3frac (problem 3.3.3)

Percentage Accurate: 85.0% → 99.1%

Time: 13.0s

Precision: binary64

Cost: 15433

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ t_1 := x \cdot x - x\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-5} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\frac{t_1 + \left(1 + x\right) \cdot \left(x + -2 \cdot \left(x + -1\right)\right)}{\left(1 + x\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \end{array} \]

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))))
        (t_1 (- (* x x) x)))
   (if (or (<= t_0 -2e-5) (not (<= t_0 0.0)))
     (/ (+ t_1 (* (+ 1.0 x) (+ x (* -2.0 (+ x -1.0))))) (* (+ 1.0 x) t_1))
     (+ (/ 2.0 (pow x 5.0)) (/ 2.0 (pow x 3.0))))))

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 t_1 = (x * x) - x;
	double tmp;
	if ((t_0 <= -2e-5) || !(t_0 <= 0.0)) {
		tmp = (t_1 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / ((1.0 + x) * t_1);
	} else {
		tmp = (2.0 / pow(x, 5.0)) + (2.0 / pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    t_1 = (x * x) - x
    if ((t_0 <= (-2d-5)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = (t_1 + ((1.0d0 + x) * (x + ((-2.0d0) * (x + (-1.0d0)))))) / ((1.0d0 + x) * t_1)
    else
        tmp = (2.0d0 / (x ** 5.0d0)) + (2.0d0 / (x ** 3.0d0))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_1 = (x * x) - x;
	double tmp;
	if ((t_0 <= -2e-5) || !(t_0 <= 0.0)) {
		tmp = (t_1 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / ((1.0 + x) * t_1);
	} else {
		tmp = (2.0 / Math.pow(x, 5.0)) + (2.0 / Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

↓

def code(x):
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))
	t_1 = (x * x) - x
	tmp = 0
	if (t_0 <= -2e-5) or not (t_0 <= 0.0):
		tmp = (t_1 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / ((1.0 + x) * t_1)
	else:
		tmp = (2.0 / math.pow(x, 5.0)) + (2.0 / math.pow(x, 3.0))
	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)))
	t_1 = Float64(Float64(x * x) - x)
	tmp = 0.0
	if ((t_0 <= -2e-5) || !(t_0 <= 0.0))
		tmp = Float64(Float64(t_1 + Float64(Float64(1.0 + x) * Float64(x + Float64(-2.0 * Float64(x + -1.0))))) / Float64(Float64(1.0 + x) * t_1));
	else
		tmp = Float64(Float64(2.0 / (x ^ 5.0)) + Float64(2.0 / (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

↓

function tmp_2 = code(x)
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	t_1 = (x * x) - x;
	tmp = 0.0;
	if ((t_0 <= -2e-5) || ~((t_0 <= 0.0)))
		tmp = (t_1 + ((1.0 + x) * (x + (-2.0 * (x + -1.0))))) / ((1.0 + x) * t_1);
	else
		tmp = (2.0 / (x ^ 5.0)) + (2.0 / (x ^ 3.0));
	end
	tmp_2 = 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]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-5], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(t$95$1 + N[(N[(1.0 + x), $MachinePrecision] * N[(x + N[(-2.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	85.0%
Target	99.5%
Herbie	99.1%

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -2.00000000000000016e-5 or 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.4%

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

   2. Simplified 99.4%

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

      [Start] 99.4%	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 99.4%	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 99.4%	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 99.4%	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 99.4%	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 99.4%	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 99.4%	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 99.4%	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 99.4%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 99.4%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 99.3%

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

      [Start] 99.4%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      frac-2neg [=>] 99.4%	\[ \frac{1}{1 + x} - \left(\color{blue}{\frac{-2}{-x}} - \frac{1}{x + -1}\right) \]
      frac-2neg [=>] 99.4%	\[ \frac{1}{1 + x} - \left(\frac{-2}{-x} - \color{blue}{\frac{-1}{-\left(x + -1\right)}}\right) \]
      metadata-eval [=>] 99.4%	\[ \frac{1}{1 + x} - \left(\frac{-2}{-x} - \frac{\color{blue}{-1}}{-\left(x + -1\right)}\right) \]
      frac-sub [=>] 99.3%	\[ \frac{1}{1 + x} - \color{blue}{\frac{\left(-2\right) \cdot \left(-\left(x + -1\right)\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)}} \]
      metadata-eval [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{\color{blue}{-2} \cdot \left(-\left(x + -1\right)\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
      +-commutative [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(-\color{blue}{\left(-1 + x\right)}\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
      distribute-neg-in [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
      metadata-eval [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(\color{blue}{1} + \left(-x\right)\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
      sub-neg [<=] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \color{blue}{\left(1 - x\right)} - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
      +-commutative [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\color{blue}{\left(-1 + x\right)}\right)} \]
      distribute-neg-in [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)}} \]
      metadata-eval [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(\color{blue}{1} + \left(-x\right)\right)} \]
      sub-neg [<=] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \color{blue}{\left(1 - x\right)}} \]

   4. Simplified 99.3%

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

      [Start] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(1 - x\right)} \]
      cancel-sign-sub [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{\color{blue}{-2 \cdot \left(1 - x\right) + x \cdot -1}}{\left(-x\right) \cdot \left(1 - x\right)} \]
      *-commutative [<=] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) + \color{blue}{-1 \cdot x}}{\left(-x\right) \cdot \left(1 - x\right)} \]
      neg-mul-1 [<=] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) + \color{blue}{\left(-x\right)}}{\left(-x\right) \cdot \left(1 - x\right)} \]
      unsub-neg [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{\color{blue}{-2 \cdot \left(1 - x\right) - x}}{\left(-x\right) \cdot \left(1 - x\right)} \]
      sub-neg [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\left(-x\right) \cdot \color{blue}{\left(1 + \left(-x\right)\right)}} \]
      +-commutative [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\left(-x\right) \cdot \color{blue}{\left(\left(-x\right) + 1\right)}} \]
      distribute-lft-in [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{\left(-x\right) \cdot \left(-x\right) + \left(-x\right) \cdot 1}} \]
      sqr-neg [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{x \cdot x} + \left(-x\right) \cdot 1} \]
      unpow2 [<=] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{{x}^{2}} + \left(-x\right) \cdot 1} \]
      *-rgt-identity [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{{x}^{2} + \color{blue}{\left(-x\right)}} \]
      sub-neg [<=] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{{x}^{2} - x}} \]
      unpow2 [=>] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{x \cdot x} - x} \]

   5. Applied egg-rr 100.0%

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

      [Start] 99.3%	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{x \cdot x - x} \]
      frac-sub [=>] 100.0%	\[ \color{blue}{\frac{1 \cdot \left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(-2 \cdot \left(1 - x\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}} \]
      *-un-lft-identity [<=] 100.0%	\[ \frac{\color{blue}{\left(x \cdot x - x\right)} - \left(1 + x\right) \cdot \left(-2 \cdot \left(1 - x\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]
      *-commutative [=>] 100.0%	\[ \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\color{blue}{\left(1 - x\right) \cdot -2} - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   if -2.00000000000000016e-5 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 65.2%

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

   2. Simplified 65.2%

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

      [Start] 65.2%	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 65.2%	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 65.2%	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 65.2%	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 65.2%	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 65.2%	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 65.2%	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 65.2%	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 65.2%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 65.2%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 99.1%

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

   4. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}} \]
      Step-by-step derivation

      [Start] 99.1%	\[ 2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]
      associate-*r/ [=>] 99.1%	\[ \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + 2 \cdot \frac{1}{{x}^{3}} \]
      metadata-eval [=>] 99.1%	\[ \frac{\color{blue}{2}}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]
      associate-*r/ [=>] 99.1%	\[ \frac{2}{{x}^{5}} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} \]
      metadata-eval [=>] 99.1%	\[ \frac{2}{{x}^{5}} + \frac{\color{blue}{2}}{{x}^{3}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -2 \cdot 10^{-5} \lor \neg \left(\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 0\right):\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(1 + x\right) \cdot \left(x + -2 \cdot \left(x + -1\right)\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	15433

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ t_1 := x \cdot x - x\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-5} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\frac{t_1 + \left(1 + x\right) \cdot \left(x + -2 \cdot \left(x + -1\right)\right)}{\left(1 + x\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	8712

\[\begin{array}{l} t_0 := \left(1 - x\right) \cdot \left(x \cdot 0.5\right)\\ t_1 := x \cdot x - x\\ t_2 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-21}:\\ \;\;\;\;\frac{t_0 + \left(1 + x\right) \cdot \left(x \cdot -0.5 + \left(x + -1\right)\right)}{\left(1 + x\right) \cdot t_0}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(1 + x\right) \cdot \left(x + -2 \cdot \left(x + -1\right)\right)}{\left(1 + x\right) \cdot t_1}\\ \end{array} \]

Alternative 3
Accuracy	85.0%
Cost	960

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

Alternative 4
Accuracy	76.6%
Cost	585

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

Alternative 5
Accuracy	83.8%
Cost	448

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

Alternative 6
Accuracy	51.7%
Cost	192

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

Alternative 7
Accuracy	3.3%
Cost	64

\[1 \]

Alternative 8
Accuracy	3.3%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023271 
(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))))