Asymptote C

Average Error: 29.8 → 0.0

Time: 11.4s

Precision: binary64

Cost: 7560

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{x \cdot -3 + -1}{\left(-1 - x\right) \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{\frac{1}{x}}{x}\right) - \frac{3}{{x}^{3}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -5e+22)
   (/ -3.0 x)
   (if (<= x 200000.0)
     (/ (+ (* x -3.0) -1.0) (* (- -1.0 x) (- 1.0 x)))
     (- (- (/ -3.0 x) (/ (/ 1.0 x) x)) (/ 3.0 (pow x 3.0))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -5e+22) {
		tmp = -3.0 / x;
	} else if (x <= 200000.0) {
		tmp = ((x * -3.0) + -1.0) / ((-1.0 - x) * (1.0 - x));
	} else {
		tmp = ((-3.0 / x) - ((1.0 / x) / x)) - (3.0 / pow(x, 3.0));
	}
	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 <= (-5d+22)) then
        tmp = (-3.0d0) / x
    else if (x <= 200000.0d0) then
        tmp = ((x * (-3.0d0)) + (-1.0d0)) / (((-1.0d0) - x) * (1.0d0 - x))
    else
        tmp = (((-3.0d0) / x) - ((1.0d0 / x) / x)) - (3.0d0 / (x ** 3.0d0))
    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 <= -5e+22) {
		tmp = -3.0 / x;
	} else if (x <= 200000.0) {
		tmp = ((x * -3.0) + -1.0) / ((-1.0 - x) * (1.0 - x));
	} else {
		tmp = ((-3.0 / x) - ((1.0 / x) / x)) - (3.0 / Math.pow(x, 3.0));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -5e+22:
		tmp = -3.0 / x
	elif x <= 200000.0:
		tmp = ((x * -3.0) + -1.0) / ((-1.0 - x) * (1.0 - x))
	else:
		tmp = ((-3.0 / x) - ((1.0 / x) / x)) - (3.0 / math.pow(x, 3.0))
	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 <= -5e+22)
		tmp = Float64(-3.0 / x);
	elseif (x <= 200000.0)
		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) - Float64(Float64(1.0 / x) / x)) - Float64(3.0 / (x ^ 3.0)));
	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 <= -5e+22)
		tmp = -3.0 / x;
	elseif (x <= 200000.0)
		tmp = ((x * -3.0) + -1.0) / ((-1.0 - x) * (1.0 - x));
	else
		tmp = ((-3.0 / x) - ((1.0 / x) / x)) - (3.0 / (x ^ 3.0));
	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, -5e+22], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 200000.0], 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] - N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - N[(3.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999996e22

   1. Initial program 60.5

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

   2. Simplified 60.5

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

      [Start] 60.5	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 60.5	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 60.5	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 60.5	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 60.5	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 60.5	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 60.5	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 60.5	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 60.5	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 60.5	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 60.5	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 60.5	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 60.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 60.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 60.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 60.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 60.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 60.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 60.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 60.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 60.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 60.5	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 60.5	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 0

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

   if -4.9999999999999996e22 < x < 2e5

   1. Initial program 1.5

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

   2. Simplified 1.5

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

      [Start] 1.5	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 1.5	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 1.5	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 1.5	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 1.5	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 1.5	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 1.5	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 1.5	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 1.5	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 1.5	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 1.5	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 1.5	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 1.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 1.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 1.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 1.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 1.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 1.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 1.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 1.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 1.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 1.5	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 1.5	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 1.5

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

   4. Simplified 1.5

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

      [Start] 1.5	\[ \frac{-1 - x}{x + -1} - \left(\frac{x}{1 - x \cdot x} \cdot -1 + \frac{x}{1 - x \cdot x} \cdot x\right) \]
      distribute-lft-in [<=] 1.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{1 - x \cdot x} \cdot \left(-1 + x\right)} \]
      associate-*l/ [=>] 1.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x \cdot \left(-1 + x\right)}{1 - x \cdot x}} \]
      +-commutative [=>] 1.5	\[ \frac{-1 - x}{x + -1} - \frac{x \cdot \color{blue}{\left(x + -1\right)}}{1 - x \cdot x} \]

   5. Applied egg-rr 1.5

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

   6. Simplified 1.5

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

      [Start] 1.5	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - x \cdot \left(1 - x\right)}{\left(1 - x\right) \cdot \left(-1 - x\right)} \]
      *-commutative [=>] 1.5	\[ \frac{\color{blue}{\left(-1 - x\right) \cdot \left(x + 1\right)} - x \cdot \left(1 - x\right)}{\left(1 - x\right) \cdot \left(-1 - x\right)} \]
      *-commutative [=>] 1.5	\[ \frac{\left(-1 - x\right) \cdot \left(x + 1\right) - x \cdot \left(1 - x\right)}{\color{blue}{\left(-1 - x\right) \cdot \left(1 - x\right)}} \]

   7. Taylor expanded in x around 0 0.0

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

   if 2e5 < x

   1. Initial program 59.6

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

   2. Simplified 59.6

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

      [Start] 59.6	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 59.6	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 59.6	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 59.6	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 59.6	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 59.6	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 59.6	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 59.6	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 59.6	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 59.6	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 59.6	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 59.6	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 59.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 59.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 59.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 59.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 59.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 59.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 59.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 59.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 59.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 59.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 59.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 0.3

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

   4. Simplified 0.0

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

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

3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{x \cdot -3 + -1}{\left(-1 - x\right) \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{\frac{1}{x}}{x}\right) - \frac{3}{{x}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\frac{x \cdot -3 + -1}{\left(-1 - x\right) \cdot \left(1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{1}{x}}{x}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\frac{x \cdot -3 + -1}{-1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{1}{x}}{x}\\ \end{array} \]

Alternative 3
Error	0.6
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 4
Error	0.4
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 5
Error	0.7
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 6
Error	0.8
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
Error	1.2
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 8
Error	31.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023045 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))