Asymptote C

Average Accuracy: 53.4% → 100.0%

Time: 17.4s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(-1 + x \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -5e+17)
   (/ -3.0 x)
   (if (<= x 100000000.0)
     (* (/ 1.0 (fma x x -1.0)) (+ -1.0 (* x -3.0)))
     (/ (+ -3.0 (/ 2.0 x)) (+ x -1.0)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -5e+17) {
		tmp = -3.0 / x;
	} else if (x <= 100000000.0) {
		tmp = (1.0 / fma(x, x, -1.0)) * (-1.0 + (x * -3.0));
	} else {
		tmp = (-3.0 + (2.0 / x)) / (x + -1.0);
	}
	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 <= -5e+17)
		tmp = Float64(-3.0 / x);
	elseif (x <= 100000000.0)
		tmp = Float64(Float64(1.0 / fma(x, x, -1.0)) * Float64(-1.0 + Float64(x * -3.0)));
	else
		tmp = Float64(Float64(-3.0 + Float64(2.0 / x)) / Float64(x + -1.0));
	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, -5e+17], N[(-3.0 / x), $MachinePrecision], If[LessEqual[x, 100000000.0], N[(N[(1.0 / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-3.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e17

   1. Initial program 5.0%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -5e17 < x < 1e8

   1. Initial program 99.1%

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

   2. Applied egg-rr 99.1%

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

      [Start] 99.1	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      clear-num [=>] 99.0	\[ \frac{x}{x + 1} - \color{blue}{\frac{1}{\frac{x - 1}{x + 1}}} \]
      frac-sub [=>] 99.1	\[ \color{blue}{\frac{x \cdot \frac{x - 1}{x + 1} - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}}} \]
      *-commutative [<=] 99.1	\[ \frac{x \cdot \frac{x - 1}{x + 1} - \color{blue}{1 \cdot \left(x + 1\right)}}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      *-un-lft-identity [<=] 99.1	\[ \frac{x \cdot \frac{x - 1}{x + 1} - \color{blue}{\left(x + 1\right)}}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      fma-neg [=>] 99.1	\[ \frac{\color{blue}{\mathsf{fma}\left(x, \frac{x - 1}{x + 1}, -\left(x + 1\right)\right)}}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      sub-neg [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{\color{blue}{x + \left(-1\right)}}{x + 1}, -\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      metadata-eval [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + \color{blue}{-1}}{x + 1}, -\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      distribute-neg-in [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \color{blue}{\left(-x\right) + \left(-1\right)}\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      neg-mul-1 [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \color{blue}{-1 \cdot x} + \left(-1\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      metadata-eval [<=] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \color{blue}{\left(-1\right)} \cdot x + \left(-1\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      fma-def [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \color{blue}{\mathsf{fma}\left(-1, x, -1\right)}\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      metadata-eval [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(\color{blue}{-1}, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      metadata-eval [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(-1, x, \color{blue}{-1}\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      sub-neg [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \]
      metadata-eval [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{x + \color{blue}{-1}}{x + 1}} \]

   3. Simplified 99.1%

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

      [Start] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      +-commutative [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{\color{blue}{1 + x}}, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      fma-udef [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, \color{blue}{-1 \cdot x + -1}\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      neg-mul-1 [<=] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, \color{blue}{\left(-x\right)} + -1\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      +-commutative [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, \color{blue}{-1 + \left(-x\right)}\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      unsub-neg [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, \color{blue}{-1 - x}\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      associate-*r/ [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + -1\right)}{x + 1}}} \]
      metadata-eval [<=] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\left(x + 1\right) \cdot \left(x + \color{blue}{\left(-1\right)}\right)}{x + 1}} \]
      sub-neg [<=] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\left(x + 1\right) \cdot \color{blue}{\left(x - 1\right)}}{x + 1}} \]
      difference-of-sqr--1 [<=] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\color{blue}{x \cdot x + -1}}{x + 1}} \]
      fma-udef [<=] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}{x + 1}} \]
      +-commutative [=>] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{1 + x}}} \]

   4. Applied egg-rr 99.1%

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

      [Start] 99.1	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      *-un-lft-identity [=>] 99.1	\[ \frac{\color{blue}{1 \cdot \mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      div-inv [=>] 99.1	\[ \frac{1 \cdot \mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\color{blue}{\mathsf{fma}\left(x, x, -1\right) \cdot \frac{1}{1 + x}}} \]
      times-frac [=>] 99.1	\[ \color{blue}{\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{1}{1 + x}}} \]
      frac-2neg [=>] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \color{blue}{\frac{-\left(x + -1\right)}{-\left(1 + x\right)}}, -1 - x\right)}{\frac{1}{1 + x}} \]
      +-commutative [=>] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{-\color{blue}{\left(-1 + x\right)}}{-\left(1 + x\right)}, -1 - x\right)}{\frac{1}{1 + x}} \]
      distribute-neg-in [=>] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{\color{blue}{\left(--1\right) + \left(-x\right)}}{-\left(1 + x\right)}, -1 - x\right)}{\frac{1}{1 + x}} \]
      metadata-eval [=>] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{\color{blue}{1} + \left(-x\right)}{-\left(1 + x\right)}, -1 - x\right)}{\frac{1}{1 + x}} \]
      sub-neg [<=] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{\color{blue}{1 - x}}{-\left(1 + x\right)}, -1 - x\right)}{\frac{1}{1 + x}} \]
      distribute-neg-in [=>] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{1 - x}{\color{blue}{\left(-1\right) + \left(-x\right)}}, -1 - x\right)}{\frac{1}{1 + x}} \]
      metadata-eval [=>] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{1 - x}{\color{blue}{-1} + \left(-x\right)}, -1 - x\right)}{\frac{1}{1 + x}} \]
      sub-neg [<=] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{1 - x}{\color{blue}{-1 - x}}, -1 - x\right)}{\frac{1}{1 + x}} \]
      frac-2neg [=>] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\color{blue}{\frac{-1}{-\left(1 + x\right)}}} \]
      metadata-eval [=>] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\frac{\color{blue}{-1}}{-\left(1 + x\right)}} \]
      distribute-neg-in [=>] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\frac{-1}{\color{blue}{\left(-1\right) + \left(-x\right)}}} \]
      metadata-eval [=>] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\frac{-1}{\color{blue}{-1} + \left(-x\right)}} \]
      sub-neg [<=] 99.1	\[ \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\frac{-1}{\color{blue}{-1 - x}}} \]

   5. Taylor expanded in x around 0 100.0%

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

   if 1e8 < x

   1. Initial program 6.0%

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

   2. Applied egg-rr 6.0%

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

      [Start] 6.0	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
      clear-num [=>] 6.0	\[ \frac{x}{x + 1} - \color{blue}{\frac{1}{\frac{x - 1}{x + 1}}} \]
      frac-sub [=>] 6.8	\[ \color{blue}{\frac{x \cdot \frac{x - 1}{x + 1} - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}}} \]
      *-commutative [<=] 6.8	\[ \frac{x \cdot \frac{x - 1}{x + 1} - \color{blue}{1 \cdot \left(x + 1\right)}}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      *-un-lft-identity [<=] 6.8	\[ \frac{x \cdot \frac{x - 1}{x + 1} - \color{blue}{\left(x + 1\right)}}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      fma-neg [=>] 6.0	\[ \frac{\color{blue}{\mathsf{fma}\left(x, \frac{x - 1}{x + 1}, -\left(x + 1\right)\right)}}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      sub-neg [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{\color{blue}{x + \left(-1\right)}}{x + 1}, -\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      metadata-eval [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + \color{blue}{-1}}{x + 1}, -\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      distribute-neg-in [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \color{blue}{\left(-x\right) + \left(-1\right)}\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      neg-mul-1 [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \color{blue}{-1 \cdot x} + \left(-1\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      metadata-eval [<=] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \color{blue}{\left(-1\right)} \cdot x + \left(-1\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      fma-def [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \color{blue}{\mathsf{fma}\left(-1, x, -1\right)}\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      metadata-eval [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(\color{blue}{-1}, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      metadata-eval [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(-1, x, \color{blue}{-1}\right)\right)}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}} \]
      sub-neg [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{\color{blue}{x + \left(-1\right)}}{x + 1}} \]
      metadata-eval [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{x + \color{blue}{-1}}{x + 1}} \]

   3. Simplified 6.0%

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

      [Start] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      +-commutative [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{\color{blue}{1 + x}}, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      fma-udef [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, \color{blue}{-1 \cdot x + -1}\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      neg-mul-1 [<=] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, \color{blue}{\left(-x\right)} + -1\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      +-commutative [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, \color{blue}{-1 + \left(-x\right)}\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      unsub-neg [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, \color{blue}{-1 - x}\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}} \]
      associate-*r/ [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\color{blue}{\frac{\left(x + 1\right) \cdot \left(x + -1\right)}{x + 1}}} \]
      metadata-eval [<=] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\left(x + 1\right) \cdot \left(x + \color{blue}{\left(-1\right)}\right)}{x + 1}} \]
      sub-neg [<=] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\left(x + 1\right) \cdot \color{blue}{\left(x - 1\right)}}{x + 1}} \]
      difference-of-sqr--1 [<=] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\color{blue}{x \cdot x + -1}}{x + 1}} \]
      fma-udef [<=] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}{x + 1}} \]
      +-commutative [=>] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{1 + x}}} \]

   4. Applied egg-rr 6.0%

      \[\leadsto \color{blue}{0 + \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{x + -1}} \]
      Proof

      [Start] 6.0	\[ \frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      add-log-exp [=>] 6.0	\[ \color{blue}{\log \left(e^{\frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}}}\right)} \]
      *-un-lft-identity [=>] 6.0	\[ \log \color{blue}{\left(1 \cdot e^{\frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}}}\right)} \]
      log-prod [=>] 6.0	\[ \color{blue}{\log 1 + \log \left(e^{\frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}}}\right)} \]
      metadata-eval [=>] 6.0	\[ \color{blue}{0} + \log \left(e^{\frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}}}\right) \]
      add-log-exp [<=] 6.0	\[ 0 + \color{blue}{\frac{\mathsf{fma}\left(x, \frac{x + -1}{1 + x}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}}} \]
      frac-2neg [=>] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \color{blue}{\frac{-\left(x + -1\right)}{-\left(1 + x\right)}}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      +-commutative [=>] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \frac{-\color{blue}{\left(-1 + x\right)}}{-\left(1 + x\right)}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      distribute-neg-in [=>] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \frac{\color{blue}{\left(--1\right) + \left(-x\right)}}{-\left(1 + x\right)}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      metadata-eval [=>] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \frac{\color{blue}{1} + \left(-x\right)}{-\left(1 + x\right)}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      sub-neg [<=] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \frac{\color{blue}{1 - x}}{-\left(1 + x\right)}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      distribute-neg-in [=>] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \frac{1 - x}{\color{blue}{\left(-1\right) + \left(-x\right)}}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      metadata-eval [=>] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \frac{1 - x}{\color{blue}{-1} + \left(-x\right)}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      sub-neg [<=] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \frac{1 - x}{\color{blue}{-1 - x}}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, -1\right)}{1 + x}} \]
      metadata-eval [<=] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\frac{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}{1 + x}} \]
      fma-neg [<=] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{\frac{\color{blue}{x \cdot x - 1}}{1 + x}} \]

   5. Simplified 6.8%

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

      [Start] 6.0	\[ 0 + \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{x + -1} \]
      +-lft-identity [=>] 6.0	\[ \color{blue}{\frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1 - x\right)}{x + -1}} \]
      fma-def [<=] 6.8	\[ \frac{\color{blue}{x \cdot \frac{1 - x}{-1 - x} + \left(-1 - x\right)}}{x + -1} \]
      associate-+r- [=>] 6.8	\[ \frac{\color{blue}{\left(x \cdot \frac{1 - x}{-1 - x} + -1\right) - x}}{x + -1} \]
      fma-udef [<=] 6.8	\[ \frac{\color{blue}{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1\right)} - x}{x + -1} \]
      +-commutative [=>] 6.8	\[ \frac{\mathsf{fma}\left(x, \frac{1 - x}{-1 - x}, -1\right) - x}{\color{blue}{-1 + x}} \]

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{x} - 3}}{-1 + x} \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{\frac{2}{x} + -3}}{-1 + x} \]
      Proof

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

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(-1 + x \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \leq -15000:\\ \;\;\;\;\frac{\frac{2}{x} - \left(3 + \frac{2}{x \cdot x}\right)}{x + -1}\\ \mathbf{elif}\;x \leq 430000:\\ \;\;\;\;\frac{\left(-1 + \frac{x \cdot \left(1 - x\right)}{-1 - x}\right) - x}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -500000:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 350000:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -31000:\\ \;\;\;\;\frac{\frac{2}{x} - \left(3 + \frac{2}{x \cdot x}\right)}{x + -1}\\ \mathbf{elif}\;x \leq 350000:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{-1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

Alternative 4
Accuracy	99.9%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -400000:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 350000:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \end{array} \]

Alternative 5
Accuracy	99.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.88\right):\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;x - \left(-1 + -2 \cdot \left(x + x \cdot x\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	99.1%
Cost	841

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

Alternative 7
Accuracy	99.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.82\right):\\ \;\;\;\;\frac{-3 + \frac{2}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array} \]

Alternative 8
Accuracy	98.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot -0.3333333333333333 + 0.1111111111111111}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array} \]

Alternative 9
Accuracy	98.7%
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	50.2%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023153 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))