3frac (problem 3.3.3)

Average Error: 84.7% → 98.9%

Time: 21.2s

Precision: binary64

Cost: 8713.00

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -10 \lor \neg \left(t_1 \leq 10^{-29}\right):\\ \;\;\;\;t_0 + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{-2 \cdot \left(x + -1\right) - x}{x \cdot \left(1 - x\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \]

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))) (t_1 (+ (- t_0 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (or (<= t_1 -10.0) (not (<= t_1 1e-29)))
     (+
      t_0
      (+
       (/ -2.0 x)
       (+ (/ -2.0 x) (/ (- (* -2.0 (+ x -1.0)) x) (* x (- 1.0 x))))))
     (/ 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);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if ((t_1 <= -10.0) || !(t_1 <= 1e-29)) {
		tmp = t_0 + ((-2.0 / x) + ((-2.0 / x) + (((-2.0 * (x + -1.0)) - x) / (x * (1.0 - x)))));
	} else {
		tmp = 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)
    t_1 = (t_0 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if ((t_1 <= (-10.0d0)) .or. (.not. (t_1 <= 1d-29))) then
        tmp = t_0 + (((-2.0d0) / x) + (((-2.0d0) / x) + ((((-2.0d0) * (x + (-1.0d0))) - x) / (x * (1.0d0 - x)))))
    else
        tmp = 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);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if ((t_1 <= -10.0) || !(t_1 <= 1e-29)) {
		tmp = t_0 + ((-2.0 / x) + ((-2.0 / x) + (((-2.0 * (x + -1.0)) - x) / (x * (1.0 - x)))));
	} else {
		tmp = 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)
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if (t_1 <= -10.0) or not (t_1 <= 1e-29):
		tmp = t_0 + ((-2.0 / x) + ((-2.0 / x) + (((-2.0 * (x + -1.0)) - x) / (x * (1.0 - x)))))
	else:
		tmp = 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(1.0 / Float64(1.0 + x))
	t_1 = Float64(Float64(t_0 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if ((t_1 <= -10.0) || !(t_1 <= 1e-29))
		tmp = Float64(t_0 + Float64(Float64(-2.0 / x) + Float64(Float64(-2.0 / x) + Float64(Float64(Float64(-2.0 * Float64(x + -1.0)) - x) / Float64(x * Float64(1.0 - x))))));
	else
		tmp = 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);
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if ((t_1 <= -10.0) || ~((t_1 <= 1e-29)))
		tmp = t_0 + ((-2.0 / x) + ((-2.0 / x) + (((-2.0 * (x + -1.0)) - x) / (x * (1.0 - x)))));
	else
		tmp = 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[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -10.0], N[Not[LessEqual[t$95$1, 1e-29]], $MachinePrecision]], N[(t$95$0 + N[(N[(-2.0 / x), $MachinePrecision] + N[(N[(-2.0 / x), $MachinePrecision] + N[(N[(N[(-2.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]]]

Target

Original	84.7%
Target	99.5%
Herbie	98.9%

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -10 or 9.99999999999999943e-30 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.1

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

   2. Simplified 99.1

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

      [Start] 99.1	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 99.1	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 99.1	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 99.1	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 99.1	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 99.1	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 99.1	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 99.1	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 99.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 99.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 99.1

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

   4. Simplified 99.1

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

      [Start] 99.1	\[ \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right) \]
      sub-neg [=>] 99.1	\[ \color{blue}{\left(\frac{1}{1 + x} + \left(-\frac{2}{x}\right)\right)} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right) \]
      mul-1-neg [<=] 99.1	\[ \left(\frac{1}{1 + x} + \color{blue}{-1 \cdot \frac{2}{x}}\right) + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right) \]
      associate-+l+ [=>] 99.1	\[ \color{blue}{\frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right)} \]
      +-commutative [=>] 99.1	\[ \frac{1}{\color{blue}{x + 1}} + \left(-1 \cdot \frac{2}{x} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      associate-*r/ [=>] 99.1	\[ \frac{1}{x + 1} + \left(\color{blue}{\frac{-1 \cdot 2}{x}} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      metadata-eval [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{\color{blue}{-2}}{x} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      sub-neg [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \color{blue}{\left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) + \left(-\frac{1}{1 - x}\right)\right)}\right) \]
      mul-1-neg [<=] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\mathsf{fma}\left(\color{blue}{-1 \cdot \frac{2}{x}}, 1, \frac{2}{x}\right) + \left(-\frac{1}{1 - x}\right)\right)\right) \]
      fma-udef [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\color{blue}{\left(\left(-1 \cdot \frac{2}{x}\right) \cdot 1 + \frac{2}{x}\right)} + \left(-\frac{1}{1 - x}\right)\right)\right) \]
      associate-+l+ [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \color{blue}{\left(\left(-1 \cdot \frac{2}{x}\right) \cdot 1 + \left(\frac{2}{x} + \left(-\frac{1}{1 - x}\right)\right)\right)}\right) \]
      associate-*r/ [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\color{blue}{\frac{-1 \cdot 2}{x}} \cdot 1 + \left(\frac{2}{x} + \left(-\frac{1}{1 - x}\right)\right)\right)\right) \]
      metadata-eval [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{\color{blue}{-2}}{x} \cdot 1 + \left(\frac{2}{x} + \left(-\frac{1}{1 - x}\right)\right)\right)\right) \]
      associate-*l/ [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\color{blue}{\frac{-2 \cdot 1}{x}} + \left(\frac{2}{x} + \left(-\frac{1}{1 - x}\right)\right)\right)\right) \]
      metadata-eval [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{\color{blue}{-2}}{x} + \left(\frac{2}{x} + \left(-\frac{1}{1 - x}\right)\right)\right)\right) \]
      distribute-neg-frac [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \left(\frac{2}{x} + \color{blue}{\frac{-1}{1 - x}}\right)\right)\right) \]

   5. Applied egg-rr 99.1

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

   6. Simplified 99.1

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

      [Start] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \mathsf{fma}\left(2, 1 - x, -x\right) \cdot \frac{1}{x \cdot \left(1 - x\right)}\right)\right) \]
      associate-*r/ [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \color{blue}{\frac{\mathsf{fma}\left(2, 1 - x, -x\right) \cdot 1}{x \cdot \left(1 - x\right)}}\right)\right) \]
      *-rgt-identity [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{\color{blue}{\mathsf{fma}\left(2, 1 - x, -x\right)}}{x \cdot \left(1 - x\right)}\right)\right) \]
      fma-udef [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{\color{blue}{2 \cdot \left(1 - x\right) + \left(-x\right)}}{x \cdot \left(1 - x\right)}\right)\right) \]
      unsub-neg [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{\color{blue}{2 \cdot \left(1 - x\right) - x}}{x \cdot \left(1 - x\right)}\right)\right) \]
      metadata-eval [<=] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{\color{blue}{\left(-2 \cdot -1\right)} \cdot \left(1 - x\right) - x}{x \cdot \left(1 - x\right)}\right)\right) \]
      associate-*r* [<=] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{\color{blue}{-2 \cdot \left(-1 \cdot \left(1 - x\right)\right)} - x}{x \cdot \left(1 - x\right)}\right)\right) \]
      neg-mul-1 [<=] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{-2 \cdot \color{blue}{\left(-\left(1 - x\right)\right)} - x}{x \cdot \left(1 - x\right)}\right)\right) \]
      sub-neg [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{-2 \cdot \left(-\color{blue}{\left(1 + \left(-x\right)\right)}\right) - x}{x \cdot \left(1 - x\right)}\right)\right) \]
      distribute-neg-in [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{-2 \cdot \color{blue}{\left(\left(-1\right) + \left(-\left(-x\right)\right)\right)} - x}{x \cdot \left(1 - x\right)}\right)\right) \]
      metadata-eval [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{-2 \cdot \left(\color{blue}{-1} + \left(-\left(-x\right)\right)\right) - x}{x \cdot \left(1 - x\right)}\right)\right) \]
      remove-double-neg [=>] 99.1	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{-2 \cdot \left(-1 + \color{blue}{x}\right) - x}{x \cdot \left(1 - x\right)}\right)\right) \]

   if -10 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 9.99999999999999943e-30

   1. Initial program 70.1

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

   2. Simplified 70.1

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

      [Start] 70.1	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 70.1	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 70.1	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 70.1	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 70.1	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 70.1	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 70.1	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 70.1	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 70.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 70.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 98.7

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -10 \lor \neg \left(\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-29}\right):\\ \;\;\;\;\frac{1}{1 + x} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \frac{-2 \cdot \left(x + -1\right) - x}{x \cdot \left(1 - x\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error	94.2%
Cost	3016.00

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 10^{-29}:\\ \;\;\;\;\frac{x \cdot -2}{\left(1 - x \cdot x\right) \cdot \left(x \cdot x - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} + x \cdot -2\\ \end{array} \]

Alternative 2
Error	95.7%
Cost	1088.00

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

Alternative 3
Error	84.7%
Cost	960.00

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

Alternative 4
Error	76.2%
Cost	585.00

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

Alternative 5
Error	76.5%
Cost	584.00

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

Alternative 6
Error	83.5%
Cost	456.00

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{0}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x}\\ \end{array} \]

Alternative 7
Error	83.5%
Cost	448.00

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

Alternative 8
Error	51.5%
Cost	192.00

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

Reproduce

herbie shell --seed 2023093 
(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))))