Asymptote C

Average Error: 45.55% → 0.04%

Time: 13.5s

Precision: binary64

Cost: 14280

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -50000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot -2}{\left(x + -1\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(\frac{-3}{{x}^{3}} + \frac{-1}{{x}^{4}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ -3.0 x) (/ (/ -1.0 x) x))))
   (if (<= x -50000000000000.0)
     t_0
     (if (<= x 10000.0)
       (+ (/ -1.0 (+ x -1.0)) (/ (* x -2.0) (* (+ x -1.0) (+ x 1.0))))
       (+ t_0 (+ (/ -3.0 (pow x 3.0)) (/ -1.0 (pow x 4.0))))))))

C

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

↓

double code(double x) {
	double t_0 = (-3.0 / x) + ((-1.0 / x) / x);
	double tmp;
	if (x <= -50000000000000.0) {
		tmp = t_0;
	} else if (x <= 10000.0) {
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / ((x + -1.0) * (x + 1.0)));
	} else {
		tmp = t_0 + ((-3.0 / pow(x, 3.0)) + (-1.0 / pow(x, 4.0)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = ((-3.0d0) / x) + (((-1.0d0) / x) / x)
    if (x <= (-50000000000000.0d0)) then
        tmp = t_0
    else if (x <= 10000.0d0) then
        tmp = ((-1.0d0) / (x + (-1.0d0))) + ((x * (-2.0d0)) / ((x + (-1.0d0)) * (x + 1.0d0)))
    else
        tmp = t_0 + (((-3.0d0) / (x ** 3.0d0)) + ((-1.0d0) / (x ** 4.0d0)))
    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 t_0 = (-3.0 / x) + ((-1.0 / x) / x);
	double tmp;
	if (x <= -50000000000000.0) {
		tmp = t_0;
	} else if (x <= 10000.0) {
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / ((x + -1.0) * (x + 1.0)));
	} else {
		tmp = t_0 + ((-3.0 / Math.pow(x, 3.0)) + (-1.0 / Math.pow(x, 4.0)));
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = (-3.0 / x) + ((-1.0 / x) / x)
	tmp = 0
	if x <= -50000000000000.0:
		tmp = t_0
	elif x <= 10000.0:
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / ((x + -1.0) * (x + 1.0)))
	else:
		tmp = t_0 + ((-3.0 / math.pow(x, 3.0)) + (-1.0 / math.pow(x, 4.0)))
	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)
	t_0 = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x))
	tmp = 0.0
	if (x <= -50000000000000.0)
		tmp = t_0;
	elseif (x <= 10000.0)
		tmp = Float64(Float64(-1.0 / Float64(x + -1.0)) + Float64(Float64(x * -2.0) / Float64(Float64(x + -1.0) * Float64(x + 1.0))));
	else
		tmp = Float64(t_0 + Float64(Float64(-3.0 / (x ^ 3.0)) + Float64(-1.0 / (x ^ 4.0))));
	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)
	t_0 = (-3.0 / x) + ((-1.0 / x) / x);
	tmp = 0.0;
	if (x <= -50000000000000.0)
		tmp = t_0;
	elseif (x <= 10000.0)
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / ((x + -1.0) * (x + 1.0)));
	else
		tmp = t_0 + ((-3.0 / (x ^ 3.0)) + (-1.0 / (x ^ 4.0)));
	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_] := Block[{t$95$0 = N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -50000000000000.0], t$95$0, If[LessEqual[x, 10000.0], N[(N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x * -2.0), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(-3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e13

   1. Initial program 94.28

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

   2. Simplified 94.28

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

      [Start] 94.28	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 94.28	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 94.28	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 94.28	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 94.28	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 94.28	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 94.28	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 94.28	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 94.28	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 94.28	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 94.28	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 94.28	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 94.28	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 94.28	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 94.28	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 94.28	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 94.28	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 94.28	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 94.28	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 94.28	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 94.28	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 94.28	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 94.28	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 0.52

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

   4. Simplified 0

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

      [Start] 0.52	\[ -\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right) \]
      neg-sub0 [=>] 0.52	\[ \color{blue}{0 - \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]
      +-commutative [=>] 0.52	\[ 0 - \color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]
      associate--r+ [=>] 0.52	\[ \color{blue}{\left(0 - 3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]
      neg-sub0 [<=] 0.52	\[ \color{blue}{\left(-3 \cdot \frac{1}{x}\right)} - \frac{1}{{x}^{2}} \]
      associate-*r/ [=>] 0	\[ \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]
      metadata-eval [=>] 0	\[ \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]
      distribute-neg-frac [=>] 0	\[ \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]
      metadata-eval [=>] 0	\[ \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]
      unpow2 [=>] 0	\[ \frac{-3}{x} - \frac{1}{\color{blue}{x \cdot x}} \]
      associate-/r* [=>] 0	\[ \frac{-3}{x} - \color{blue}{\frac{\frac{1}{x}}{x}} \]

   if -5e13 < x < 1e4

   1. Initial program 0.78

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

   2. Simplified 0.78

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

      [Start] 0.78	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 0.78	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 0.78	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 0.78	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 0.78	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 0.78	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 0.78	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 0.78	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 0.78	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 0.78	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 0.78	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 0.78	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 0.78	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 0.78	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 0.78	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 0.78	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 0.78	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 0.78	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 0.78	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 0.78	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 0.78	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 0.78	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 0.78	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 0.78

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

   4. Applied egg-rr 0.76

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

   5. Simplified 0.06

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

      [Start] 0.76	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      *-commutative [=>] 0.76	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \color{blue}{\left(-x\right) \cdot \left(x + -1\right)}}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      distribute-lft-out-- [=>] 0.06	\[ \frac{-1}{-1 + x} + \frac{\color{blue}{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}}{\left(x + -1\right) \cdot \left(x + 1\right)} \]

   6. Taylor expanded in x around 0 0.06

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

   if 1e4 < x

   1. Initial program 92.67

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

   2. Simplified 92.67

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

      [Start] 92.67	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 92.67	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 92.67	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 92.67	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 92.67	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 92.67	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 92.67	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 92.67	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 92.67	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 92.67	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 92.67	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 92.67	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 92.67	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 92.67	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 92.67	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 92.67	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 92.67	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 92.67	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 92.67	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 92.67	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 92.67	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 92.67	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 92.67	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 0.49

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

   4. Simplified 0.02

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

      [Start] 0.49	\[ -\left(\frac{1}{{x}^{4}} + \left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)\right)\right) \]
      neg-sub0 [=>] 0.49	\[ \color{blue}{0 - \left(\frac{1}{{x}^{4}} + \left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)\right)\right)} \]
      +-commutative [=>] 0.49	\[ 0 - \color{blue}{\left(\left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)\right) + \frac{1}{{x}^{4}}\right)} \]
      +-commutative [=>] 0.49	\[ 0 - \left(\left(\frac{1}{{x}^{2}} + \color{blue}{\left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)}\right) + \frac{1}{{x}^{4}}\right) \]
      associate-+r+ [=>] 0.49	\[ 0 - \left(\color{blue}{\left(\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right) + 3 \cdot \frac{1}{{x}^{3}}\right)} + \frac{1}{{x}^{4}}\right) \]
      +-commutative [<=] 0.49	\[ 0 - \left(\color{blue}{\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)} + \frac{1}{{x}^{4}}\right) \]
      associate--r+ [=>] 0.49	\[ \color{blue}{\left(0 - \left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)\right) - \frac{1}{{x}^{4}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.04

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50000000000000:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot -2}{\left(x + -1\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\right) + \left(\frac{-3}{{x}^{3}} + \frac{-1}{{x}^{4}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.03%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;x \leq -50000000000000 \lor \neg \left(x \leq 100000000\right):\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot -2}{\left(x + -1\right) \cdot \left(x + 1\right)}\\ \end{array} \]

Alternative 2
Error	0.12%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \leq -450000:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 14500:\\ \;\;\;\;\left(\frac{-1}{x + -1} - \frac{x}{-1 - x}\right) - \frac{x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-3 + \frac{2}{x}\right) + \frac{-2}{x \cdot x}}{x + -1}\\ \end{array} \]

Alternative 3
Error	0.12%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \leq -450000:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 11500:\\ \;\;\;\;\frac{-1}{x + -1} - \left(\frac{x}{x + -1} + \frac{x}{-1 - x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-3 + \frac{2}{x}\right) + \frac{-2}{x \cdot x}}{x + -1}\\ \end{array} \]

Alternative 4
Error	0.12%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \leq -620000:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 11500:\\ \;\;\;\;\frac{\frac{x \cdot x - x}{x + 1} + \left(-1 - x\right)}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-3 + \frac{2}{x}\right) + \frac{-2}{x \cdot x}}{x + -1}\\ \end{array} \]

Alternative 5
Error	0.12%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -470000:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 15500:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-3 + \frac{2}{x}\right) + \frac{-2}{x \cdot x}}{x + -1}\\ \end{array} \]

Alternative 6
Error	0.15%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -470000 \lor \neg \left(x \leq 350000\right):\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 7
Error	1.05%
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 8
Error	0.83%
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 9
Error	1.33%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;\frac{-1}{x + -1} + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 10
Error	1.51%
Cost	585

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

Alternative 11
Error	2.2%
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 12
Error	49.05%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))