 55.3% → 100.0%

Time: 13.5s

Precision: binary64

Cost: 8328

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+15}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\frac{-1 + {x}^{4}}{1 + x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{3 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}{-1 - x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -6e+15)
   (/ -3.0 x)
   (if (<= x 200000.0)
     (+
      (/ -1.0 (+ x -1.0))
      (/
       (* x (+ (+ x -1.0) (- -1.0 x)))
       (/ (+ -1.0 (pow x 4.0)) (+ 1.0 (* x x)))))
     (/ (+ 3.0 (+ (/ 4.0 x) (/ 4.0 (* x x)))) (- -1.0 x)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -6e+15) {
		tmp = -3.0 / x;
	} else if (x <= 200000.0) {
		tmp = (-1.0 / (x + -1.0)) + ((x * ((x + -1.0) + (-1.0 - x))) / ((-1.0 + pow(x, 4.0)) / (1.0 + (x * x))));
	} else {
		tmp = (3.0 + ((4.0 / x) + (4.0 / (x * x)))) / (-1.0 - 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) :: tmp
    if (x <= (-6d+15)) then
        tmp = (-3.0d0) / x
    else if (x <= 200000.0d0) then
        tmp = ((-1.0d0) / (x + (-1.0d0))) + ((x * ((x + (-1.0d0)) + ((-1.0d0) - x))) / (((-1.0d0) + (x ** 4.0d0)) / (1.0d0 + (x * x))))
    else
        tmp = (3.0d0 + ((4.0d0 / x) + (4.0d0 / (x * x)))) / ((-1.0d0) - 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 tmp;
	if (x <= -6e+15) {
		tmp = -3.0 / x;
	} else if (x <= 200000.0) {
		tmp = (-1.0 / (x + -1.0)) + ((x * ((x + -1.0) + (-1.0 - x))) / ((-1.0 + Math.pow(x, 4.0)) / (1.0 + (x * x))));
	} else {
		tmp = (3.0 + ((4.0 / x) + (4.0 / (x * x)))) / (-1.0 - x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -6e+15:
		tmp = -3.0 / x
	elif x <= 200000.0:
		tmp = (-1.0 / (x + -1.0)) + ((x * ((x + -1.0) + (-1.0 - x))) / ((-1.0 + math.pow(x, 4.0)) / (1.0 + (x * x))))
	else:
		tmp = (3.0 + ((4.0 / x) + (4.0 / (x * x)))) / (-1.0 - 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)
	tmp = 0.0
	if (x <= -6e+15)
		tmp = Float64(-3.0 / x);
	elseif (x <= 200000.0)
		tmp = Float64(Float64(-1.0 / Float64(x + -1.0)) + Float64(Float64(x * Float64(Float64(x + -1.0) + Float64(-1.0 - x))) / Float64(Float64(-1.0 + (x ^ 4.0)) / Float64(1.0 + Float64(x * x)))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(4.0 / x) + Float64(4.0 / Float64(x * x)))) / Float64(-1.0 - 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)
	tmp = 0.0;
	if (x <= -6e+15)
		tmp = -3.0 / x;
	elseif (x <= 200000.0)
		tmp = (-1.0 / (x + -1.0)) + ((x * ((x + -1.0) + (-1.0 - x))) / ((-1.0 + (x ^ 4.0)) / (1.0 + (x * x))));
	else
		tmp = (3.0 + ((4.0 / x) + (4.0 / (x * x)))) / (-1.0 - 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_] := If[LessEqual[x, -6e+15], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 200000.0], N[(N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(N[(x + -1.0), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 + N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(4.0 / x), $MachinePrecision] + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6e15

   1. Initial program 5.4%

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

   2. Simplified 5.4%

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

      [Start] 5.4	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 5.4	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 5.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 5.4	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 5.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 5.4	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 5.4	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 5.4	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 5.4	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 5.4	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 5.4	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 5.4	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 5.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 5.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 5.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 5.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 5.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 5.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 5.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 5.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 5.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 5.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 5.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{-3}{x}} \]

   if -6e15 < x < 2e5

   1. Initial program 99.0%

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

   2. Simplified 99.0%

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

      [Start] 99.0	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 99.0	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 99.0	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 99.0	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 99.0	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 99.0	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 99.0	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 99.0	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 99.0	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 99.0	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 99.0	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 99.0	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 99.0	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 99.0	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 99.0	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 99.0	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 99.0	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 99.0	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 99.0	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 99.0	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 99.0	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 99.0	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 99.0	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 99.0%

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

   4. Applied egg-rr 99.0%

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

   5. Simplified 99.9%

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

      [Start] 99.0	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      *-commutative [=>] 99.0	\[ \frac{-1}{-1 + x} + \frac{\color{blue}{\left(x + 1\right) \cdot \left(-x\right)} - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      distribute-rgt-out-- [=>] 99.9	\[ \frac{-1}{-1 + x} + \frac{\color{blue}{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}}{\left(x + -1\right) \cdot \left(x + 1\right)} \]

   6. Applied egg-rr 99.9%

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

   7. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1}{1 + x \cdot x}} \]
      sub-neg [=>] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}{\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(-1\right)}}{1 + x \cdot x}} \]
      associate-*r* [=>] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}{\frac{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} + \left(-1\right)}{1 + x \cdot x}} \]
      unpow3 [<=] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}{\frac{\color{blue}{{x}^{3}} \cdot x + \left(-1\right)}{1 + x \cdot x}} \]
      pow-plus [=>] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}{\frac{\color{blue}{{x}^{\left(3 + 1\right)}} + \left(-1\right)}{1 + x \cdot x}} \]
      metadata-eval [=>] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}{\frac{{x}^{\color{blue}{4}} + \left(-1\right)}{1 + x \cdot x}} \]
      metadata-eval [=>] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}{\frac{{x}^{4} + \color{blue}{-1}}{1 + x \cdot x}} \]

   if 2e5 < x

   1. Initial program 7.4%

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

   2. Simplified 7.4%

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

      [Start] 7.4	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 7.4	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 7.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 7.4	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 7.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 7.4	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 7.4	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 7.4	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 7.4	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 7.4	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 7.4	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 7.4	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 7.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 7.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 7.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 7.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 7.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 7.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 7.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 7.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 7.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 7.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 7.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 20.1%

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

   4. Simplified 4.8%

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

      [Start] 20.1	\[ \frac{-1 - \left(x + \frac{-1 + x}{-1 - x} \cdot x\right)}{\frac{-1 + x}{-1 - x} \cdot \left(-1 - x\right)} \]
      associate-/r* [=>] 20.1	\[ \color{blue}{\frac{\frac{-1 - \left(x + \frac{-1 + x}{-1 - x} \cdot x\right)}{\frac{-1 + x}{-1 - x}}}{-1 - x}} \]
      associate--r+ [=>] 8.4	\[ \frac{\frac{\color{blue}{\left(-1 - x\right) - \frac{-1 + x}{-1 - x} \cdot x}}{\frac{-1 + x}{-1 - x}}}{-1 - x} \]
      associate-*l/ [=>] 4.8	\[ \frac{\frac{\left(-1 - x\right) - \color{blue}{\frac{\left(-1 + x\right) \cdot x}{-1 - x}}}{\frac{-1 + x}{-1 - x}}}{-1 - x} \]
      *-commutative [<=] 4.8	\[ \frac{\frac{\left(-1 - x\right) - \frac{\color{blue}{x \cdot \left(-1 + x\right)}}{-1 - x}}{\frac{-1 + x}{-1 - x}}}{-1 - x} \]
      +-commutative [=>] 4.8	\[ \frac{\frac{\left(-1 - x\right) - \frac{x \cdot \color{blue}{\left(x + -1\right)}}{-1 - x}}{\frac{-1 + x}{-1 - x}}}{-1 - x} \]
      +-commutative [=>] 4.8	\[ \frac{\frac{\left(-1 - x\right) - \frac{x \cdot \left(x + -1\right)}{-1 - x}}{\frac{\color{blue}{x + -1}}{-1 - x}}}{-1 - x} \]

   5. Taylor expanded in x around inf 100.0%

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

   6. Simplified 100.0%

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

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

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+15}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot \left(\left(x + -1\right) + \left(-1 - x\right)\right)}{\frac{-1 + {x}^{4}}{1 + x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{3 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}{-1 - x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	1736

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+15}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x}{x + -1} \cdot \frac{\left(1 + \left(x + 1\right)\right) - x}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{3 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}{-1 - x}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;x \leq -15000 \lor \neg \left(x \leq 18000\right):\\ \;\;\;\;\frac{3 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -440000:\\ \;\;\;\;\frac{-3}{x} + \frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 340000:\\ \;\;\;\;\left(x + 1\right) \cdot \frac{1}{1 - x} - \frac{x}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

Alternative 4
Accuracy	99.9%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -11400:\\ \;\;\;\;\frac{\frac{2}{x} + \left(-3 + \frac{-2}{x \cdot x}\right)}{x + -1}\\ \mathbf{elif}\;x \leq 18000:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{3 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}{-1 - x}\\ \end{array} \]

Alternative 5
Accuracy	100.0%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 230000:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot -2}{-1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{3 + \left(\frac{4}{x} + \frac{4}{x \cdot x}\right)}{-1 - x}\\ \end{array} \]

Alternative 6
Accuracy	99.8%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -500000:\\ \;\;\;\;\frac{-3}{x} + \frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 340000:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

Alternative 7
Accuracy	98.9%
Cost	841

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

Alternative 8
Accuracy	99.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{-3}{x} + \frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} + x \cdot 2\\ \end{array} \]

Alternative 9
Accuracy	99.0%
Cost	841

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

Alternative 10
Accuracy	98.3%
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 11
Accuracy	97.7%
Cost	456

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

Alternative 12
Accuracy	51.6%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023125 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))