3frac (problem 3.3.3)

Average Error: 15.76% → 0.85%

Time: 14.1s

Precision: binary64

Cost: 9092

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot x + \left(\left(1 + x\right) \cdot \left(\left(x + 2\right) + x \cdot -2\right) - x\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{\left(1 + x\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{1 - x \cdot x} + \frac{-2}{x}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (+ (/ 1.0 (+ 1.0 x)) (/ -2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_0 -5e-5)
     (/
      (+ (* x x) (- (* (+ 1.0 x) (+ (+ x 2.0) (* x -2.0))) x))
      (* (+ 1.0 x) (fma x x (- x))))
     (if (<= t_0 0.0)
       (/
        (+ (/ 2.0 x) (/ 4.0 (* x x)))
        (* (+ 1.0 x) (/ (+ x -1.0) (/ (+ x -2.0) x))))
       (+ (/ (* x -2.0) (- 1.0 (* x x))) (/ -2.0 x))))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) + (-2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_0 <= -5e-5) {
		tmp = ((x * x) + (((1.0 + x) * ((x + 2.0) + (x * -2.0))) - x)) / ((1.0 + x) * fma(x, x, -x));
	} else if (t_0 <= 0.0) {
		tmp = ((2.0 / x) + (4.0 / (x * x))) / ((1.0 + x) * ((x + -1.0) / ((x + -2.0) / x)));
	} else {
		tmp = ((x * -2.0) / (1.0 - (x * x))) + (-2.0 / x);
	}
	return tmp;
}

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

↓

function code(x)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= -5e-5)
		tmp = Float64(Float64(Float64(x * x) + Float64(Float64(Float64(1.0 + x) * Float64(Float64(x + 2.0) + Float64(x * -2.0))) - x)) / Float64(Float64(1.0 + x) * fma(x, x, Float64(-x))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(2.0 / x) + Float64(4.0 / Float64(x * x))) / Float64(Float64(1.0 + x) * Float64(Float64(x + -1.0) / Float64(Float64(x + -2.0) / x))));
	else
		tmp = Float64(Float64(Float64(x * -2.0) / Float64(1.0 - Float64(x * x))) + Float64(-2.0 / x));
	end
	return tmp
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-5], N[(N[(N[(x * x), $MachinePrecision] + N[(N[(N[(1.0 + x), $MachinePrecision] * N[(N[(x + 2.0), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * N[(x * x + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(2.0 / x), $MachinePrecision] + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(x + -1.0), $MachinePrecision] / N[(N[(x + -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * -2.0), $MachinePrecision] / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	15.76%
Target	0.45%
Herbie	0.85%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 0.05

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

   2. Simplified 0.05

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

      [Start] 0.05	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 0.05	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.05	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 0.05	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 0.05	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 0.05	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 0.05	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 0.05	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.05	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 0.05	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 0.05

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

   4. Simplified 0.05

      \[\leadsto \color{blue}{\frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(x \cdot 2 - \left(x - -2\right)\right)\right)}{\left(x + 1\right) \cdot \mathsf{fma}\left(x, x, -x\right)}} \]
      Proof

      [Start] 0.05	\[ \frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.05	\[ \frac{x \cdot x - \left(x + \color{blue}{\left(x + 1\right)} \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.05	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \color{blue}{\left(\left(2 \cdot x - x\right) + -2\right)}\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      associate-+l- [=>] 0.05	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \color{blue}{\left(2 \cdot x - \left(x - -2\right)\right)}\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      *-commutative [=>] 0.05	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(\color{blue}{x \cdot 2} - \left(x - -2\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.05	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(x \cdot 2 - \left(x - -2\right)\right)\right)}{\color{blue}{\left(x + 1\right)} \cdot \mathsf{fma}\left(x, x, -x\right)} \]

   if -5.00000000000000024e-5 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 31

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

   2. Simplified 31

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

      [Start] 31	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 31	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 31	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 31	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 31	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 31	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 31	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 31	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 31	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 31	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 31.08

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

   4. Simplified 81.89

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

      [Start] 31.08	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{x + -1} \]
      associate-/l/ [=>] 81.89	\[ \frac{1}{1 + x} - \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{\left(x + -1\right) \cdot x}} \]
      +-commutative [=>] 81.89	\[ \frac{1}{1 + x} - \frac{\color{blue}{\left(2 \cdot x - x\right) + -2}}{\left(x + -1\right) \cdot x} \]
      associate-+l- [=>] 81.89	\[ \frac{1}{1 + x} - \frac{\color{blue}{2 \cdot x - \left(x - -2\right)}}{\left(x + -1\right) \cdot x} \]
      *-commutative [=>] 81.89	\[ \frac{1}{1 + x} - \frac{\color{blue}{x \cdot 2} - \left(x - -2\right)}{\left(x + -1\right) \cdot x} \]
      *-commutative [<=] 81.89	\[ \frac{1}{1 + x} - \frac{x \cdot 2 - \left(x - -2\right)}{\color{blue}{x \cdot \left(x + -1\right)}} \]

   5. Applied egg-rr 45.52

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

   6. Taylor expanded in x around inf 0.53

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

   7. Simplified 0.53

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

      [Start] 0.53	\[ \frac{4 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{x}}{\left(x + 1\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}} \]
      +-commutative [<=] 0.53	\[ \frac{\color{blue}{2 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}}}{\left(x + 1\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}} \]
      associate-*r/ [=>] 0.53	\[ \frac{\color{blue}{\frac{2 \cdot 1}{x}} + 4 \cdot \frac{1}{{x}^{2}}}{\left(x + 1\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}} \]
      metadata-eval [=>] 0.53	\[ \frac{\frac{\color{blue}{2}}{x} + 4 \cdot \frac{1}{{x}^{2}}}{\left(x + 1\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}} \]
      associate-*r/ [=>] 0.53	\[ \frac{\frac{2}{x} + \color{blue}{\frac{4 \cdot 1}{{x}^{2}}}}{\left(x + 1\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}} \]
      metadata-eval [=>] 0.53	\[ \frac{\frac{2}{x} + \frac{\color{blue}{4}}{{x}^{2}}}{\left(x + 1\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}} \]
      unpow2 [=>] 0.53	\[ \frac{\frac{2}{x} + \frac{4}{\color{blue}{x \cdot x}}}{\left(x + 1\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}} \]

   if 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 2.15

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

   2. Simplified 2.15

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

      [Start] 2.15	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 2.15	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 2.15	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 2.15	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 2.15	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 2.15	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 2.15	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 2.15	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 2.15	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 2.15	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 2.19

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

   4. Applied egg-rr 2.25

      \[\leadsto \color{blue}{\frac{-\left(x + x\right)}{-\mathsf{fma}\left(x, x, -1\right)}} + \left(-\frac{2}{x}\right) \]

   5. Simplified 2.24

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

      [Start] 2.25	\[ \frac{-\left(x + x\right)}{-\mathsf{fma}\left(x, x, -1\right)} + \left(-\frac{2}{x}\right) \]
      neg-mul-1 [=>] 2.25	\[ \frac{\color{blue}{-1 \cdot \left(x + x\right)}}{-\mathsf{fma}\left(x, x, -1\right)} + \left(-\frac{2}{x}\right) \]
      count-2 [=>] 2.25	\[ \frac{-1 \cdot \color{blue}{\left(2 \cdot x\right)}}{-\mathsf{fma}\left(x, x, -1\right)} + \left(-\frac{2}{x}\right) \]
      associate-*r* [=>] 2.25	\[ \frac{\color{blue}{\left(-1 \cdot 2\right) \cdot x}}{-\mathsf{fma}\left(x, x, -1\right)} + \left(-\frac{2}{x}\right) \]
      metadata-eval [=>] 2.25	\[ \frac{\color{blue}{-2} \cdot x}{-\mathsf{fma}\left(x, x, -1\right)} + \left(-\frac{2}{x}\right) \]
      neg-mul-1 [=>] 2.25	\[ \frac{-2 \cdot x}{\color{blue}{-1 \cdot \mathsf{fma}\left(x, x, -1\right)}} + \left(-\frac{2}{x}\right) \]
      fma-udef [=>] 2.24	\[ \frac{-2 \cdot x}{-1 \cdot \color{blue}{\left(x \cdot x + -1\right)}} + \left(-\frac{2}{x}\right) \]
      distribute-lft-in [=>] 2.24	\[ \frac{-2 \cdot x}{\color{blue}{-1 \cdot \left(x \cdot x\right) + -1 \cdot -1}} + \left(-\frac{2}{x}\right) \]
      associate-*l* [<=] 2.24	\[ \frac{-2 \cdot x}{\color{blue}{\left(-1 \cdot x\right) \cdot x} + -1 \cdot -1} + \left(-\frac{2}{x}\right) \]
      neg-mul-1 [<=] 2.24	\[ \frac{-2 \cdot x}{\color{blue}{\left(-x\right)} \cdot x + -1 \cdot -1} + \left(-\frac{2}{x}\right) \]
      metadata-eval [=>] 2.24	\[ \frac{-2 \cdot x}{\left(-x\right) \cdot x + \color{blue}{1}} + \left(-\frac{2}{x}\right) \]
      +-commutative [<=] 2.24	\[ \frac{-2 \cdot x}{\color{blue}{1 + \left(-x\right) \cdot x}} + \left(-\frac{2}{x}\right) \]
      cancel-sign-sub-inv [<=] 2.24	\[ \frac{-2 \cdot x}{\color{blue}{1 - x \cdot x}} + \left(-\frac{2}{x}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.85

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot x + \left(\left(1 + x\right) \cdot \left(\left(x + 2\right) + x \cdot -2\right) - x\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{\left(1 + x\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{1 - x \cdot x} + \frac{-2}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.85%
Cost	3528

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ t_2 := 1 - x \cdot x\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;t_0 + \left(\frac{-2}{x} + \left(1 + x\right) \cdot \frac{-1}{t_2}\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{\left(1 + x\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{t_2} + \frac{-2}{x}\\ \end{array} \]

Alternative 2
Error	0.85%
Cost	3528

\[\begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ t_1 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t_0 + \left(1 + x\right) \cdot \left(2 - x\right)}{1 + x}}{t_0}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{\left(1 + x\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{1 - x \cdot x} + \frac{-2}{x}\\ \end{array} \]

Alternative 3
Error	1.12%
Cost	3144

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{2}{x}}{\left(1 + x\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{1 - x \cdot x} + \frac{-2}{x}\\ \end{array} \]

Alternative 4
Error	0.73%
Cost	1609

\[\begin{array}{l} \mathbf{if}\;x \leq -310000 \lor \neg \left(x \leq 280000\right):\\ \;\;\;\;\frac{\frac{2}{x}}{\left(1 + x\right) \cdot \frac{x + -1}{\frac{x + -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \left(\frac{-2}{x} + \left(1 + x\right) \cdot \frac{-1}{1 - x \cdot x}\right)\\ \end{array} \]

Alternative 5
Error	15.78%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+41} \lor \neg \left(x \leq 10^{+21}\right):\\ \;\;\;\;1 + \left(-1 + \frac{-2}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{1 - x \cdot x} + \frac{-2}{x}\\ \end{array} \]

Alternative 6
Error	15.76%
Cost	960

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

Alternative 7
Error	23.94%
Cost	585

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

Alternative 8
Error	23.54%
Cost	584

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

Alternative 9
Error	16.95%
Cost	448

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

Alternative 10
Error	47.99%
Cost	192

\[\frac{-2}{x} \]

Alternative 11
Error	96.73%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))