3frac (problem 3.3.3)

Average Accuracy: 84.6% → 99.2%

Time: 16.1s

Precision: binary64

Cost: 16328

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -0.0005:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + x\right) + \left(x + -1\right)\right) - 2 \cdot \mathsf{fma}\left(x, x, -1\right)}{x \cdot \mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + \left(-2 + x \cdot -2\right)\right) \cdot \left(1 - x\right) - \mathsf{fma}\left(x, x, x\right)}{\left(1 - x\right) \cdot \mathsf{fma}\left(x, x, x\right)}\\ \end{array} \]

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 -0.0005)
     (/
      (- (* x (+ (+ 1.0 x) (+ x -1.0))) (* 2.0 (fma x x -1.0)))
      (* x (fma x x -1.0)))
     (if (<= t_0 0.0)
       (+ (/ 2.0 (pow x 5.0)) (/ 2.0 (pow x 3.0)))
       (/
        (- (* (+ x (+ -2.0 (* x -2.0))) (- 1.0 x)) (fma x x x))
        (* (- 1.0 x) (fma x x 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 <= -0.0005) {
		tmp = ((x * ((1.0 + x) + (x + -1.0))) - (2.0 * fma(x, x, -1.0))) / (x * fma(x, x, -1.0));
	} else if (t_0 <= 0.0) {
		tmp = (2.0 / pow(x, 5.0)) + (2.0 / pow(x, 3.0));
	} else {
		tmp = (((x + (-2.0 + (x * -2.0))) * (1.0 - x)) - fma(x, x, x)) / ((1.0 - x) * fma(x, x, 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 <= -0.0005)
		tmp = Float64(Float64(Float64(x * Float64(Float64(1.0 + x) + Float64(x + -1.0))) - Float64(2.0 * fma(x, x, -1.0))) / Float64(x * fma(x, x, -1.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(2.0 / (x ^ 5.0)) + Float64(2.0 / (x ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(x + Float64(-2.0 + Float64(x * -2.0))) * Float64(1.0 - x)) - fma(x, x, x)) / Float64(Float64(1.0 - x) * fma(x, x, 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, -0.0005], N[(N[(N[(x * N[(N[(1.0 + x), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + N[(-2.0 + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] - N[(x * x + x), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - x), $MachinePrecision] * N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	84.6%
Target	99.6%
Herbie	99.2%

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -5.0000000000000001e-4

   1. Initial program 99.8%

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

   2. Applied egg-rr 99.8%

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

      [Start] 99.8	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      +-commutative [=>] 99.8	\[ \color{blue}{\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)} \]
      associate-+r- [=>] 99.8	\[ \color{blue}{\left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) - \frac{2}{x}} \]
      sub-neg [=>] 99.8	\[ \left(\frac{1}{\color{blue}{x + \left(-1\right)}} + \frac{1}{x + 1}\right) - \frac{2}{x} \]
      metadata-eval [=>] 99.8	\[ \left(\frac{1}{x + \color{blue}{-1}} + \frac{1}{x + 1}\right) - \frac{2}{x} \]
      +-commutative [=>] 99.8	\[ \left(\frac{1}{x + -1} + \frac{1}{\color{blue}{1 + x}}\right) - \frac{2}{x} \]

   3. Applied egg-rr 100.0%

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

      [Start] 99.8	\[ \left(\frac{1}{x + -1} + \frac{1}{1 + x}\right) - \frac{2}{x} \]
      frac-add [=>] 99.8	\[ \color{blue}{\frac{1 \cdot \left(1 + x\right) + \left(x + -1\right) \cdot 1}{\left(x + -1\right) \cdot \left(1 + x\right)}} - \frac{2}{x} \]
      frac-sub [=>] 99.9	\[ \color{blue}{\frac{\left(1 \cdot \left(1 + x\right) + \left(x + -1\right) \cdot 1\right) \cdot x - \left(\left(x + -1\right) \cdot \left(1 + x\right)\right) \cdot 2}{\left(\left(x + -1\right) \cdot \left(1 + x\right)\right) \cdot x}} \]

   if -5.0000000000000001e-4 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 69.5%

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

   2. Taylor expanded in x around inf 99.3%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}} \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}} \]
      Proof

      [Start] 99.3	\[ 2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]
      associate-*r/ [=>] 99.3	\[ \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + 2 \cdot \frac{1}{{x}^{3}} \]
      metadata-eval [=>] 99.3	\[ \frac{\color{blue}{2}}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]
      associate-*r/ [=>] 99.3	\[ \frac{2}{{x}^{5}} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} \]
      metadata-eval [=>] 99.3	\[ \frac{2}{{x}^{5}} + \frac{\color{blue}{2}}{{x}^{3}} \]

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

   1. Initial program 98.9%

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

   2. Applied egg-rr 98.9%

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

      [Start] 98.9	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      frac-sub [=>] 98.9	\[ \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1} \]
      frac-2neg [=>] 98.9	\[ \frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \color{blue}{\frac{-1}{-\left(x - 1\right)}} \]
      metadata-eval [=>] 98.9	\[ \frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{\color{blue}{-1}}{-\left(x - 1\right)} \]
      frac-add [=>] 99.9	\[ \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(-\left(x - 1\right)\right) + \left(\left(x + 1\right) \cdot x\right) \cdot -1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(-\left(x - 1\right)\right)}} \]

   3. Simplified 99.9%

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

      [Start] 98.9	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, \mathsf{fma}\left(-1, x, 1\right), \left(x \cdot \left(1 + x\right)\right) \cdot -1\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      *-commutative [=>] 98.9	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, \mathsf{fma}\left(-1, x, 1\right), \color{blue}{-1 \cdot \left(x \cdot \left(1 + x\right)\right)}\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      neg-mul-1 [<=] 98.9	\[ \frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, \mathsf{fma}\left(-1, x, 1\right), \color{blue}{-x \cdot \left(1 + x\right)}\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      fma-neg [<=] 99.9	\[ \frac{\color{blue}{\left(x - \left(1 + x\right) \cdot 2\right) \cdot \mathsf{fma}\left(-1, x, 1\right) - x \cdot \left(1 + x\right)}}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      *-commutative [=>] 99.9	\[ \frac{\left(x - \color{blue}{2 \cdot \left(1 + x\right)}\right) \cdot \mathsf{fma}\left(-1, x, 1\right) - x \cdot \left(1 + x\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      cancel-sign-sub-inv [=>] 99.9	\[ \frac{\color{blue}{\left(x + \left(-2\right) \cdot \left(1 + x\right)\right)} \cdot \mathsf{fma}\left(-1, x, 1\right) - x \cdot \left(1 + x\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      metadata-eval [=>] 99.9	\[ \frac{\left(x + \color{blue}{-2} \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right) - x \cdot \left(1 + x\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      distribute-lft-in [=>] 99.9	\[ \frac{\left(x + \color{blue}{\left(-2 \cdot 1 + -2 \cdot x\right)}\right) \cdot \mathsf{fma}\left(-1, x, 1\right) - x \cdot \left(1 + x\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      metadata-eval [=>] 99.9	\[ \frac{\left(x + \left(\color{blue}{-2} + -2 \cdot x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right) - x \cdot \left(1 + x\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      fma-udef [=>] 99.9	\[ \frac{\left(x + \left(-2 + -2 \cdot x\right)\right) \cdot \color{blue}{\left(-1 \cdot x + 1\right)} - x \cdot \left(1 + x\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      neg-mul-1 [<=] 99.9	\[ \frac{\left(x + \left(-2 + -2 \cdot x\right)\right) \cdot \left(\color{blue}{\left(-x\right)} + 1\right) - x \cdot \left(1 + x\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      +-commutative [<=] 99.9	\[ \frac{\left(x + \left(-2 + -2 \cdot x\right)\right) \cdot \color{blue}{\left(1 + \left(-x\right)\right)} - x \cdot \left(1 + x\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      sub-neg [<=] 99.9	\[ \frac{\left(x + \left(-2 + -2 \cdot x\right)\right) \cdot \color{blue}{\left(1 - x\right)} - x \cdot \left(1 + x\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      *-commutative [=>] 99.9	\[ \frac{\left(x + \left(-2 + -2 \cdot x\right)\right) \cdot \left(1 - x\right) - \color{blue}{\left(1 + x\right) \cdot x}}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      +-commutative [=>] 99.9	\[ \frac{\left(x + \left(-2 + -2 \cdot x\right)\right) \cdot \left(1 - x\right) - \color{blue}{\left(x + 1\right)} \cdot x}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      distribute-lft1-in [<=] 99.4	\[ \frac{\left(x + \left(-2 + -2 \cdot x\right)\right) \cdot \left(1 - x\right) - \color{blue}{\left(x \cdot x + x\right)}}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]
      fma-def [=>] 99.9	\[ \frac{\left(x + \left(-2 + -2 \cdot x\right)\right) \cdot \left(1 - x\right) - \color{blue}{\mathsf{fma}\left(x, x, x\right)}}{\left(x \cdot \left(1 + x\right)\right) \cdot \mathsf{fma}\left(-1, x, 1\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -0.0005:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + x\right) + \left(x + -1\right)\right) - 2 \cdot \mathsf{fma}\left(x, x, -1\right)}{x \cdot \mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + \left(-2 + x \cdot -2\right)\right) \cdot \left(1 - x\right) - \mathsf{fma}\left(x, x, x\right)}{\left(1 - x\right) \cdot \mathsf{fma}\left(x, x, x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	16201

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -0.0005 \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\frac{x \cdot \left(\left(1 + x\right) + \left(x + -1\right)\right) - 2 \cdot \mathsf{fma}\left(x, x, -1\right)}{x \cdot \mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	15432

\[\begin{array}{l} t_0 := \frac{1}{x + -1}\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + t_0\\ \mathbf{if}\;t_1 \leq -0.0005:\\ \;\;\;\;\frac{\mathsf{fma}\left(x + 2 \cdot \left(-1 - x\right), x + -1, x \cdot \left(1 + x\right)\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(x + -1\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 - x, \frac{\frac{1}{x}}{1 + x}, t_0\right)\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	9416

\[\begin{array}{l} t_0 := \frac{1}{x + -1}\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + t_0\\ \mathbf{if}\;t_1 \leq -0.0005:\\ \;\;\;\;t_0 + \frac{x + 2 \cdot \left(-1 - x\right)}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 - x, \frac{\frac{1}{x}}{1 + x}, t_0\right)\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	9416

\[\begin{array}{l} t_0 := \frac{1}{x + -1}\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + t_0\\ \mathbf{if}\;t_1 \leq -0.0005:\\ \;\;\;\;\frac{\mathsf{fma}\left(x + 2 \cdot \left(-1 - x\right), x + -1, x \cdot \left(1 + x\right)\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(x + -1\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 - x, \frac{\frac{1}{x}}{1 + x}, t_0\right)\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	8712

\[\begin{array}{l} t_0 := x + 2 \cdot \left(-1 - x\right)\\ t_1 := \frac{1}{x + -1}\\ t_2 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + t_1\\ \mathbf{if}\;t_2 \leq -0.0005:\\ \;\;\;\;t_1 + \frac{t_0}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1 + t_0 \cdot \frac{\frac{1}{x}}{1 + x}\\ \end{array} \]

Alternative 6
Accuracy	84.6%
Cost	1216

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

Alternative 7
Accuracy	84.6%
Cost	1216

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

Alternative 8
Accuracy	84.6%
Cost	960

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

Alternative 9
Accuracy	84.6%
Cost	960

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

Alternative 10
Accuracy	75.9%
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 11
Accuracy	76.4%
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 12
Accuracy	83.5%
Cost	448

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

Alternative 13
Accuracy	52.7%
Cost	192

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

Reproduce

herbie shell --seed 2023159 
(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))))