Asymptote C

Average Error: 46.45% → 0.28%

Time: 14.7s

Precision: binary64

Cost: 14280

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -150000000:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 2400:\\ \;\;\;\;\frac{-1 - \mathsf{fma}\left(x, \frac{x + -1}{-1 - x}, x\right)}{\frac{\frac{x + -1}{x + -1}}{\frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\right) + \frac{-3}{{x}^{3}}\right) + \frac{-1}{{x}^{4}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -150000000.0)
   (/ -3.0 x)
   (if (<= x 2400.0)
     (/
      (- -1.0 (fma x (/ (+ x -1.0) (- -1.0 x)) x))
      (/ (/ (+ x -1.0) (+ x -1.0)) (/ 1.0 (+ x -1.0))))
     (+
      (+ (+ (/ -3.0 x) (/ (/ -1.0 x) x)) (/ -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 tmp;
	if (x <= -150000000.0) {
		tmp = -3.0 / x;
	} else if (x <= 2400.0) {
		tmp = (-1.0 - fma(x, ((x + -1.0) / (-1.0 - x)), x)) / (((x + -1.0) / (x + -1.0)) / (1.0 / (x + -1.0)));
	} else {
		tmp = (((-3.0 / x) + ((-1.0 / x) / x)) + (-3.0 / pow(x, 3.0))) + (-1.0 / 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)
	tmp = 0.0
	if (x <= -150000000.0)
		tmp = Float64(-3.0 / x);
	elseif (x <= 2400.0)
		tmp = Float64(Float64(-1.0 - fma(x, Float64(Float64(x + -1.0) / Float64(-1.0 - x)), x)) / Float64(Float64(Float64(x + -1.0) / Float64(x + -1.0)) / Float64(1.0 / Float64(x + -1.0))));
	else
		tmp = Float64(Float64(Float64(Float64(-3.0 / x) + Float64(Float64(-1.0 / x) / x)) + Float64(-3.0 / (x ^ 3.0))) + Float64(-1.0 / (x ^ 4.0)));
	end
	return 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, -150000000.0], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 2400.0], N[(N[(-1.0 - N[(x * N[(N[(x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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] + N[(-1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e8

   1. Initial program 93.59

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

   2. Simplified 93.59

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

      [Start] 93.59	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 93.59	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 93.59	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 93.59	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 93.59	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 93.59	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 93.59	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 93.59	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 93.59	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 93.59	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 93.59	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 93.59	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 93.59	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 93.59	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 93.59	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 93.59	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 93.59	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 93.59	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 93.59	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 93.59	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 93.59	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 93.59	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 93.59	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 0.39

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

   if -1.5e8 < x < 2400

   1. Initial program 0.36

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

   2. Simplified 0.36

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

      [Start] 0.36	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 0.36	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 0.36	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 0.36	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 0.36	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 0.36	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 0.36	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 0.36	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 0.36	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 0.36	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 0.36	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 0.36	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 0.36	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 0.36	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 0.36	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 0.36	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 0.36	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 0.36	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 0.36	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 0.36	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 0.36	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 0.36	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 0.36	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Applied egg-rr 0.36

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

   4. Simplified 0.35

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

      [Start] 0.36	\[ \frac{-1 - \left(x + \frac{-1 + x}{-1 - x} \cdot x\right)}{\frac{-1 + x}{-1 - x} \cdot \left(-1 - x\right)} \]
      +-commutative [=>] 0.36	\[ \frac{-1 - \color{blue}{\left(\frac{-1 + x}{-1 - x} \cdot x + x\right)}}{\frac{-1 + x}{-1 - x} \cdot \left(-1 - x\right)} \]
      *-commutative [=>] 0.36	\[ \frac{-1 - \left(\color{blue}{x \cdot \frac{-1 + x}{-1 - x}} + x\right)}{\frac{-1 + x}{-1 - x} \cdot \left(-1 - x\right)} \]
      fma-def [=>] 0.35	\[ \frac{-1 - \color{blue}{\mathsf{fma}\left(x, \frac{-1 + x}{-1 - x}, x\right)}}{\frac{-1 + x}{-1 - x} \cdot \left(-1 - x\right)} \]
      +-commutative [=>] 0.35	\[ \frac{-1 - \mathsf{fma}\left(x, \frac{\color{blue}{x + -1}}{-1 - x}, x\right)}{\frac{-1 + x}{-1 - x} \cdot \left(-1 - x\right)} \]
      *-commutative [=>] 0.35	\[ \frac{-1 - \mathsf{fma}\left(x, \frac{x + -1}{-1 - x}, x\right)}{\color{blue}{\left(-1 - x\right) \cdot \frac{-1 + x}{-1 - x}}} \]
      +-commutative [=>] 0.35	\[ \frac{-1 - \mathsf{fma}\left(x, \frac{x + -1}{-1 - x}, x\right)}{\left(-1 - x\right) \cdot \frac{\color{blue}{x + -1}}{-1 - x}} \]

   5. Applied egg-rr 0.35

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

   if 2400 < x

   1. Initial program 92.52

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

   2. Simplified 92.52

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

      [Start] 92.52	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      sub-neg [=>] 92.52	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
      +-commutative [=>] 92.52	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
      remove-double-neg [<=] 92.52	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
      sub-neg [<=] 92.52	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
      distribute-neg-frac [=>] 92.52	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
      neg-sub0 [=>] 92.52	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      +-commutative [=>] 92.52	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      associate--r+ [=>] 92.52	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 92.52	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
      sub-neg [=>] 92.52	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
      metadata-eval [=>] 92.52	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
      /-rgt-identity [<=] 92.52	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
      neg-mul-1 [=>] 92.52	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
      metadata-eval [<=] 92.52	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
      *-commutative [=>] 92.52	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
      associate-/l* [=>] 92.52	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
      metadata-eval [=>] 92.52	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
      metadata-eval [=>] 92.52	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      metadata-eval [<=] 92.52	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
      associate-/l/ [=>] 92.52	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
      metadata-eval [=>] 92.52	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
      neg-mul-1 [<=] 92.52	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

   3. Taylor expanded in x around inf 0.5

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

   4. Simplified 0.05

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

      [Start] 0.5	\[ -\left(\frac{1}{{x}^{4}} + \left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)\right)\right) \]
      distribute-neg-in [=>] 0.5	\[ \color{blue}{\left(-\frac{1}{{x}^{4}}\right) + \left(-\left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)\right)\right)} \]
      +-commutative [=>] 0.5	\[ \color{blue}{\left(-\left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)\right)\right) + \left(-\frac{1}{{x}^{4}}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.28

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -150000000:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 2400:\\ \;\;\;\;\frac{-1 - \mathsf{fma}\left(x, \frac{x + -1}{-1 - x}, x\right)}{\frac{\frac{x + -1}{x + -1}}{\frac{1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\right) + \frac{-3}{{x}^{3}}\right) + \frac{-1}{{x}^{4}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.04%
Cost	1737

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+15} \lor \neg \left(x \leq 5 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{\frac{x \cdot \left(\left(x + 1\right) + \left(1 - x\right)\right)}{x + -1}}{-1 - x}\\ \end{array} \]

Alternative 2
Error	0.04%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+22} \lor \neg \left(x \leq 10^{+16}\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x + -1} + \frac{x \cdot -2}{-1 + x \cdot x}\\ \end{array} \]

Alternative 3
Error	0.29%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -115000000:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 10500:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x} + \frac{\frac{\frac{-3}{x}}{x}}{x}\\ \end{array} \]

Alternative 4
Error	0.47%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -115000000 \lor \neg \left(x \leq 165000000\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Alternative 5
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 6
Error	1.22%
Cost	840

\[\begin{array}{l} t_0 := \frac{-1}{x + -1}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t_0 + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{-2}{x}\\ \end{array} \]

Alternative 7
Error	1.11%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1.18:\\ \;\;\;\;\frac{-1}{x + -1} + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \]

Alternative 8
Error	1.48%
Cost	712

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

Alternative 9
Error	1.46%
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 10
Error	2.02%
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 11
Error	97.27%
Cost	64

\[-2 \]

Alternative 12
Error	90.05%
Cost	64

\[0.125 \]

Alternative 13
Error	49.85%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023104 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))