Asymptote C

Average Accuracy: 53.5% → 100.0%

Time: 12.2s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{-1 + x \cdot -3}{-1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -5e+37)
   (/ -3.0 x)
   (if (<= x 100000000.0)
     (/ (+ -1.0 (* x -3.0)) (+ -1.0 (* x x)))
     (+ (/ -3.0 x) (/ (/ -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 <= -5e+37) {
		tmp = -3.0 / x;
	} else if (x <= 100000000.0) {
		tmp = (-1.0 + (x * -3.0)) / (-1.0 + (x * x));
	} else {
		tmp = (-3.0 / x) + ((-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 <= (-5d+37)) then
        tmp = (-3.0d0) / x
    else if (x <= 100000000.0d0) then
        tmp = ((-1.0d0) + (x * (-3.0d0))) / ((-1.0d0) + (x * x))
    else
        tmp = ((-3.0d0) / x) + (((-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 <= -5e+37) {
		tmp = -3.0 / x;
	} else if (x <= 100000000.0) {
		tmp = (-1.0 + (x * -3.0)) / (-1.0 + (x * x));
	} else {
		tmp = (-3.0 / x) + ((-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 <= -5e+37:
		tmp = -3.0 / x
	elif x <= 100000000.0:
		tmp = (-1.0 + (x * -3.0)) / (-1.0 + (x * x))
	else:
		tmp = (-3.0 / x) + ((-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 <= -5e+37)
		tmp = Float64(-3.0 / x);
	elseif (x <= 100000000.0)
		tmp = Float64(Float64(-1.0 + Float64(x * -3.0)) / Float64(-1.0 + Float64(x * x)));
	else
		tmp = Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / 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 <= -5e+37)
		tmp = -3.0 / x;
	elseif (x <= 100000000.0)
		tmp = (-1.0 + (x * -3.0)) / (-1.0 + (x * x));
	else
		tmp = (-3.0 / x) + ((-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[LessEqual[x, -5e+37], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 100000000.0], N[(N[(-1.0 + N[(x * -3.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999989e37

   1. Initial program 5.6%

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

   2. Simplified 5.6%

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

      [Start] 5.6	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 5.6	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 5.6	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 5.6	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 5.6	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 5.6	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 5.6	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 5.6	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 5.6	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 5.6	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 5.6	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 5.6	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 5.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 5.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 5.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 5.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 5.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 5.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 5.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 5.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 5.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 5.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 5.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 100.0%

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

   if -4.99999999999999989e37 < x < 1e8

   1. Initial program 95.5%

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

   2. Simplified 95.5%

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

      [Start] 95.5	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 95.5	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 95.5	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 95.5	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 95.5	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 95.5	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 95.5	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 95.5	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 95.5	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 95.5	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 95.5	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 95.5	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 95.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 95.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 95.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 95.5	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 95.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 95.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 95.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 95.5	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 95.5	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 95.5	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 95.5	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 95.5%

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

      [Start] 95.5	\[ \frac{-1 - x}{x + -1} - \frac{x}{-1 - x} \]
      frac-2neg [=>] 95.5	\[ \color{blue}{\frac{-\left(-1 - x\right)}{-\left(x + -1\right)}} - \frac{x}{-1 - x} \]
      frac-sub [=>] 95.5	\[ \color{blue}{\frac{\left(-\left(-1 - x\right)\right) \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)}} \]
      neg-sub0 [=>] 95.5	\[ \frac{\color{blue}{\left(0 - \left(-1 - x\right)\right)} \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [<=] 95.5	\[ \frac{\left(\color{blue}{\log 1} - \left(-1 - x\right)\right) \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      associate--r- [=>] 95.5	\[ \frac{\color{blue}{\left(\left(\log 1 - -1\right) + x\right)} \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [=>] 95.5	\[ \frac{\left(\left(\color{blue}{0} - -1\right) + x\right) \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [=>] 95.5	\[ \frac{\left(\color{blue}{1} + x\right) \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      +-commutative [=>] 95.5	\[ \frac{\color{blue}{\left(x + 1\right)} \cdot \left(-1 - x\right) - \left(-\left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      neg-sub0 [=>] 95.5	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \color{blue}{\left(0 - \left(x + -1\right)\right)} \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [<=] 95.5	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \left(\color{blue}{\log 1} - \left(x + -1\right)\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      +-commutative [=>] 95.5	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \left(\log 1 - \color{blue}{\left(-1 + x\right)}\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      associate--r+ [=>] 95.5	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \color{blue}{\left(\left(\log 1 - -1\right) - x\right)} \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [=>] 95.5	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \left(\left(\color{blue}{0} - -1\right) - x\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]
      metadata-eval [=>] 95.5	\[ \frac{\left(x + 1\right) \cdot \left(-1 - x\right) - \left(\color{blue}{1} - x\right) \cdot x}{\left(-\left(x + -1\right)\right) \cdot \left(-1 - x\right)} \]

   4. Taylor expanded in x around 0 99.9%

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

   5. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ \frac{-3 \cdot x - 1}{\left(1 - x\right) \cdot \left(-1 - x\right)} \]
      *-commutative [=>] 99.9	\[ \frac{-3 \cdot x - 1}{\color{blue}{\left(-1 - x\right) \cdot \left(1 - x\right)}} \]
      sub-neg [=>] 99.9	\[ \frac{-3 \cdot x - 1}{\left(-1 - x\right) \cdot \color{blue}{\left(1 + \left(-x\right)\right)}} \]
      metadata-eval [<=] 99.9	\[ \frac{-3 \cdot x - 1}{\left(-1 - x\right) \cdot \left(\color{blue}{\left(--1\right)} + \left(-x\right)\right)} \]
      distribute-neg-in [<=] 99.9	\[ \frac{-3 \cdot x - 1}{\left(-1 - x\right) \cdot \color{blue}{\left(-\left(-1 + x\right)\right)}} \]
      distribute-rgt-neg-in [<=] 99.9	\[ \frac{-3 \cdot x - 1}{\color{blue}{-\left(-1 - x\right) \cdot \left(-1 + x\right)}} \]
      distribute-lft-neg-in [=>] 99.9	\[ \frac{-3 \cdot x - 1}{\color{blue}{\left(-\left(-1 - x\right)\right) \cdot \left(-1 + x\right)}} \]
      sub-neg [=>] 99.9	\[ \frac{-3 \cdot x - 1}{\left(-\color{blue}{\left(-1 + \left(-x\right)\right)}\right) \cdot \left(-1 + x\right)} \]
      distribute-neg-in [=>] 99.9	\[ \frac{-3 \cdot x - 1}{\color{blue}{\left(\left(--1\right) + \left(-\left(-x\right)\right)\right)} \cdot \left(-1 + x\right)} \]
      metadata-eval [=>] 99.9	\[ \frac{-3 \cdot x - 1}{\left(\color{blue}{1} + \left(-\left(-x\right)\right)\right) \cdot \left(-1 + x\right)} \]
      add-sqr-sqrt [=>] 47.1	\[ \frac{-3 \cdot x - 1}{\left(1 + \left(-\left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right)\right) \cdot \left(-1 + x\right)} \]
      distribute-lft-neg-in [=>] 47.1	\[ \frac{-3 \cdot x - 1}{\left(1 + \left(-\color{blue}{\left(-\sqrt{x}\right) \cdot \sqrt{x}}\right)\right) \cdot \left(-1 + x\right)} \]
      distribute-lft-neg-in [=>] 47.1	\[ \frac{-3 \cdot x - 1}{\left(1 + \color{blue}{\left(-\left(-\sqrt{x}\right)\right) \cdot \sqrt{x}}\right) \cdot \left(-1 + x\right)} \]
      cancel-sign-sub-inv [<=] 47.1	\[ \frac{-3 \cdot x - 1}{\color{blue}{\left(1 - \left(-\sqrt{x}\right) \cdot \sqrt{x}\right)} \cdot \left(-1 + x\right)} \]
      cancel-sign-sub [=>] 47.1	\[ \frac{-3 \cdot x - 1}{\color{blue}{\left(1 + \sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(-1 + x\right)} \]
      add-sqr-sqrt [<=] 99.9	\[ \frac{-3 \cdot x - 1}{\left(1 + \color{blue}{x}\right) \cdot \left(-1 + x\right)} \]
      +-commutative [<=] 99.9	\[ \frac{-3 \cdot x - 1}{\color{blue}{\left(x + 1\right)} \cdot \left(-1 + x\right)} \]
      +-commutative [=>] 99.9	\[ \frac{-3 \cdot x - 1}{\left(x + 1\right) \cdot \color{blue}{\left(x + -1\right)}} \]
      metadata-eval [<=] 99.9	\[ \frac{-3 \cdot x - 1}{\left(x + 1\right) \cdot \left(x + \color{blue}{\left(-1\right)}\right)} \]
      sub-neg [<=] 99.9	\[ \frac{-3 \cdot x - 1}{\left(x + 1\right) \cdot \color{blue}{\left(x - 1\right)}} \]
      difference-of-sqr-1 [<=] 99.9	\[ \frac{-3 \cdot x - 1}{\color{blue}{x \cdot x - 1}} \]

   if 1e8 < x

   1. Initial program 6.6%

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

   2. Simplified 6.6%

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

      [Start] 6.6	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 6.6	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 6.6	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 6.6	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 6.6	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 6.6	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 6.6	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 6.6	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 6.6	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 6.6	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 6.6	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 6.6	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 6.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 6.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 6.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 6.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 6.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 6.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 6.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 6.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 6.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 6.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 6.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   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%

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

      [Start] 99.5	\[ -\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right) \]
      neg-sub0 [=>] 99.5	\[ \color{blue}{0 - \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)} \]
      +-commutative [=>] 99.5	\[ 0 - \color{blue}{\left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)} \]
      associate--r+ [=>] 99.5	\[ \color{blue}{\left(0 - 3 \cdot \frac{1}{x}\right) - \frac{1}{{x}^{2}}} \]
      neg-sub0 [<=] 99.5	\[ \color{blue}{\left(-3 \cdot \frac{1}{x}\right)} - \frac{1}{{x}^{2}} \]
      associate-*r/ [=>] 100.0	\[ \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) - \frac{1}{{x}^{2}} \]
      metadata-eval [=>] 100.0	\[ \left(-\frac{\color{blue}{3}}{x}\right) - \frac{1}{{x}^{2}} \]
      distribute-neg-frac [=>] 100.0	\[ \color{blue}{\frac{-3}{x}} - \frac{1}{{x}^{2}} \]
      metadata-eval [=>] 100.0	\[ \frac{\color{blue}{-3}}{x} - \frac{1}{{x}^{2}} \]
      unpow2 [=>] 100.0	\[ \frac{-3}{x} - \frac{1}{\color{blue}{x \cdot x}} \]
      associate-/r* [=>] 100.0	\[ \frac{-3}{x} - \color{blue}{\frac{\frac{1}{x}}{x}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{-1 + x \cdot -3}{-1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	960

\[\frac{\frac{-1}{x + 1} \cdot \left(1 + x \cdot 3\right)}{-1 + x} \]

Alternative 2
Accuracy	98.7%
Cost	841

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

Alternative 3
Accuracy	99.0%
Cost	841

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

Alternative 4
Accuracy	98.5%
Cost	840

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

Alternative 5
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 6
Accuracy	97.8%
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 7
Accuracy	2.7%
Cost	64

\[-1 \]

Alternative 8
Accuracy	50.0%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023131 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))