Asymptote C

Percentage Accurate: 53.8% → 99.8%

Time: 12.3s

Precision: binary64

Cost: 14660

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 + t_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{3 + \left(\frac{2}{x \cdot x} - \left(\frac{2}{x} + \frac{2}{{x}^{3}}\right)\right)}{t_0 \cdot \left(-1 - x\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, t_1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (/ (- -1.0 x) (+ x -1.0))))
   (if (<= (+ t_0 t_1) 5e-5)
     (/
      (/
       (+ 3.0 (- (/ 2.0 (* x x)) (+ (/ 2.0 x) (/ 2.0 (pow x 3.0)))))
       (* t_0 (- -1.0 x)))
      (* (/ (+ x 1.0) x) (/ (+ x -1.0) (+ x 1.0))))
     (fma (+ x -1.0) (/ x (fma x x -1.0)) t_1))))

C

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

↓

double code(double x) {
	double t_0 = x / (x + 1.0);
	double t_1 = (-1.0 - x) / (x + -1.0);
	double tmp;
	if ((t_0 + t_1) <= 5e-5) {
		tmp = ((3.0 + ((2.0 / (x * x)) - ((2.0 / x) + (2.0 / pow(x, 3.0))))) / (t_0 * (-1.0 - x))) / (((x + 1.0) / x) * ((x + -1.0) / (x + 1.0)));
	} else {
		tmp = fma((x + -1.0), (x / fma(x, x, -1.0)), t_1);
	}
	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)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = Float64(Float64(-1.0 - x) / Float64(x + -1.0))
	tmp = 0.0
	if (Float64(t_0 + t_1) <= 5e-5)
		tmp = Float64(Float64(Float64(3.0 + Float64(Float64(2.0 / Float64(x * x)) - Float64(Float64(2.0 / x) + Float64(2.0 / (x ^ 3.0))))) / Float64(t_0 * Float64(-1.0 - x))) / Float64(Float64(Float64(x + 1.0) / x) * Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	else
		tmp = fma(Float64(x + -1.0), Float64(x / fma(x, x, -1.0)), t_1);
	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_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + t$95$1), $MachinePrecision], 5e-5], N[(N[(N[(3.0 + N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] * N[(x / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.00000000000000024e-5

   1. Initial program 7.4%

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

   2. Applied egg-rr 7.3%

      \[\leadsto \color{blue}{\frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x} \cdot 1}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}} \]
      Step-by-step derivation

      [Start] 7.4%	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      clear-num [=>] 7.3%	\[ \color{blue}{\frac{1}{\frac{x + 1}{x}}} - \frac{x + 1}{x - 1} \]
      clear-num [=>] 7.3%	\[ \frac{1}{\frac{x + 1}{x}} - \color{blue}{\frac{1}{\frac{x - 1}{x + 1}}} \]
      frac-sub [=>] 7.3%	\[ \color{blue}{\frac{1 \cdot \frac{x - 1}{x + 1} - \frac{x + 1}{x} \cdot 1}{\frac{x + 1}{x} \cdot \frac{x - 1}{x + 1}}} \]
      *-un-lft-identity [<=] 7.3%	\[ \frac{\color{blue}{\frac{x - 1}{x + 1}} - \frac{x + 1}{x} \cdot 1}{\frac{x + 1}{x} \cdot \frac{x - 1}{x + 1}} \]
      sub-neg [=>] 7.3%	\[ \frac{\frac{\color{blue}{x + \left(-1\right)}}{x + 1} - \frac{x + 1}{x} \cdot 1}{\frac{x + 1}{x} \cdot \frac{x - 1}{x + 1}} \]
      metadata-eval [=>] 7.3%	\[ \frac{\frac{x + \color{blue}{-1}}{x + 1} - \frac{x + 1}{x} \cdot 1}{\frac{x + 1}{x} \cdot \frac{x - 1}{x + 1}} \]
      sub-neg [=>] 7.3%	\[ \frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x} \cdot 1}{\frac{x + 1}{x} \cdot \frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \]
      metadata-eval [=>] 7.3%	\[ \frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x} \cdot 1}{\frac{x + 1}{x} \cdot \frac{x + \color{blue}{-1}}{x + 1}} \]

   3. Applied egg-rr 7.3%

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

      [Start] 7.3%	\[ \frac{\frac{x + -1}{x + 1} - \frac{x + 1}{x} \cdot 1}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      *-rgt-identity [=>] 7.3%	\[ \frac{\frac{x + -1}{x + 1} - \color{blue}{\frac{x + 1}{x}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      frac-2neg [=>] 7.3%	\[ \frac{\color{blue}{\frac{-\left(x + -1\right)}{-\left(x + 1\right)}} - \frac{x + 1}{x}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      clear-num [=>] 7.3%	\[ \frac{\frac{-\left(x + -1\right)}{-\left(x + 1\right)} - \color{blue}{\frac{1}{\frac{x}{x + 1}}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      frac-sub [=>] 7.3%	\[ \frac{\color{blue}{\frac{\left(-\left(x + -1\right)\right) \cdot \frac{x}{x + 1} - \left(-\left(x + 1\right)\right) \cdot 1}{\left(-\left(x + 1\right)\right) \cdot \frac{x}{x + 1}}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]

   4. Simplified 7.3%

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

      [Start] 7.3%	\[ \frac{\frac{\left(-\left(x + -1\right)\right) \cdot \frac{x}{x + 1} - \left(-\left(x + 1\right)\right) \cdot 1}{\left(-\left(x + 1\right)\right) \cdot \frac{x}{x + 1}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      cancel-sign-sub [=>] 7.3%	\[ \frac{\frac{\color{blue}{\left(-\left(x + -1\right)\right) \cdot \frac{x}{x + 1} + \left(x + 1\right) \cdot 1}}{\left(-\left(x + 1\right)\right) \cdot \frac{x}{x + 1}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      *-commutative [=>] 7.3%	\[ \frac{\frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-\left(x + -1\right)\right)} + \left(x + 1\right) \cdot 1}{\left(-\left(x + 1\right)\right) \cdot \frac{x}{x + 1}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      *-rgt-identity [=>] 7.3%	\[ \frac{\frac{\frac{x}{x + 1} \cdot \left(-\left(x + -1\right)\right) + \color{blue}{\left(x + 1\right)}}{\left(-\left(x + 1\right)\right) \cdot \frac{x}{x + 1}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      fma-def [=>] 7.3%	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{x}{x + 1}, -\left(x + -1\right), x + 1\right)}}{\left(-\left(x + 1\right)\right) \cdot \frac{x}{x + 1}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      neg-sub0 [=>] 7.3%	\[ \frac{\frac{\mathsf{fma}\left(\frac{x}{x + 1}, \color{blue}{0 - \left(x + -1\right)}, x + 1\right)}{\left(-\left(x + 1\right)\right) \cdot \frac{x}{x + 1}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      +-commutative [=>] 7.3%	\[ \frac{\frac{\mathsf{fma}\left(\frac{x}{x + 1}, 0 - \color{blue}{\left(-1 + x\right)}, x + 1\right)}{\left(-\left(x + 1\right)\right) \cdot \frac{x}{x + 1}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      associate--r+ [=>] 7.3%	\[ \frac{\frac{\mathsf{fma}\left(\frac{x}{x + 1}, \color{blue}{\left(0 - -1\right) - x}, x + 1\right)}{\left(-\left(x + 1\right)\right) \cdot \frac{x}{x + 1}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      metadata-eval [=>] 7.3%	\[ \frac{\frac{\mathsf{fma}\left(\frac{x}{x + 1}, \color{blue}{1} - x, x + 1\right)}{\left(-\left(x + 1\right)\right) \cdot \frac{x}{x + 1}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      *-commutative [=>] 7.3%	\[ \frac{\frac{\mathsf{fma}\left(\frac{x}{x + 1}, 1 - x, x + 1\right)}{\color{blue}{\frac{x}{x + 1} \cdot \left(-\left(x + 1\right)\right)}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      neg-sub0 [=>] 7.3%	\[ \frac{\frac{\mathsf{fma}\left(\frac{x}{x + 1}, 1 - x, x + 1\right)}{\frac{x}{x + 1} \cdot \color{blue}{\left(0 - \left(x + 1\right)\right)}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      +-commutative [=>] 7.3%	\[ \frac{\frac{\mathsf{fma}\left(\frac{x}{x + 1}, 1 - x, x + 1\right)}{\frac{x}{x + 1} \cdot \left(0 - \color{blue}{\left(1 + x\right)}\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      associate--r+ [=>] 7.3%	\[ \frac{\frac{\mathsf{fma}\left(\frac{x}{x + 1}, 1 - x, x + 1\right)}{\frac{x}{x + 1} \cdot \color{blue}{\left(\left(0 - 1\right) - x\right)}}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]
      metadata-eval [=>] 7.3%	\[ \frac{\frac{\mathsf{fma}\left(\frac{x}{x + 1}, 1 - x, x + 1\right)}{\frac{x}{x + 1} \cdot \left(\color{blue}{-1} - x\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}} \]

   5. Taylor expanded in x around inf 100.0%

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

   6. Simplified 100.0%

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

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

   if 5.00000000000000024e-5 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

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

      [Start] 100.0%	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      flip-+ [=>] 100.0%	\[ \frac{x}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}} - \frac{x + 1}{x - 1} \]
      associate-/r/ [=>] 100.0%	\[ \color{blue}{\frac{x}{x \cdot x - 1 \cdot 1} \cdot \left(x - 1\right)} - \frac{x + 1}{x - 1} \]
      fma-neg [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\frac{x + 1}{x - 1}\right)} \]
      metadata-eval [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{x \cdot x - \color{blue}{1}}, x - 1, -\frac{x + 1}{x - 1}\right) \]
      fma-neg [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}, x - 1, -\frac{x + 1}{x - 1}\right) \]
      metadata-eval [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}, x - 1, -\frac{x + 1}{x - 1}\right) \]
      sub-neg [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \color{blue}{x + \left(-1\right)}, -\frac{x + 1}{x - 1}\right) \]
      metadata-eval [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + \color{blue}{-1}, -\frac{x + 1}{x - 1}\right) \]
      distribute-neg-frac [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \color{blue}{\frac{-\left(x + 1\right)}{x - 1}}\right) \]
      distribute-neg-in [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{\color{blue}{\left(-x\right) + \left(-1\right)}}{x - 1}\right) \]
      neg-mul-1 [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{\color{blue}{-1 \cdot x} + \left(-1\right)}{x - 1}\right) \]
      metadata-eval [<=] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{\color{blue}{\left(-1\right)} \cdot x + \left(-1\right)}{x - 1}\right) \]
      fma-def [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{\color{blue}{\mathsf{fma}\left(-1, x, -1\right)}}{x - 1}\right) \]
      metadata-eval [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{\mathsf{fma}\left(\color{blue}{-1}, x, -1\right)}{x - 1}\right) \]
      metadata-eval [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{\mathsf{fma}\left(-1, x, \color{blue}{-1}\right)}{x - 1}\right) \]
      sub-neg [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{\mathsf{fma}\left(-1, x, -1\right)}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{\mathsf{fma}\left(-1, x, -1\right)}{x + \color{blue}{-1}}\right) \]

   3. Simplified 100.0%

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

      [Start] 100.0%	\[ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, x, -1\right)}, x + -1, \frac{\mathsf{fma}\left(-1, x, -1\right)}{x + -1}\right) \]
      fma-udef [=>] 100.0%	\[ \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + -1\right) + \frac{\mathsf{fma}\left(-1, x, -1\right)}{x + -1}} \]
      *-commutative [=>] 100.0%	\[ \color{blue}{\left(x + -1\right) \cdot \frac{x}{\mathsf{fma}\left(x, x, -1\right)}} + \frac{\mathsf{fma}\left(-1, x, -1\right)}{x + -1} \]
      fma-def [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \frac{\mathsf{fma}\left(-1, x, -1\right)}{x + -1}\right)} \]
      fma-udef [=>] 100.0%	\[ \mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \frac{\color{blue}{-1 \cdot x + -1}}{x + -1}\right) \]
      neg-mul-1 [<=] 100.0%	\[ \mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \frac{\color{blue}{\left(-x\right)} + -1}{x + -1}\right) \]
      metadata-eval [<=] 100.0%	\[ \mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \frac{\left(-x\right) + \color{blue}{\left(-1\right)}}{x + -1}\right) \]
      distribute-neg-in [<=] 100.0%	\[ \mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \frac{\color{blue}{-\left(x + 1\right)}}{x + -1}\right) \]
      +-commutative [=>] 100.0%	\[ \mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \frac{-\color{blue}{\left(1 + x\right)}}{x + -1}\right) \]
      distribute-neg-in [=>] 100.0%	\[ \mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \frac{\color{blue}{\left(-1\right) + \left(-x\right)}}{x + -1}\right) \]
      metadata-eval [=>] 100.0%	\[ \mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \frac{\color{blue}{-1} + \left(-x\right)}{x + -1}\right) \]
      unsub-neg [=>] 100.0%	\[ \mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \frac{\color{blue}{-1 - x}}{x + -1}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{3 + \left(\frac{2}{x \cdot x} - \left(\frac{2}{x} + \frac{2}{{x}^{3}}\right)\right)}{\frac{x}{x + 1} \cdot \left(-1 - x\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, \frac{-1 - x}{x + -1}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	14660

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 + t_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{3 + \left(\frac{2}{x \cdot x} - \left(\frac{2}{x} + \frac{2}{{x}^{3}}\right)\right)}{t_0 \cdot \left(-1 - x\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x + -1, \frac{x}{\mathsf{fma}\left(x, x, -1\right)}, t_1\right)\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	9860

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;t_0 + \frac{-1 - x}{x + -1} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{3 + \left(\frac{2}{x \cdot x} - \left(\frac{2}{x} + \frac{2}{{x}^{3}}\right)\right)}{t_0 \cdot \left(-1 - x\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + x \cdot -3}{x \cdot \left(x + 1\right)}}{1 - \frac{1}{x}}\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	8196

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + x \cdot -3}{x \cdot \left(x + 1\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4
Accuracy	99.8%
Cost	2500

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{\frac{2}{x \cdot x} - \frac{3}{x}}{1 - \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + x \cdot -3}{x \cdot \left(x + 1\right)}}{\frac{x + 1}{x} \cdot \frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	1988

\[\begin{array}{l} t_0 := 1 - \frac{1}{x}\\ \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0:\\ \;\;\;\;\frac{\frac{2}{x \cdot x} - \frac{3}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + x \cdot -3}{x \cdot \left(x + 1\right)}}{t_0}\\ \end{array} \]

Alternative 6
Accuracy	99.7%
Cost	1860

\[\begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x} - \frac{3}{x}}{1 - \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	99.7%
Cost	1732

\[\begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-3 - \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
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 9
Accuracy	98.6%
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
Accuracy	98.1%
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
Accuracy	50.5%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))