3frac (problem 3.3.3)

Average Accuracy: 84.3% → 99.6%

Time: 13.8s

Precision: binary64

Cost: 6788

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ \mathbf{if}\;x \leq -200000000:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{elif}\;x \leq 132000000:\\ \;\;\;\;\frac{t_0 + \left(x + 1\right) \cdot \left(x - \left(-2 + x \cdot 2\right)\right)}{t_0 \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x \cdot x}}{-1 - x}\\ \end{array} \]

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 (* x (+ x -1.0))))
   (if (<= x -200000000.0)
     (/ 2.0 (pow x 3.0))
     (if (<= x 132000000.0)
       (/ (+ t_0 (* (+ x 1.0) (- x (+ -2.0 (* x 2.0))))) (* t_0 (+ x 1.0)))
       (/ (/ -2.0 (* x 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 tmp;
	if (x <= -200000000.0) {
		tmp = 2.0 / pow(x, 3.0);
	} else if (x <= 132000000.0) {
		tmp = (t_0 + ((x + 1.0) * (x - (-2.0 + (x * 2.0))))) / (t_0 * (x + 1.0));
	} else {
		tmp = (-2.0 / (x * 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) :: tmp
    t_0 = x * (x + (-1.0d0))
    if (x <= (-200000000.0d0)) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else if (x <= 132000000.0d0) then
        tmp = (t_0 + ((x + 1.0d0) * (x - ((-2.0d0) + (x * 2.0d0))))) / (t_0 * (x + 1.0d0))
    else
        tmp = ((-2.0d0) / (x * 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 tmp;
	if (x <= -200000000.0) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else if (x <= 132000000.0) {
		tmp = (t_0 + ((x + 1.0) * (x - (-2.0 + (x * 2.0))))) / (t_0 * (x + 1.0));
	} else {
		tmp = (-2.0 / (x * 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)
	tmp = 0
	if x <= -200000000.0:
		tmp = 2.0 / math.pow(x, 3.0)
	elif x <= 132000000.0:
		tmp = (t_0 + ((x + 1.0) * (x - (-2.0 + (x * 2.0))))) / (t_0 * (x + 1.0))
	else:
		tmp = (-2.0 / (x * 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))
	tmp = 0.0
	if (x <= -200000000.0)
		tmp = Float64(2.0 / (x ^ 3.0));
	elseif (x <= 132000000.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(x + 1.0) * Float64(x - Float64(-2.0 + Float64(x * 2.0))))) / Float64(t_0 * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(-2.0 / Float64(x * 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);
	tmp = 0.0;
	if (x <= -200000000.0)
		tmp = 2.0 / (x ^ 3.0);
	elseif (x <= 132000000.0)
		tmp = (t_0 + ((x + 1.0) * (x - (-2.0 + (x * 2.0))))) / (t_0 * (x + 1.0));
	else
		tmp = (-2.0 / (x * 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]}, If[LessEqual[x, -200000000.0], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 132000000.0], N[(N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] * N[(x - N[(-2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	84.3%
Target	99.6%
Herbie	99.6%

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

Derivation

1. Split input into 3 regimes

2. if x < -2e8

   1. Initial program 69.2%

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

   2. Simplified 69.2%

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

      [Start] 69.2	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 69.2	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 69.2	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 69.2	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 69.2	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 69.2	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 69.2	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 69.2	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 69.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 69.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 99.3%

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

   if -2e8 < x < 1.32e8

   1. Initial program 98.9%

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

   2. Simplified 98.9%

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

      [Start] 98.9	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 98.9	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 98.9	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 98.9	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 98.9	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 98.9	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 98.9	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 98.9	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 98.9	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 98.9	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 98.9%

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

      [Start] 98.9	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      flip-+ [=>] 98.9	\[ \frac{1}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      sub-neg [=>] 98.9	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{1 + \left(-x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      metadata-eval [<=] 98.9	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{\left(--1\right)} + \left(-x\right)}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      distribute-neg-in [<=] 98.9	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{-\left(-1 + x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      +-commutative [<=] 98.9	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{-\color{blue}{\left(x + -1\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      associate-/r/ [=>] 98.9	\[ \color{blue}{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(-\left(x + -1\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      metadata-eval [=>] 98.9	\[ \frac{1}{\color{blue}{1} - x \cdot x} \cdot \left(-\left(x + -1\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      +-commutative [=>] 98.9	\[ \frac{1}{1 - x \cdot x} \cdot \left(-\color{blue}{\left(-1 + x\right)}\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      distribute-neg-in [=>] 98.9	\[ \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      metadata-eval [=>] 98.9	\[ \frac{1}{1 - x \cdot x} \cdot \left(\color{blue}{1} + \left(-x\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      sub-neg [<=] 98.9	\[ \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(1 - x\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   4. Simplified 98.9%

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

      [Start] 98.9	\[ \frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      associate-*l/ [=>] 98.9	\[ \color{blue}{\frac{1 \cdot \left(1 - x\right)}{1 - x \cdot x}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      *-lft-identity [=>] 98.9	\[ \frac{\color{blue}{1 - x}}{1 - x \cdot x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   5. Applied egg-rr 99.9%

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

      [Start] 98.9	\[ \frac{1 - x}{1 - x \cdot x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      clear-num [=>] 98.9	\[ \color{blue}{\frac{1}{\frac{1 - x \cdot x}{1 - x}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      frac-sub [=>] 98.9	\[ \frac{1}{\frac{1 - x \cdot x}{1 - x}} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]
      frac-sub [=>] 99.8	\[ \color{blue}{\frac{1 \cdot \left(x \cdot \left(x + -1\right)\right) - \frac{1 - x \cdot x}{1 - x} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)}} \]
      *-un-lft-identity [<=] 99.8	\[ \frac{\color{blue}{x \cdot \left(x + -1\right)} - \frac{1 - x \cdot x}{1 - x} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      metadata-eval [<=] 99.8	\[ \frac{x \cdot \left(x + -1\right) - \frac{\color{blue}{1 \cdot 1} - x \cdot x}{1 - x} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      flip-+ [<=] 99.9	\[ \frac{x \cdot \left(x + -1\right) - \color{blue}{\left(1 + x\right)} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      *-rgt-identity [=>] 99.9	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - \color{blue}{x}\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      +-commutative [=>] 99.9	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \left(2 \cdot \color{blue}{\left(-1 + x\right)} - x\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      distribute-lft-in [=>] 99.9	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \left(\color{blue}{\left(2 \cdot -1 + 2 \cdot x\right)} - x\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      metadata-eval [=>] 99.9	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \left(\left(\color{blue}{-2} + 2 \cdot x\right) - x\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   if 1.32e8 < x

   1. Initial program 68.8%

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

   2. Simplified 68.8%

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

      [Start] 68.8	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 68.8	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 68.8	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 68.8	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 68.8	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 68.8	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 68.8	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 68.8	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 68.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 68.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 11.5%

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

      [Start] 68.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      sub-neg [=>] 68.8	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} + \left(-\frac{1}{x + -1}\right)\right)} \]
      flip-+ [=>] 16.1	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{2}{x} \cdot \frac{2}{x} - \left(-\frac{1}{x + -1}\right) \cdot \left(-\frac{1}{x + -1}\right)}{\frac{2}{x} - \left(-\frac{1}{x + -1}\right)}} \]

   4. Simplified 11.6%

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

      [Start] 11.5	\[ \frac{1}{1 + x} - \frac{\frac{4}{x \cdot x} - \frac{1}{1 - x} \cdot \frac{1}{1 - x}}{\frac{2}{x} - \frac{1}{1 - x}} \]
      associate-*r/ [=>] 11.6	\[ \frac{1}{1 + x} - \frac{\frac{4}{x \cdot x} - \color{blue}{\frac{\frac{1}{1 - x} \cdot 1}{1 - x}}}{\frac{2}{x} - \frac{1}{1 - x}} \]
      *-rgt-identity [=>] 11.6	\[ \frac{1}{1 + x} - \frac{\frac{4}{x \cdot x} - \frac{\color{blue}{\frac{1}{1 - x}}}{1 - x}}{\frac{2}{x} - \frac{1}{1 - x}} \]
      sub-neg [=>] 11.6	\[ \frac{1}{1 + x} - \frac{\frac{4}{x \cdot x} - \frac{\frac{1}{1 - x}}{1 - x}}{\color{blue}{\frac{2}{x} + \left(-\frac{1}{1 - x}\right)}} \]
      distribute-neg-frac [=>] 11.6	\[ \frac{1}{1 + x} - \frac{\frac{4}{x \cdot x} - \frac{\frac{1}{1 - x}}{1 - x}}{\frac{2}{x} + \color{blue}{\frac{-1}{1 - x}}} \]
      metadata-eval [=>] 11.6	\[ \frac{1}{1 + x} - \frac{\frac{4}{x \cdot x} - \frac{\frac{1}{1 - x}}{1 - x}}{\frac{2}{x} + \frac{\color{blue}{-1}}{1 - x}} \]

   5. Applied egg-rr 66.2%

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

      [Start] 11.6	\[ \frac{1}{1 + x} - \frac{\frac{4}{x \cdot x} - \frac{\frac{1}{1 - x}}{1 - x}}{\frac{2}{x} + \frac{-1}{1 - x}} \]
      frac-2neg [=>] 11.6	\[ \color{blue}{\frac{-1}{-\left(1 + x\right)}} - \frac{\frac{4}{x \cdot x} - \frac{\frac{1}{1 - x}}{1 - x}}{\frac{2}{x} + \frac{-1}{1 - x}} \]
      metadata-eval [=>] 11.6	\[ \frac{\color{blue}{-1}}{-\left(1 + x\right)} - \frac{\frac{4}{x \cdot x} - \frac{\frac{1}{1 - x}}{1 - x}}{\frac{2}{x} + \frac{-1}{1 - x}} \]
      clear-num [=>] 13.1	\[ \frac{-1}{-\left(1 + x\right)} - \color{blue}{\frac{1}{\frac{\frac{2}{x} + \frac{-1}{1 - x}}{\frac{4}{x \cdot x} - \frac{\frac{1}{1 - x}}{1 - x}}}} \]
      frac-sub [=>] 11.2	\[ \color{blue}{\frac{-1 \cdot \frac{\frac{2}{x} + \frac{-1}{1 - x}}{\frac{4}{x \cdot x} - \frac{\frac{1}{1 - x}}{1 - x}} - \left(-\left(1 + x\right)\right) \cdot 1}{\left(-\left(1 + x\right)\right) \cdot \frac{\frac{2}{x} + \frac{-1}{1 - x}}{\frac{4}{x \cdot x} - \frac{\frac{1}{1 - x}}{1 - x}}}} \]

   6. Simplified 66.2%

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

      [Start] 66.2	\[ \frac{-1 \cdot \frac{1}{\frac{2}{x} + \frac{1}{1 - x}} - \left(-1 + \left(-x\right)\right) \cdot 1}{\left(-1 + \left(-x\right)\right) \cdot \frac{1}{\frac{2}{x} + \frac{1}{1 - x}}} \]
      associate-*r/ [=>] 66.2	\[ \frac{\color{blue}{\frac{-1 \cdot 1}{\frac{2}{x} + \frac{1}{1 - x}}} - \left(-1 + \left(-x\right)\right) \cdot 1}{\left(-1 + \left(-x\right)\right) \cdot \frac{1}{\frac{2}{x} + \frac{1}{1 - x}}} \]
      metadata-eval [=>] 66.2	\[ \frac{\frac{\color{blue}{-1}}{\frac{2}{x} + \frac{1}{1 - x}} - \left(-1 + \left(-x\right)\right) \cdot 1}{\left(-1 + \left(-x\right)\right) \cdot \frac{1}{\frac{2}{x} + \frac{1}{1 - x}}} \]
      *-rgt-identity [=>] 66.2	\[ \frac{\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} - \color{blue}{\left(-1 + \left(-x\right)\right)}}{\left(-1 + \left(-x\right)\right) \cdot \frac{1}{\frac{2}{x} + \frac{1}{1 - x}}} \]
      unsub-neg [=>] 66.2	\[ \frac{\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} - \color{blue}{\left(-1 - x\right)}}{\left(-1 + \left(-x\right)\right) \cdot \frac{1}{\frac{2}{x} + \frac{1}{1 - x}}} \]
      associate-*r/ [=>] 66.2	\[ \frac{\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} - \left(-1 - x\right)}{\color{blue}{\frac{\left(-1 + \left(-x\right)\right) \cdot 1}{\frac{2}{x} + \frac{1}{1 - x}}}} \]
      *-rgt-identity [=>] 66.2	\[ \frac{\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} - \left(-1 - x\right)}{\frac{\color{blue}{-1 + \left(-x\right)}}{\frac{2}{x} + \frac{1}{1 - x}}} \]
      unsub-neg [=>] 66.2	\[ \frac{\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} - \left(-1 - x\right)}{\frac{\color{blue}{-1 - x}}{\frac{2}{x} + \frac{1}{1 - x}}} \]

   7. Applied egg-rr 67.1%

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

      [Start] 66.2	\[ \frac{\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} - \left(-1 - x\right)}{\frac{-1 - x}{\frac{2}{x} + \frac{1}{1 - x}}} \]
      div-sub [=>] 66.9	\[ \color{blue}{\frac{\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}}}{\frac{-1 - x}{\frac{2}{x} + \frac{1}{1 - x}}} - \frac{-1 - x}{\frac{-1 - x}{\frac{2}{x} + \frac{1}{1 - x}}}} \]
      sub-neg [=>] 66.9	\[ \color{blue}{\frac{\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}}}{\frac{-1 - x}{\frac{2}{x} + \frac{1}{1 - x}}} + \left(-\frac{-1 - x}{\frac{-1 - x}{\frac{2}{x} + \frac{1}{1 - x}}}\right)} \]
      div-inv [=>] 64.2	\[ \color{blue}{\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} \cdot \frac{1}{\frac{-1 - x}{\frac{2}{x} + \frac{1}{1 - x}}}} + \left(-\frac{-1 - x}{\frac{-1 - x}{\frac{2}{x} + \frac{1}{1 - x}}}\right) \]
      clear-num [<=] 63.2	\[ \frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} \cdot \color{blue}{\frac{\frac{2}{x} + \frac{1}{1 - x}}{-1 - x}} + \left(-\frac{-1 - x}{\frac{-1 - x}{\frac{2}{x} + \frac{1}{1 - x}}}\right) \]
      div-inv [=>] 63.0	\[ \frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} \cdot \frac{\frac{2}{x} + \frac{1}{1 - x}}{-1 - x} + \left(-\color{blue}{\left(-1 - x\right) \cdot \frac{1}{\frac{-1 - x}{\frac{2}{x} + \frac{1}{1 - x}}}}\right) \]
      clear-num [<=] 67.1	\[ \frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} \cdot \frac{\frac{2}{x} + \frac{1}{1 - x}}{-1 - x} + \left(-\left(-1 - x\right) \cdot \color{blue}{\frac{\frac{2}{x} + \frac{1}{1 - x}}{-1 - x}}\right) \]

   8. Simplified 60.1%

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

      [Start] 67.1	\[ \frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} \cdot \frac{\frac{2}{x} + \frac{1}{1 - x}}{-1 - x} + \left(-\left(-1 - x\right) \cdot \frac{\frac{2}{x} + \frac{1}{1 - x}}{-1 - x}\right) \]
      sub-neg [<=] 67.1	\[ \color{blue}{\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} \cdot \frac{\frac{2}{x} + \frac{1}{1 - x}}{-1 - x} - \left(-1 - x\right) \cdot \frac{\frac{2}{x} + \frac{1}{1 - x}}{-1 - x}} \]
      distribute-rgt-out-- [=>] 66.0	\[ \color{blue}{\frac{\frac{2}{x} + \frac{1}{1 - x}}{-1 - x} \cdot \left(\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} - \left(-1 - x\right)\right)} \]
      associate-*l/ [=>] 58.7	\[ \color{blue}{\frac{\left(\frac{2}{x} + \frac{1}{1 - x}\right) \cdot \left(\frac{-1}{\frac{2}{x} + \frac{1}{1 - x}} - \left(-1 - x\right)\right)}{-1 - x}} \]

   9. Taylor expanded in x around inf 99.3%

      \[\leadsto \frac{\color{blue}{\frac{-2}{{x}^{2}}}}{-1 - x} \]

   10. Simplified 99.3%

       \[\leadsto \frac{\color{blue}{\frac{-2}{x \cdot x}}}{-1 - x} \]
       Proof

       [Start] 99.3	\[ \frac{\frac{-2}{{x}^{2}}}{-1 - x} \]
       unpow2 [=>] 99.3	\[ \frac{\frac{-2}{\color{blue}{x \cdot x}}}{-1 - x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200000000:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{elif}\;x \leq 132000000:\\ \;\;\;\;\frac{x \cdot \left(x + -1\right) + \left(x + 1\right) \cdot \left(x - \left(-2 + x \cdot 2\right)\right)}{\left(x \cdot \left(x + -1\right)\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x \cdot x}}{-1 - x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.5%
Cost	3272

\[\begin{array}{l} t_0 := \frac{1}{x + -1}\\ t_1 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + t_0\\ \mathbf{if}\;t_1 \leq -1000000:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\frac{-2}{x \cdot x}}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{1 - x \cdot x} + \left(t_0 - \frac{2}{x}\right)\\ \end{array} \]

Alternative 2
Accuracy	98.6%
Cost	3272

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -1000000:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{-2}{x \cdot x}}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \left(x + 1\right) \cdot \left(\frac{2}{x} + \frac{1}{1 - x}\right)}{-1 - x}\\ \end{array} \]

Alternative 3
Accuracy	98.6%
Cost	3016

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -1000000:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{-2}{x \cdot x}}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	1992

\[\begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ \mathbf{if}\;x \leq -200000000:\\ \;\;\;\;\frac{\frac{-2}{x} + \frac{-4}{x \cdot x}}{\frac{-1 - x}{\frac{2}{x} + \frac{1}{1 - x}}}\\ \mathbf{elif}\;x \leq 132000000:\\ \;\;\;\;\frac{t_0 + \left(x + 1\right) \cdot \left(x - \left(-2 + x \cdot 2\right)\right)}{t_0 \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x \cdot x}}{-1 - x}\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	1604

\[\begin{array}{l} t_0 := \frac{2}{x} + \frac{1}{1 - x}\\ \mathbf{if}\;x \leq -23000:\\ \;\;\;\;\frac{\frac{-2}{x} + \frac{-4}{x \cdot x}}{\frac{-1 - x}{t_0}}\\ \mathbf{elif}\;x \leq 300000:\\ \;\;\;\;\frac{-1 + \left(x + 1\right) \cdot t_0}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x \cdot x}}{-1 - x}\\ \end{array} \]

Alternative 6
Accuracy	98.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-2}{x \cdot x}}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 7
Accuracy	75.4%
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 8
Accuracy	75.8%
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
Accuracy	82.9%
Cost	448

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

Alternative 10
Accuracy	51.7%
Cost	192

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

Reproduce

herbie shell --seed 2023151 
(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))))