Asymptote C

Average Error: 29.4 → 0.0

Time: 14.2s

Precision: binary64

Cost: 1736

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -100000000000:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 200000000:\\ \;\;\;\;\frac{-1}{1 - x \cdot x} \cdot \left(-1 - x\right) + \frac{x \cdot -2}{\left(x + -1\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -100000000000.0)
   (+ (/ -3.0 x) (/ (/ -1.0 x) x))
   (if (<= x 200000000.0)
     (+
      (* (/ -1.0 (- 1.0 (* x x))) (- -1.0 x))
      (/ (* x -2.0) (* (+ x -1.0) (+ x 1.0))))
     (/ (+ -3.0 (/ 2.0 x)) (+ x -1.0)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -100000000000.0) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else if (x <= 200000000.0) {
		tmp = ((-1.0 / (1.0 - (x * x))) * (-1.0 - x)) + ((x * -2.0) / ((x + -1.0) * (x + 1.0)));
	} else {
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0);
	}
	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 <= (-100000000000.0d0)) then
        tmp = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    else if (x <= 200000000.0d0) then
        tmp = (((-1.0d0) / (1.0d0 - (x * x))) * ((-1.0d0) - x)) + ((x * (-2.0d0)) / ((x + (-1.0d0)) * (x + 1.0d0)))
    else
        tmp = ((-3.0d0) + (2.0d0 / x)) / (x + (-1.0d0))
    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 <= -100000000000.0) {
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	} else if (x <= 200000000.0) {
		tmp = ((-1.0 / (1.0 - (x * x))) * (-1.0 - x)) + ((x * -2.0) / ((x + -1.0) * (x + 1.0)));
	} else {
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -100000000000.0:
		tmp = (-3.0 / x) + ((-1.0 / x) / x)
	elif x <= 200000000.0:
		tmp = ((-1.0 / (1.0 - (x * x))) * (-1.0 - x)) + ((x * -2.0) / ((x + -1.0) * (x + 1.0)))
	else:
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0)
	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 <= -100000000000.0)
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x));
	elseif (x <= 200000000.0)
		tmp = Float64(Float64(Float64(-1.0 / Float64(1.0 - Float64(x * x))) * Float64(-1.0 - x)) + Float64(Float64(x * -2.0) / Float64(Float64(x + -1.0) * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(-3.0 + Float64(2.0 / x)) / Float64(x + -1.0));
	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 <= -100000000000.0)
		tmp = (-3.0 / x) + ((-1.0 / x) / x);
	elseif (x <= 200000000.0)
		tmp = ((-1.0 / (1.0 - (x * x))) * (-1.0 - x)) + ((x * -2.0) / ((x + -1.0) * (x + 1.0)));
	else
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0);
	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, -100000000000.0], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 200000000.0], N[(N[(N[(-1.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(N[(x * -2.0), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e11

   1. Initial program 60.1

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

   2. Simplified 60.1

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

      [Start] 60.1	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 60.1	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 60.1	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 60.1	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 60.1	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 60.1	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 60.1	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 60.1	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 60.1	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 60.1	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 60.1	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 60.1	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 60.1	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 60.1	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 60.1	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 60.1	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 60.1	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 60.1	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 60.1	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 60.1	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 60.1	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 60.1	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 60.1	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 0.3

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

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{\frac{1}{x}}{x}} \]
      Proof

      [Start] 0.3	\[ -\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right) \]
      neg-sub0 [=>] 0.3	\[ \color{blue}{0 - \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]
      +-commutative [=>] 0.3	\[ 0 - \color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]
      associate--r+ [=>] 0.3	\[ \color{blue}{\left(0 - 3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]
      neg-sub0 [<=] 0.3	\[ \color{blue}{\left(-3 \cdot \frac{1}{x}\right)} - \frac{1}{{x}^{2}} \]
      associate-*r/ [=>] 0.0	\[ \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]
      metadata-eval [=>] 0.0	\[ \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]
      distribute-neg-frac [=>] 0.0	\[ \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]
      metadata-eval [=>] 0.0	\[ \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]
      unpow2 [=>] 0.0	\[ \frac{-3}{x} - \frac{1}{\color{blue}{x \cdot x}} \]
      associate-/r* [=>] 0.0	\[ \frac{-3}{x} - \color{blue}{\frac{\frac{1}{x}}{x}} \]

   if -1e11 < x < 2e8

   1. Initial program 0.5

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

   2. Simplified 0.5

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

      [Start] 0.5	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 0.5	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 0.5	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 0.5	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 0.5	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 0.5	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 0.5	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 0.5	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 0.5	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 0.5	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 0.5	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 0.5	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 0.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 0.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 0.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 0.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 0.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 0.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 0.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 0.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 0.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 0.5	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 0.5	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 0.5

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

   4. Applied egg-rr 0.5

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

   5. Simplified 0.0

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

      [Start] 0.5	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      *-commutative [=>] 0.5	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \color{blue}{\left(-x\right) \cdot \left(x + -1\right)}}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      distribute-lft-out-- [=>] 0.0	\[ \frac{-1}{-1 + x} + \frac{\color{blue}{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}}{\left(x + -1\right) \cdot \left(x + 1\right)} \]

   6. Taylor expanded in x around 0 0.0

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

   7. Applied egg-rr 0.0

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

   if 2e8 < x

   1. Initial program 59.8

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

   2. Simplified 59.8

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

      [Start] 59.8	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 59.8	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 59.8	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 59.8	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 59.8	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 59.8	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 59.8	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 59.8	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 59.8	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 59.8	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 59.8	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 59.8	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 59.8	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 59.8	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 59.8	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 59.8	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 59.8	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 59.8	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 59.8	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 59.8	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 59.8	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 59.8	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 59.8	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 51.4

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

   4. Simplified 61.4

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

      [Start] 51.4	\[ \frac{-1 - \left(x + \frac{-1 + x}{-1 - x} \cdot x\right)}{\frac{-1 + x}{-1 - x} \cdot \left(-1 - x\right)} \]
      associate-/r* [=>] 51.4	\[ \color{blue}{\frac{\frac{-1 - \left(x + \frac{-1 + x}{-1 - x} \cdot x\right)}{\frac{-1 + x}{-1 - x}}}{-1 - x}} \]
      associate--r+ [=>] 58.9	\[ \frac{\frac{\color{blue}{\left(-1 - x\right) - \frac{-1 + x}{-1 - x} \cdot x}}{\frac{-1 + x}{-1 - x}}}{-1 - x} \]
      *-commutative [=>] 58.9	\[ \frac{\frac{\left(-1 - x\right) - \color{blue}{x \cdot \frac{-1 + x}{-1 - x}}}{\frac{-1 + x}{-1 - x}}}{-1 - x} \]
      associate-*r/ [=>] 61.4	\[ \frac{\frac{\left(-1 - x\right) - \color{blue}{\frac{x \cdot \left(-1 + x\right)}{-1 - x}}}{\frac{-1 + x}{-1 - x}}}{-1 - x} \]
      +-commutative [=>] 61.4	\[ \frac{\frac{\left(-1 - x\right) - \frac{x \cdot \color{blue}{\left(x + -1\right)}}{-1 - x}}{\frac{-1 + x}{-1 - x}}}{-1 - x} \]
      +-commutative [=>] 61.4	\[ \frac{\frac{\left(-1 - x\right) - \frac{x \cdot \left(x + -1\right)}{-1 - x}}{\frac{\color{blue}{x + -1}}{-1 - x}}}{-1 - x} \]

   5. Applied egg-rr 61.4

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

   6. Simplified 61.4

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

      [Start] 61.4	\[ \frac{\left(-1 - x\right) + \frac{x \cdot \left(1 - x\right)}{-1 - x}}{1} \cdot \frac{\frac{-1 - x}{x + -1}}{-1 - x} \]
      *-commutative [=>] 61.4	\[ \color{blue}{\frac{\frac{-1 - x}{x + -1}}{-1 - x} \cdot \frac{\left(-1 - x\right) + \frac{x \cdot \left(1 - x\right)}{-1 - x}}{1}} \]
      associate-/l/ [=>] 61.4	\[ \color{blue}{\frac{-1 - x}{\left(-1 - x\right) \cdot \left(x + -1\right)}} \cdot \frac{\left(-1 - x\right) + \frac{x \cdot \left(1 - x\right)}{-1 - x}}{1} \]
      associate-/r* [=>] 61.4	\[ \color{blue}{\frac{\frac{-1 - x}{-1 - x}}{x + -1}} \cdot \frac{\left(-1 - x\right) + \frac{x \cdot \left(1 - x\right)}{-1 - x}}{1} \]
      *-inverses [=>] 61.4	\[ \frac{\color{blue}{1}}{x + -1} \cdot \frac{\left(-1 - x\right) + \frac{x \cdot \left(1 - x\right)}{-1 - x}}{1} \]
      /-rgt-identity [=>] 61.4	\[ \frac{1}{x + -1} \cdot \color{blue}{\left(\left(-1 - x\right) + \frac{x \cdot \left(1 - x\right)}{-1 - x}\right)} \]
      associate-*l/ [=>] 61.4	\[ \color{blue}{\frac{1 \cdot \left(\left(-1 - x\right) + \frac{x \cdot \left(1 - x\right)}{-1 - x}\right)}{x + -1}} \]
      *-lft-identity [=>] 61.4	\[ \frac{\color{blue}{\left(-1 - x\right) + \frac{x \cdot \left(1 - x\right)}{-1 - x}}}{x + -1} \]

   7. Taylor expanded in x around inf 0.0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} - 3}}{x + -1} \]

   8. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\frac{2}{x} + -3}}{x + -1} \]
      Proof

      [Start] 0.0	\[ \frac{2 \cdot \frac{1}{x} - 3}{x + -1} \]
      sub-neg [=>] 0.0	\[ \frac{\color{blue}{2 \cdot \frac{1}{x} + \left(-3\right)}}{x + -1} \]
      associate-*r/ [=>] 0.0	\[ \frac{\color{blue}{\frac{2 \cdot 1}{x}} + \left(-3\right)}{x + -1} \]
      metadata-eval [=>] 0.0	\[ \frac{\frac{\color{blue}{2}}{x} + \left(-3\right)}{x + -1} \]
      metadata-eval [=>] 0.0	\[ \frac{\frac{2}{x} + \color{blue}{-3}}{x + -1} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -100000000000:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 200000000:\\ \;\;\;\;\frac{-1}{1 - x \cdot x} \cdot \left(-1 - x\right) + \frac{x \cdot -2}{\left(x + -1\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	1225

\[\begin{array}{l} \mathbf{if}\;x \leq -18500 \lor \neg \left(x \leq 21000\right):\\ \;\;\;\;\frac{\frac{4}{x} + \left(3 + \frac{4}{x \cdot x}\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -18500:\\ \;\;\;\;\frac{\frac{4}{x} + \left(3 + \frac{4}{x \cdot x}\right)}{-1 - x}\\ \mathbf{elif}\;x \leq 21000:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} + \left(-3 - \frac{2}{x \cdot x}\right)}{x + -1}\\ \end{array} \]

Alternative 3
Error	0.0
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -100000000000:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;\frac{-1}{x + -1} - \frac{x \cdot 2}{-1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -280000 \lor \neg \left(x \leq 290000\right):\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 5
Error	0.7
Cost	841

\[\begin{array}{l} t_0 := \frac{-1}{x + -1}\\ \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;t_0 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 + x \cdot 2\\ \end{array} \]

Alternative 6
Error	0.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} + x \cdot 2\\ \end{array} \]

Alternative 7
Error	0.5
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 8
Error	0.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1.22:\\ \;\;\;\;\frac{-1}{x + -1} + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 9
Error	1.0
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 10
Error	1.4
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 11
Error	31.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))