3frac (problem 3.3.3)

Average Error: 9.7 → 0.3

Time: 13.2s

Precision: binary64

Cost: 8712

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ t_1 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{t_0 + \left(1 + x\right) \cdot \left(2 - x\right)}{1 + x}}{t_0}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(1 + x\right) \cdot \left(x + -2\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(1 - x\right)\right)}\\ \end{array} \]

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 (* x (+ x -1.0)))
        (t_1 (+ (+ (/ 1.0 (+ 1.0 x)) (/ -2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -5e-12)
     (/ (/ (+ t_0 (* (+ 1.0 x) (- 2.0 x))) (+ 1.0 x)) t_0)
     (if (<= t_1 0.0)
       (* 2.0 (pow x -3.0))
       (/
        (+ (* x (- 1.0 x)) (* (+ 1.0 x) (+ x -2.0)))
        (* x (* (+ 1.0 x) (- 1.0 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 = x * (x + -1.0);
	double t_1 = ((1.0 / (1.0 + x)) + (-2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -5e-12) {
		tmp = ((t_0 + ((1.0 + x) * (2.0 - x))) / (1.0 + x)) / t_0;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * pow(x, -3.0);
	} else {
		tmp = ((x * (1.0 - x)) + ((1.0 + x) * (x + -2.0))) / (x * ((1.0 + x) * (1.0 - x)));
	}
	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 = x * (x + (-1.0d0))
    t_1 = ((1.0d0 / (1.0d0 + x)) + ((-2.0d0) / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_1 <= (-5d-12)) then
        tmp = ((t_0 + ((1.0d0 + x) * (2.0d0 - x))) / (1.0d0 + x)) / t_0
    else if (t_1 <= 0.0d0) then
        tmp = 2.0d0 * (x ** (-3.0d0))
    else
        tmp = ((x * (1.0d0 - x)) + ((1.0d0 + x) * (x + (-2.0d0)))) / (x * ((1.0d0 + x) * (1.0d0 - x)))
    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 = x * (x + -1.0);
	double t_1 = ((1.0 / (1.0 + x)) + (-2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -5e-12) {
		tmp = ((t_0 + ((1.0 + x) * (2.0 - x))) / (1.0 + x)) / t_0;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 * Math.pow(x, -3.0);
	} else {
		tmp = ((x * (1.0 - x)) + ((1.0 + x) * (x + -2.0))) / (x * ((1.0 + x) * (1.0 - x)));
	}
	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 = x * (x + -1.0)
	t_1 = ((1.0 / (1.0 + x)) + (-2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_1 <= -5e-12:
		tmp = ((t_0 + ((1.0 + x) * (2.0 - x))) / (1.0 + x)) / t_0
	elif t_1 <= 0.0:
		tmp = 2.0 * math.pow(x, -3.0)
	else:
		tmp = ((x * (1.0 - x)) + ((1.0 + x) * (x + -2.0))) / (x * ((1.0 + x) * (1.0 - 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(x * Float64(x + -1.0))
	t_1 = 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_1 <= -5e-12)
		tmp = Float64(Float64(Float64(t_0 + Float64(Float64(1.0 + x) * Float64(2.0 - x))) / Float64(1.0 + x)) / t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 * (x ^ -3.0));
	else
		tmp = Float64(Float64(Float64(x * Float64(1.0 - x)) + Float64(Float64(1.0 + x) * Float64(x + -2.0))) / Float64(x * Float64(Float64(1.0 + x) * Float64(1.0 - x))));
	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 = x * (x + -1.0);
	t_1 = ((1.0 / (1.0 + x)) + (-2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= -5e-12)
		tmp = ((t_0 + ((1.0 + x) * (2.0 - x))) / (1.0 + x)) / t_0;
	elseif (t_1 <= 0.0)
		tmp = 2.0 * (x ^ -3.0);
	else
		tmp = ((x * (1.0 - x)) + ((1.0 + x) * (x + -2.0))) / (x * ((1.0 + x) * (1.0 - x)));
	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[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1, -5e-12], N[(N[(N[(t$95$0 + N[(N[(1.0 + x), $MachinePrecision] * N[(2.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	9.7
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -4.9999999999999997e-12

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 0.1	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.1	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 0.1	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 0.1	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 0.1	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 0.1	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 0.1	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 0.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 0.1

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

   4. Simplified 0.1

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

      [Start] 0.1	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]
      *-commutative [=>] 0.1	\[ \frac{1}{1 + x} - \color{blue}{\frac{1}{x + -1} \cdot \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1}} \]
      /-rgt-identity [=>] 0.1	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{x}} \]
      associate-*r/ [=>] 0.1	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{1}{x + -1} \cdot \left(-2 + \left(2 \cdot x - x\right)\right)}{x}} \]
      *-commutative [=>] 0.1	\[ \frac{1}{1 + x} - \frac{\frac{1}{x + -1} \cdot \left(-2 + \left(\color{blue}{x \cdot 2} - x\right)\right)}{x} \]

   5. Applied egg-rr 0.1

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

   6. Simplified 0.1

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

      [Start] 0.1	\[ \frac{1}{1 + x} + \left(-\frac{\frac{1}{x + -1} \cdot \left(-2 + \left(x \cdot 2 - x\right)\right)}{x}\right) \]
      sub-neg [<=] 0.1	\[ \color{blue}{\frac{1}{1 + x} - \frac{\frac{1}{x + -1} \cdot \left(-2 + \left(x \cdot 2 - x\right)\right)}{x}} \]
      associate-*l/ [<=] 0.1	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{1}{x + -1}}{x} \cdot \left(-2 + \left(x \cdot 2 - x\right)\right)} \]
      +-commutative [=>] 0.1	\[ \frac{1}{\color{blue}{x + 1}} - \frac{\frac{1}{x + -1}}{x} \cdot \left(-2 + \left(x \cdot 2 - x\right)\right) \]
      associate-/l/ [=>] 0.1	\[ \frac{1}{x + 1} - \color{blue}{\frac{1}{x \cdot \left(x + -1\right)}} \cdot \left(-2 + \left(x \cdot 2 - x\right)\right) \]
      associate-*l/ [=>] 0.1	\[ \frac{1}{x + 1} - \color{blue}{\frac{1 \cdot \left(-2 + \left(x \cdot 2 - x\right)\right)}{x \cdot \left(x + -1\right)}} \]
      *-lft-identity [=>] 0.1	\[ \frac{1}{x + 1} - \frac{\color{blue}{-2 + \left(x \cdot 2 - x\right)}}{x \cdot \left(x + -1\right)} \]
      sub-neg [=>] 0.1	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{\left(x \cdot 2 + \left(-x\right)\right)}}{x \cdot \left(x + -1\right)} \]
      *-commutative [=>] 0.1	\[ \frac{1}{x + 1} - \frac{-2 + \left(\color{blue}{2 \cdot x} + \left(-x\right)\right)}{x \cdot \left(x + -1\right)} \]
      neg-mul-1 [=>] 0.1	\[ \frac{1}{x + 1} - \frac{-2 + \left(2 \cdot x + \color{blue}{-1 \cdot x}\right)}{x \cdot \left(x + -1\right)} \]
      distribute-rgt-out [=>] 0.1	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{x \cdot \left(2 + -1\right)}}{x \cdot \left(x + -1\right)} \]
      metadata-eval [=>] 0.1	\[ \frac{1}{x + 1} - \frac{-2 + x \cdot \color{blue}{1}}{x \cdot \left(x + -1\right)} \]
      *-rgt-identity [=>] 0.1	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{x}}{x \cdot \left(x + -1\right)} \]
      remove-double-neg [<=] 0.1	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{\left(-\left(-x\right)\right)}}{x \cdot \left(x + -1\right)} \]
      mul-1-neg [<=] 0.1	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{-1 \cdot \left(-x\right)}}{x \cdot \left(x + -1\right)} \]
      mul-1-neg [=>] 0.1	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{\left(-\left(-x\right)\right)}}{x \cdot \left(x + -1\right)} \]
      remove-double-neg [=>] 0.1	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{x}}{x \cdot \left(x + -1\right)} \]

   7. Applied egg-rr 0.0

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

   if -4.9999999999999997e-12 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 19.3

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

   2. Simplified 19.3

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

      [Start] 19.3	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 19.3	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.3	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 19.3	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 19.3	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 19.3	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 19.3	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 19.3	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.3	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 19.3	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 0.6

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

   4. Applied egg-rr 0.1

      \[\leadsto \color{blue}{{x}^{-3} \cdot 2} \]

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

   1. Initial program 1.5

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

   2. Simplified 1.5

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

      [Start] 1.5	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 1.5	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 1.5	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 1.5	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 1.5	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 1.5	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 1.5	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 1.5	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 1.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 1.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 1.5

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

   4. Simplified 1.5

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

      [Start] 1.5	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]
      *-commutative [=>] 1.5	\[ \frac{1}{1 + x} - \color{blue}{\frac{1}{x + -1} \cdot \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1}} \]
      /-rgt-identity [=>] 1.5	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{x}} \]
      associate-*r/ [=>] 1.5	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{1}{x + -1} \cdot \left(-2 + \left(2 \cdot x - x\right)\right)}{x}} \]
      *-commutative [=>] 1.5	\[ \frac{1}{1 + x} - \frac{\frac{1}{x + -1} \cdot \left(-2 + \left(\color{blue}{x \cdot 2} - x\right)\right)}{x} \]

   5. Applied egg-rr 1.5

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

   6. Simplified 1.6

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

      [Start] 1.5	\[ \frac{1}{1 + x} + \left(-\frac{\frac{1}{x + -1} \cdot \left(-2 + \left(x \cdot 2 - x\right)\right)}{x}\right) \]
      sub-neg [<=] 1.5	\[ \color{blue}{\frac{1}{1 + x} - \frac{\frac{1}{x + -1} \cdot \left(-2 + \left(x \cdot 2 - x\right)\right)}{x}} \]
      associate-*l/ [<=] 1.6	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{1}{x + -1}}{x} \cdot \left(-2 + \left(x \cdot 2 - x\right)\right)} \]
      +-commutative [=>] 1.6	\[ \frac{1}{\color{blue}{x + 1}} - \frac{\frac{1}{x + -1}}{x} \cdot \left(-2 + \left(x \cdot 2 - x\right)\right) \]
      associate-/l/ [=>] 1.6	\[ \frac{1}{x + 1} - \color{blue}{\frac{1}{x \cdot \left(x + -1\right)}} \cdot \left(-2 + \left(x \cdot 2 - x\right)\right) \]
      associate-*l/ [=>] 1.6	\[ \frac{1}{x + 1} - \color{blue}{\frac{1 \cdot \left(-2 + \left(x \cdot 2 - x\right)\right)}{x \cdot \left(x + -1\right)}} \]
      *-lft-identity [=>] 1.6	\[ \frac{1}{x + 1} - \frac{\color{blue}{-2 + \left(x \cdot 2 - x\right)}}{x \cdot \left(x + -1\right)} \]
      sub-neg [=>] 1.6	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{\left(x \cdot 2 + \left(-x\right)\right)}}{x \cdot \left(x + -1\right)} \]
      *-commutative [=>] 1.6	\[ \frac{1}{x + 1} - \frac{-2 + \left(\color{blue}{2 \cdot x} + \left(-x\right)\right)}{x \cdot \left(x + -1\right)} \]
      neg-mul-1 [=>] 1.6	\[ \frac{1}{x + 1} - \frac{-2 + \left(2 \cdot x + \color{blue}{-1 \cdot x}\right)}{x \cdot \left(x + -1\right)} \]
      distribute-rgt-out [=>] 1.6	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{x \cdot \left(2 + -1\right)}}{x \cdot \left(x + -1\right)} \]
      metadata-eval [=>] 1.6	\[ \frac{1}{x + 1} - \frac{-2 + x \cdot \color{blue}{1}}{x \cdot \left(x + -1\right)} \]
      *-rgt-identity [=>] 1.6	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{x}}{x \cdot \left(x + -1\right)} \]
      remove-double-neg [<=] 1.6	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{\left(-\left(-x\right)\right)}}{x \cdot \left(x + -1\right)} \]
      mul-1-neg [<=] 1.6	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{-1 \cdot \left(-x\right)}}{x \cdot \left(x + -1\right)} \]
      mul-1-neg [=>] 1.6	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{\left(-\left(-x\right)\right)}}{x \cdot \left(x + -1\right)} \]
      remove-double-neg [=>] 1.6	\[ \frac{1}{x + 1} - \frac{-2 + \color{blue}{x}}{x \cdot \left(x + -1\right)} \]

   7. Applied egg-rr 0.9

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

   8. Simplified 0.9

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

      [Start] 0.9	\[ \frac{\left(x + -1\right) \cdot \left(-x\right) - \left(-1 - x\right) \cdot \left(-2 + x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      *-commutative [=>] 0.9	\[ \frac{\color{blue}{\left(-x\right) \cdot \left(x + -1\right)} - \left(-1 - x\right) \cdot \left(-2 + x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      mul-1-neg [<=] 0.9	\[ \frac{\color{blue}{\left(-1 \cdot x\right)} \cdot \left(x + -1\right) - \left(-1 - x\right) \cdot \left(-2 + x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      *-commutative [=>] 0.9	\[ \frac{\color{blue}{\left(x \cdot -1\right)} \cdot \left(x + -1\right) - \left(-1 - x\right) \cdot \left(-2 + x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      associate-*l* [=>] 0.9	\[ \frac{\color{blue}{x \cdot \left(-1 \cdot \left(x + -1\right)\right)} - \left(-1 - x\right) \cdot \left(-2 + x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      distribute-lft-in [=>] 0.9	\[ \frac{x \cdot \color{blue}{\left(-1 \cdot x + -1 \cdot -1\right)} - \left(-1 - x\right) \cdot \left(-2 + x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      metadata-eval [=>] 0.9	\[ \frac{x \cdot \left(-1 \cdot x + \color{blue}{1}\right) - \left(-1 - x\right) \cdot \left(-2 + x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      +-commutative [<=] 0.9	\[ \frac{x \cdot \color{blue}{\left(1 + -1 \cdot x\right)} - \left(-1 - x\right) \cdot \left(-2 + x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      mul-1-neg [=>] 0.9	\[ \frac{x \cdot \left(1 + \color{blue}{\left(-x\right)}\right) - \left(-1 - x\right) \cdot \left(-2 + x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      sub-neg [<=] 0.9	\[ \frac{x \cdot \color{blue}{\left(1 - x\right)} - \left(-1 - x\right) \cdot \left(-2 + x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      *-commutative [=>] 0.9	\[ \frac{x \cdot \left(1 - x\right) - \color{blue}{\left(-2 + x\right) \cdot \left(-1 - x\right)}}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      +-commutative [=>] 0.9	\[ \frac{x \cdot \left(1 - x\right) - \color{blue}{\left(x + -2\right)} \cdot \left(-1 - x\right)}{\left(-1 - x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      *-commutative [=>] 0.9	\[ \frac{x \cdot \left(1 - x\right) - \left(x + -2\right) \cdot \left(-1 - x\right)}{\color{blue}{\left(x \cdot \left(x + -1\right)\right) \cdot \left(-1 - x\right)}} \]
      associate-*l* [=>] 0.9	\[ \frac{x \cdot \left(1 - x\right) - \left(x + -2\right) \cdot \left(-1 - x\right)}{\color{blue}{x \cdot \left(\left(x + -1\right) \cdot \left(-1 - x\right)\right)}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x + -1\right) + \left(1 + x\right) \cdot \left(2 - x\right)}{1 + x}}{x \cdot \left(x + -1\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(1 + x\right) \cdot \left(x + -2\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(1 - x\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	3528

\[\begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ t_1 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{t_0 + \left(1 + x\right) \cdot \left(2 - x\right)}{1 + x}}{t_0}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(1 + x\right) \cdot \left(x + -2\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(1 - x\right)\right)}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	2504

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} - x \cdot 2\\ \end{array} \]

Alternative 3
Error	0.6
Cost	2504

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;t_0 + \frac{\frac{x + -2}{1 - x}}{x}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} - x \cdot 2\\ \end{array} \]

Alternative 4
Error	0.6
Cost	2504

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;t_0 + \frac{2 - x}{x \cdot \left(x + -1\right)}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} - x \cdot 2\\ \end{array} \]

Alternative 5
Error	0.6
Cost	2504

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{\left(1 + x\right) \cdot \left(x + -2\right)}{x + -1} - x}{x \cdot \left(-1 - x\right)}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} - x \cdot 2\\ \end{array} \]

Alternative 6
Error	0.1
Cost	1737

\[\begin{array}{l} \mathbf{if}\;x \leq -100000000 \lor \neg \left(x \leq 50000000\right):\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(1 + x\right) \cdot \left(x + -2\right)}{x \cdot \left(\left(1 + x\right) \cdot \left(1 - x\right)\right)}\\ \end{array} \]

Alternative 7
Error	0.6
Cost	713

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

Alternative 8
Error	15.4
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 9
Error	10.6
Cost	448

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

Alternative 10
Error	30.4
Cost	192

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

Alternative 11
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023187 
(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))))