Asymptote C

Average Accuracy: 54.9% → 100.0%

Time: 12.5s

Precision: binary64

Cost: 1225

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1000000000 \lor \neg \left(x \leq 20000000000\right):\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot -2}{-1 + x \cdot x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (or (<= x -1000000000.0) (not (<= x 20000000000.0)))
   (+ (/ -1.0 (* x x)) (/ -3.0 x))
   (+ (/ -1.0 (+ x -1.0)) (/ (* x -2.0) (+ -1.0 (* x 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 <= -1000000000.0) || !(x <= 20000000000.0)) {
		tmp = (-1.0 / (x * x)) + (-3.0 / x);
	} else {
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / (-1.0 + (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) :: tmp
    if ((x <= (-1000000000.0d0)) .or. (.not. (x <= 20000000000.0d0))) then
        tmp = ((-1.0d0) / (x * x)) + ((-3.0d0) / x)
    else
        tmp = ((-1.0d0) / (x + (-1.0d0))) + ((x * (-2.0d0)) / ((-1.0d0) + (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 tmp;
	if ((x <= -1000000000.0) || !(x <= 20000000000.0)) {
		tmp = (-1.0 / (x * x)) + (-3.0 / x);
	} else {
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / (-1.0 + (x * 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 <= -1000000000.0) or not (x <= 20000000000.0):
		tmp = (-1.0 / (x * x)) + (-3.0 / x)
	else:
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / (-1.0 + (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)
	tmp = 0.0
	if ((x <= -1000000000.0) || !(x <= 20000000000.0))
		tmp = Float64(Float64(-1.0 / Float64(x * x)) + Float64(-3.0 / x));
	else
		tmp = Float64(Float64(-1.0 / Float64(x + -1.0)) + Float64(Float64(x * -2.0) / Float64(-1.0 + 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)
	tmp = 0.0;
	if ((x <= -1000000000.0) || ~((x <= 20000000000.0)))
		tmp = (-1.0 / (x * x)) + (-3.0 / x);
	else
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / (-1.0 + (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_] := If[Or[LessEqual[x, -1000000000.0], N[Not[LessEqual[x, 20000000000.0]], $MachinePrecision]], N[(N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-3.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x * -2.0), $MachinePrecision] / N[(-1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e9 or 2e10 < x

   1. Initial program 6.1%

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

   2. Simplified 6.1%

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

      [Start] 6.1	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 6.1	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 6.1	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 6.1	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 6.1	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 6.1	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 6.1	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 6.1	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 6.1	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 6.1	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 6.1	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 6.1	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 6.1	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 6.1	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 6.1	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 6.1	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 6.1	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 6.1	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 6.1	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 6.1	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 6.1	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 6.1	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 6.1	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 99.5%

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

   4. Simplified 100.0%

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

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

   if -1e9 < x < 2e10

   1. Initial program 99.4%

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

   2. Simplified 99.4%

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

      [Start] 99.4	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 99.4	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 99.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 99.4	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 99.4	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 99.4	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 99.4	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 99.4	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 99.4	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 99.4	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 99.4	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 99.4	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 99.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ \frac{-1 - x}{x + -1} - \frac{x}{-1 - x} \]
      div-sub [=>] 99.4	\[ \color{blue}{\left(\frac{-1}{x + -1} - \frac{x}{x + -1}\right)} - \frac{x}{-1 - x} \]
      sub-neg [=>] 99.4	\[ \color{blue}{\left(\frac{-1}{x + -1} + \left(-\frac{x}{x + -1}\right)\right)} - \frac{x}{-1 - x} \]
      associate--l+ [=>] 99.4	\[ \color{blue}{\frac{-1}{x + -1} + \left(\left(-\frac{x}{x + -1}\right) - \frac{x}{-1 - x}\right)} \]
      +-commutative [=>] 99.4	\[ \frac{-1}{\color{blue}{-1 + x}} + \left(\left(-\frac{x}{x + -1}\right) - \frac{x}{-1 - x}\right) \]
      +-commutative [=>] 99.4	\[ \frac{-1}{-1 + x} + \left(\left(-\frac{x}{\color{blue}{-1 + x}}\right) - \frac{x}{-1 - x}\right) \]

   4. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ \frac{-1}{-1 + x} + \left(\left(-\frac{x}{-1 + x}\right) - \frac{x}{-1 - x}\right) \]
      distribute-neg-frac [=>] 99.4	\[ \frac{-1}{-1 + x} + \left(\color{blue}{\frac{-x}{-1 + x}} - \frac{x}{-1 - x}\right) \]
      frac-2neg [=>] 99.4	\[ \frac{-1}{-1 + x} + \left(\frac{-x}{-1 + x} - \color{blue}{\frac{-x}{-\left(-1 - x\right)}}\right) \]
      frac-sub [=>] 99.4	\[ \frac{-1}{-1 + x} + \color{blue}{\frac{\left(-x\right) \cdot \left(-\left(-1 - x\right)\right) - \left(-1 + x\right) \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)}} \]
      neg-sub0 [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \color{blue}{\left(0 - \left(-1 - x\right)\right)} - \left(-1 + x\right) \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)} \]
      associate--r- [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \color{blue}{\left(\left(0 - -1\right) + x\right)} - \left(-1 + x\right) \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)} \]
      metadata-eval [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(\color{blue}{1} + x\right) - \left(-1 + x\right) \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)} \]
      +-commutative [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \color{blue}{\left(x + 1\right)} - \left(-1 + x\right) \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)} \]
      +-commutative [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \color{blue}{\left(x + -1\right)} \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)} \]
      +-commutative [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\color{blue}{\left(x + -1\right)} \cdot \left(-\left(-1 - x\right)\right)} \]
      neg-sub0 [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \color{blue}{\left(0 - \left(-1 - x\right)\right)}} \]
      associate--r- [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \color{blue}{\left(\left(0 - -1\right) + x\right)}} \]
      metadata-eval [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \left(\color{blue}{1} + x\right)} \]
      +-commutative [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \color{blue}{\left(x + 1\right)}} \]

   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] 99.4	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      *-commutative [=>] 99.4	\[ \frac{-1}{-1 + x} + \frac{\color{blue}{\left(x + 1\right) \cdot \left(-x\right)} - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      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. Taylor expanded in x around 0 99.9%

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

   7. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot 2}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      *-commutative [=>] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot 2}{\color{blue}{\left(x + 1\right) \cdot \left(x + -1\right)}} \]
      metadata-eval [<=] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot 2}{\left(x + 1\right) \cdot \left(x + \color{blue}{\left(-1\right)}\right)} \]
      sub-neg [<=] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot 2}{\left(x + 1\right) \cdot \color{blue}{\left(x - 1\right)}} \]
      difference-of-sqr-1 [<=] 99.9	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot 2}{\color{blue}{x \cdot x - 1}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000000 \lor \neg \left(x \leq 20000000000\right):\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot -2}{-1 + x \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -360000 \lor \neg \left(x \leq 430000\right):\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	969

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

Alternative 3
Accuracy	98.9%
Cost	841

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

Alternative 4
Accuracy	99.1%
Cost	841

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

Alternative 5
Accuracy	99.0%
Cost	841

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

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	97.7%
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	51.5%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))