Asymptote C

Average Accuracy: 54.0% → 100.0%

Time: 13.8s

Precision: binary64

Cost: 1224

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 2000000:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot -2}{-1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 - \frac{1}{x}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1e+25)
   (/ -3.0 x)
   (if (<= x 2000000.0)
     (+ (/ -1.0 (+ x -1.0)) (/ (* x -2.0) (+ -1.0 (* x x))))
     (/ (- -3.0 (/ 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 <= -1e+25) {
		tmp = -3.0 / x;
	} else if (x <= 2000000.0) {
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / (-1.0 + (x * x)));
	} else {
		tmp = (-3.0 - (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 <= (-1d+25)) then
        tmp = (-3.0d0) / x
    else if (x <= 2000000.0d0) then
        tmp = ((-1.0d0) / (x + (-1.0d0))) + ((x * (-2.0d0)) / ((-1.0d0) + (x * x)))
    else
        tmp = ((-3.0d0) - (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 <= -1e+25) {
		tmp = -3.0 / x;
	} else if (x <= 2000000.0) {
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / (-1.0 + (x * x)));
	} else {
		tmp = (-3.0 - (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 <= -1e+25:
		tmp = -3.0 / x
	elif x <= 2000000.0:
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / (-1.0 + (x * x)))
	else:
		tmp = (-3.0 - (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 <= -1e+25)
		tmp = Float64(-3.0 / x);
	elseif (x <= 2000000.0)
		tmp = Float64(Float64(-1.0 / Float64(x + -1.0)) + Float64(Float64(x * -2.0) / Float64(-1.0 + Float64(x * x))));
	else
		tmp = Float64(Float64(-3.0 - 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 <= -1e+25)
		tmp = -3.0 / x;
	elseif (x <= 2000000.0)
		tmp = (-1.0 / (x + -1.0)) + ((x * -2.0) / (-1.0 + (x * x)));
	else
		tmp = (-3.0 - (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[LessEqual[x, -1e+25], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 2000000.0], 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], N[(N[(-3.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000009e25

   1. Initial program 5.7%

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

   2. Simplified 5.7%

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

      [Start] 5.7	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 5.7	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 5.7	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 5.7	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 5.7	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 5.7	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 5.7	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 5.7	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 5.7	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 5.7	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 5.7	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 5.7	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 5.7	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 5.7	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 5.7	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 5.7	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 5.7	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 5.7	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 5.7	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 5.7	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 5.7	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 5.7	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 5.7	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 100.0%

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

   if -1.00000000000000009e25 < x < 2e6

   1. Initial program 97.6%

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

   2. Simplified 97.6%

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

      [Start] 97.6	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 97.6	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 97.6	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 97.6	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 97.6	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 97.6	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 97.6	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 97.6	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 97.6	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 97.6	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 97.6	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 97.6	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 97.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 97.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 97.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 97.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 97.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 97.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 97.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 97.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 97.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 97.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 97.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 97.8%

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

      [Start] 97.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{-1 - x} \]
      div-sub [=>] 97.6	\[ \color{blue}{\left(\frac{-1}{x + -1} - \frac{x}{x + -1}\right)} - \frac{x}{-1 - x} \]
      sub-neg [=>] 97.6	\[ \color{blue}{\left(\frac{-1}{x + -1} + \left(-\frac{x}{x + -1}\right)\right)} - \frac{x}{-1 - x} \]
      associate--l+ [=>] 97.8	\[ \color{blue}{\frac{-1}{x + -1} + \left(\left(-\frac{x}{x + -1}\right) - \frac{x}{-1 - x}\right)} \]
      +-commutative [=>] 97.8	\[ \frac{-1}{\color{blue}{-1 + x}} + \left(\left(-\frac{x}{x + -1}\right) - \frac{x}{-1 - x}\right) \]
      +-commutative [=>] 97.8	\[ \frac{-1}{-1 + x} + \left(\left(-\frac{x}{\color{blue}{-1 + x}}\right) - \frac{x}{-1 - x}\right) \]

   4. Applied egg-rr 97.8%

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

      [Start] 97.8	\[ \frac{-1}{-1 + x} + \left(\left(-\frac{x}{-1 + x}\right) - \frac{x}{-1 - x}\right) \]
      distribute-neg-frac [=>] 97.8	\[ \frac{-1}{-1 + x} + \left(\color{blue}{\frac{-x}{-1 + x}} - \frac{x}{-1 - x}\right) \]
      frac-2neg [=>] 97.8	\[ \frac{-1}{-1 + x} + \left(\frac{-x}{-1 + x} - \color{blue}{\frac{-x}{-\left(-1 - x\right)}}\right) \]
      frac-sub [=>] 97.8	\[ \frac{-1}{-1 + x} + \color{blue}{\frac{\left(-x\right) \cdot \left(-\left(-1 - x\right)\right) - \left(-1 + x\right) \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)}} \]
      neg-sub0 [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \color{blue}{\left(0 - \left(-1 - x\right)\right)} - \left(-1 + x\right) \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)} \]
      associate--r- [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \color{blue}{\left(\left(0 - -1\right) + x\right)} - \left(-1 + x\right) \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)} \]
      metadata-eval [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(\color{blue}{1} + x\right) - \left(-1 + x\right) \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)} \]
      +-commutative [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \color{blue}{\left(x + 1\right)} - \left(-1 + x\right) \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)} \]
      +-commutative [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \color{blue}{\left(x + -1\right)} \cdot \left(-x\right)}{\left(-1 + x\right) \cdot \left(-\left(-1 - x\right)\right)} \]
      +-commutative [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\color{blue}{\left(x + -1\right)} \cdot \left(-\left(-1 - x\right)\right)} \]
      neg-sub0 [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \color{blue}{\left(0 - \left(-1 - x\right)\right)}} \]
      associate--r- [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \color{blue}{\left(\left(0 - -1\right) + x\right)}} \]
      metadata-eval [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \left(\color{blue}{1} + x\right)} \]
      +-commutative [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \color{blue}{\left(x + 1\right)}} \]

   5. Simplified 98.8%

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

      [Start] 97.8	\[ \frac{-1}{-1 + x} + \frac{\left(-x\right) \cdot \left(x + 1\right) - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      *-commutative [=>] 97.8	\[ \frac{-1}{-1 + x} + \frac{\color{blue}{\left(x + 1\right) \cdot \left(-x\right)} - \left(x + -1\right) \cdot \left(-x\right)}{\left(x + -1\right) \cdot \left(x + 1\right)} \]
      distribute-rgt-out-- [=>] 98.8	\[ \frac{-1}{-1 + x} + \frac{\color{blue}{\left(-x\right) \cdot \left(\left(x + 1\right) - \left(x + -1\right)\right)}}{\left(x + -1\right) \cdot \left(x + 1\right)} \]

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

   if 2e6 < x

   1. Initial program 6.8%

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

   2. Simplified 6.8%

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

      [Start] 6.8	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 6.8	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 6.8	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 6.8	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 6.8	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 6.8	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 6.8	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 6.8	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 6.8	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 6.8	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 6.8	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 6.8	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 6.8	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 6.8	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 6.8	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 6.8	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 6.8	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 6.8	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 6.8	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 6.8	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 6.8	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 6.8	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 6.8	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 99.4%

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

   4. Simplified 99.9%

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

      [Start] 99.4	\[ -\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right) \]
      distribute-neg-in [=>] 99.4	\[ \color{blue}{\left(-\frac{1}{{x}^{2}}\right) + \left(-3 \cdot \frac{1}{x}\right)} \]
      unpow2 [=>] 99.4	\[ \left(-\frac{1}{\color{blue}{x \cdot x}}\right) + \left(-3 \cdot \frac{1}{x}\right) \]
      distribute-neg-frac [=>] 99.4	\[ \color{blue}{\frac{-1}{x \cdot x}} + \left(-3 \cdot \frac{1}{x}\right) \]
      metadata-eval [=>] 99.4	\[ \frac{\color{blue}{-1}}{x \cdot x} + \left(-3 \cdot \frac{1}{x}\right) \]
      associate-*r/ [=>] 99.9	\[ \frac{-1}{x \cdot x} + \left(-\color{blue}{\frac{3 \cdot 1}{x}}\right) \]
      metadata-eval [=>] 99.9	\[ \frac{-1}{x \cdot x} + \left(-\frac{\color{blue}{3}}{x}\right) \]
      distribute-neg-frac [=>] 99.9	\[ \frac{-1}{x \cdot x} + \color{blue}{\frac{-3}{x}} \]
      metadata-eval [=>] 99.9	\[ \frac{-1}{x \cdot x} + \frac{\color{blue}{-3}}{x} \]

   5. Applied egg-rr 32.4%

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

      [Start] 99.9	\[ \frac{-1}{x \cdot x} + \frac{-3}{x} \]
      +-commutative [=>] 99.9	\[ \color{blue}{\frac{-3}{x} + \frac{-1}{x \cdot x}} \]
      frac-2neg [=>] 99.9	\[ \color{blue}{\frac{--3}{-x}} + \frac{-1}{x \cdot x} \]
      metadata-eval [=>] 99.9	\[ \frac{\color{blue}{3}}{-x} + \frac{-1}{x \cdot x} \]
      frac-add [=>] 32.5	\[ \color{blue}{\frac{3 \cdot \left(x \cdot x\right) + \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(x \cdot x\right)}} \]
      +-commutative [=>] 32.5	\[ \frac{\color{blue}{\left(-x\right) \cdot -1 + 3 \cdot \left(x \cdot x\right)}}{\left(-x\right) \cdot \left(x \cdot x\right)} \]
      distribute-lft-neg-in [<=] 32.5	\[ \frac{\color{blue}{\left(-x \cdot -1\right)} + 3 \cdot \left(x \cdot x\right)}{\left(-x\right) \cdot \left(x \cdot x\right)} \]
      distribute-rgt-neg-in [=>] 32.5	\[ \frac{\color{blue}{x \cdot \left(--1\right)} + 3 \cdot \left(x \cdot x\right)}{\left(-x\right) \cdot \left(x \cdot x\right)} \]
      metadata-eval [=>] 32.5	\[ \frac{x \cdot \color{blue}{1} + 3 \cdot \left(x \cdot x\right)}{\left(-x\right) \cdot \left(x \cdot x\right)} \]
      *-commutative [<=] 32.5	\[ \frac{\color{blue}{1 \cdot x} + 3 \cdot \left(x \cdot x\right)}{\left(-x\right) \cdot \left(x \cdot x\right)} \]
      *-un-lft-identity [<=] 32.5	\[ \frac{\color{blue}{x} + 3 \cdot \left(x \cdot x\right)}{\left(-x\right) \cdot \left(x \cdot x\right)} \]
      *-commutative [=>] 32.5	\[ \frac{x + \color{blue}{\left(x \cdot x\right) \cdot 3}}{\left(-x\right) \cdot \left(x \cdot x\right)} \]
      *-commutative [=>] 32.5	\[ \frac{x + \left(x \cdot x\right) \cdot 3}{\color{blue}{\left(x \cdot x\right) \cdot \left(-x\right)}} \]
      distribute-rgt-neg-in [<=] 32.5	\[ \frac{x + \left(x \cdot x\right) \cdot 3}{\color{blue}{-\left(x \cdot x\right) \cdot x}} \]
      pow3 [=>] 32.4	\[ \frac{x + \left(x \cdot x\right) \cdot 3}{-\color{blue}{{x}^{3}}} \]

   6. Simplified 99.9%

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

      [Start] 32.4	\[ \frac{x + \left(x \cdot x\right) \cdot 3}{-{x}^{3}} \]
      *-commutative [=>] 32.4	\[ \frac{x + \color{blue}{3 \cdot \left(x \cdot x\right)}}{-{x}^{3}} \]
      metadata-eval [<=] 32.4	\[ \frac{x + \color{blue}{\left(--3\right)} \cdot \left(x \cdot x\right)}{-{x}^{3}} \]
      distribute-lft-neg-in [<=] 32.4	\[ \frac{x + \color{blue}{\left(--3 \cdot \left(x \cdot x\right)\right)}}{-{x}^{3}} \]
      distribute-rgt-neg-out [<=] 32.4	\[ \frac{x + \color{blue}{-3 \cdot \left(-x \cdot x\right)}}{-{x}^{3}} \]
      distribute-rgt-neg-out [<=] 32.4	\[ \frac{x + -3 \cdot \color{blue}{\left(x \cdot \left(-x\right)\right)}}{-{x}^{3}} \]
      *-lft-identity [<=] 32.4	\[ \frac{\color{blue}{1 \cdot x} + -3 \cdot \left(x \cdot \left(-x\right)\right)}{-{x}^{3}} \]
      metadata-eval [<=] 32.4	\[ \frac{\color{blue}{\left(-1 \cdot -1\right)} \cdot x + -3 \cdot \left(x \cdot \left(-x\right)\right)}{-{x}^{3}} \]
      associate-*r* [<=] 32.4	\[ \frac{\color{blue}{-1 \cdot \left(-1 \cdot x\right)} + -3 \cdot \left(x \cdot \left(-x\right)\right)}{-{x}^{3}} \]
      neg-mul-1 [<=] 32.4	\[ \frac{-1 \cdot \color{blue}{\left(-x\right)} + -3 \cdot \left(x \cdot \left(-x\right)\right)}{-{x}^{3}} \]
      associate-*r* [=>] 32.4	\[ \frac{-1 \cdot \left(-x\right) + \color{blue}{\left(-3 \cdot x\right) \cdot \left(-x\right)}}{-{x}^{3}} \]
      *-commutative [<=] 32.4	\[ \frac{-1 \cdot \left(-x\right) + \color{blue}{\left(x \cdot -3\right)} \cdot \left(-x\right)}{-{x}^{3}} \]
      distribute-rgt-out [=>] 32.4	\[ \frac{\color{blue}{\left(-x\right) \cdot \left(-1 + x \cdot -3\right)}}{-{x}^{3}} \]
      +-commutative [<=] 32.4	\[ \frac{\left(-x\right) \cdot \color{blue}{\left(x \cdot -3 + -1\right)}}{-{x}^{3}} \]
      distribute-lft-neg-in [<=] 32.4	\[ \frac{\color{blue}{-x \cdot \left(x \cdot -3 + -1\right)}}{-{x}^{3}} \]
      distribute-neg-frac [<=] 32.4	\[ \color{blue}{-\frac{x \cdot \left(x \cdot -3 + -1\right)}{-{x}^{3}}} \]
      cube-neg [<=] 32.4	\[ -\frac{x \cdot \left(x \cdot -3 + -1\right)}{\color{blue}{{\left(-x\right)}^{3}}} \]
      unpow3 [=>] 32.4	\[ -\frac{x \cdot \left(x \cdot -3 + -1\right)}{\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right) \cdot \left(-x\right)}} \]
      associate-/r* [=>] 49.2	\[ -\color{blue}{\frac{\frac{x \cdot \left(x \cdot -3 + -1\right)}{\left(-x\right) \cdot \left(-x\right)}}{-x}} \]

   7. Taylor expanded in x around 0 99.4%

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

   8. Simplified 99.9%

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

      [Start] 99.4	\[ -\left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right) \]
      unpow2 [=>] 99.4	\[ -\left(\frac{1}{\color{blue}{x \cdot x}} + 3 \cdot \frac{1}{x}\right) \]
      associate-/r* [=>] 99.4	\[ -\left(\color{blue}{\frac{\frac{1}{x}}{x}} + 3 \cdot \frac{1}{x}\right) \]
      metadata-eval [<=] 99.4	\[ -\left(\frac{\frac{\color{blue}{-1 \cdot -1}}{x}}{x} + 3 \cdot \frac{1}{x}\right) \]
      associate-*r/ [<=] 99.4	\[ -\left(\frac{\color{blue}{-1 \cdot \frac{-1}{x}}}{x} + 3 \cdot \frac{1}{x}\right) \]
      associate-*l/ [<=] 99.4	\[ -\left(\color{blue}{\frac{-1}{x} \cdot \frac{-1}{x}} + 3 \cdot \frac{1}{x}\right) \]
      associate-*r/ [=>] 99.9	\[ -\left(\frac{-1}{x} \cdot \frac{-1}{x} + \color{blue}{\frac{3 \cdot 1}{x}}\right) \]
      metadata-eval [=>] 99.9	\[ -\left(\frac{-1}{x} \cdot \frac{-1}{x} + \frac{\color{blue}{3}}{x}\right) \]
      metadata-eval [<=] 99.9	\[ -\left(\frac{-1}{x} \cdot \frac{-1}{x} + \frac{\color{blue}{-3 \cdot -1}}{x}\right) \]
      associate-*r/ [<=] 99.4	\[ -\left(\frac{-1}{x} \cdot \frac{-1}{x} + \color{blue}{-3 \cdot \frac{-1}{x}}\right) \]
      +-commutative [<=] 99.4	\[ -\color{blue}{\left(-3 \cdot \frac{-1}{x} + \frac{-1}{x} \cdot \frac{-1}{x}\right)} \]
      distribute-rgt-in [<=] 99.4	\[ -\color{blue}{\frac{-1}{x} \cdot \left(-3 + \frac{-1}{x}\right)} \]
      associate-*l/ [=>] 99.9	\[ -\color{blue}{\frac{-1 \cdot \left(-3 + \frac{-1}{x}\right)}{x}} \]
      distribute-lft-in [=>] 99.9	\[ -\frac{\color{blue}{-1 \cdot -3 + -1 \cdot \frac{-1}{x}}}{x} \]
      metadata-eval [=>] 99.9	\[ -\frac{\color{blue}{3} + -1 \cdot \frac{-1}{x}}{x} \]
      mul-1-neg [=>] 99.9	\[ -\frac{3 + \color{blue}{\left(-\frac{-1}{x}\right)}}{x} \]
      unsub-neg [=>] 99.9	\[ -\frac{\color{blue}{3 - \frac{-1}{x}}}{x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 2000000:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot -2}{-1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 - \frac{1}{x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -160000000:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 600000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 - \frac{1}{x}}{x}\\ \end{array} \]

Alternative 2
Accuracy	99.0%
Cost	713

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

Alternative 3
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 4
Accuracy	97.9%
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 5
Accuracy	50.5%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))