3frac (problem 3.3.3)

Average Error: 9.7 → 0.2

Time: 14.3s

Precision: binary64

Cost: 2248

Math

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

↓

\[\begin{array}{l} t_0 := \left(-1 - x\right) \cdot \left(x + -1\right)\\ t_1 := x \cdot \left(x + 1\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{2}{x}}{t_1}\\ \mathbf{elif}\;x \leq 230000000:\\ \;\;\;\;\frac{\frac{-2}{1 - x} \cdot t_0 + x \cdot \left(x + \left(2 - x\right)\right)}{x \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{2}{x \cdot x}}{t_1}\\ \end{array} \]

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 x) (+ x -1.0))) (t_1 (* x (+ x 1.0))))
   (if (<= x -6.2e+15)
     (/ (/ 2.0 x) t_1)
     (if (<= x 230000000.0)
       (/ (+ (* (/ -2.0 (- 1.0 x)) t_0) (* x (+ x (- 2.0 x)))) (* x t_0))
       (/ (+ (/ 2.0 x) (/ 2.0 (* x x))) t_1)))))

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 - x) * (x + -1.0);
	double t_1 = x * (x + 1.0);
	double tmp;
	if (x <= -6.2e+15) {
		tmp = (2.0 / x) / t_1;
	} else if (x <= 230000000.0) {
		tmp = (((-2.0 / (1.0 - x)) * t_0) + (x * (x + (2.0 - x)))) / (x * t_0);
	} else {
		tmp = ((2.0 / x) + (2.0 / (x * x))) / t_1;
	}
	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) - x) * (x + (-1.0d0))
    t_1 = x * (x + 1.0d0)
    if (x <= (-6.2d+15)) then
        tmp = (2.0d0 / x) / t_1
    else if (x <= 230000000.0d0) then
        tmp = ((((-2.0d0) / (1.0d0 - x)) * t_0) + (x * (x + (2.0d0 - x)))) / (x * t_0)
    else
        tmp = ((2.0d0 / x) + (2.0d0 / (x * x))) / t_1
    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 - x) * (x + -1.0);
	double t_1 = x * (x + 1.0);
	double tmp;
	if (x <= -6.2e+15) {
		tmp = (2.0 / x) / t_1;
	} else if (x <= 230000000.0) {
		tmp = (((-2.0 / (1.0 - x)) * t_0) + (x * (x + (2.0 - x)))) / (x * t_0);
	} else {
		tmp = ((2.0 / x) + (2.0 / (x * x))) / t_1;
	}
	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 - x) * (x + -1.0)
	t_1 = x * (x + 1.0)
	tmp = 0
	if x <= -6.2e+15:
		tmp = (2.0 / x) / t_1
	elif x <= 230000000.0:
		tmp = (((-2.0 / (1.0 - x)) * t_0) + (x * (x + (2.0 - x)))) / (x * t_0)
	else:
		tmp = ((2.0 / x) + (2.0 / (x * x))) / t_1
	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(-1.0 - x) * Float64(x + -1.0))
	t_1 = Float64(x * Float64(x + 1.0))
	tmp = 0.0
	if (x <= -6.2e+15)
		tmp = Float64(Float64(2.0 / x) / t_1);
	elseif (x <= 230000000.0)
		tmp = Float64(Float64(Float64(Float64(-2.0 / Float64(1.0 - x)) * t_0) + Float64(x * Float64(x + Float64(2.0 - x)))) / Float64(x * t_0));
	else
		tmp = Float64(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x))) / t_1);
	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 - x) * (x + -1.0);
	t_1 = x * (x + 1.0);
	tmp = 0.0;
	if (x <= -6.2e+15)
		tmp = (2.0 / x) / t_1;
	elseif (x <= 230000000.0)
		tmp = (((-2.0 / (1.0 - x)) * t_0) + (x * (x + (2.0 - x)))) / (x * t_0);
	else
		tmp = ((2.0 / x) + (2.0 / (x * x))) / t_1;
	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[(-1.0 - x), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e+15], N[(N[(2.0 / x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 230000000.0], N[(N[(N[(N[(-2.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(x * N[(x + N[(2.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Target

Original	9.7
Target	0.3
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if x < -6.2e15

   1. Initial program 18.1

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

   2. Simplified 18.1

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

      [Start] 18.1	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 18.1	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 18.1	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 18.1	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 18.1	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 18.1	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 18.1	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 18.1	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 18.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 18.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 18.1

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

   4. Simplified 18.1

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

      [Start] 18.1	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]
      *-commutative [=>] 18.1	\[ \frac{1}{1 + x} - \color{blue}{\frac{1}{x + -1} \cdot \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1}} \]
      /-rgt-identity [=>] 18.1	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{x}} \]
      *-commutative [=>] 18.1	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{-2 + \left(\color{blue}{x \cdot 2} - x\right)}{x} \]

   5. Applied egg-rr 18.1

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

   6. Simplified 18.1

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

      [Start] 18.1	\[ \frac{x - \left(1 + x\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      metadata-eval [<=] 18.1	\[ \frac{x - \left(\color{blue}{\left(--1\right)} + x\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      remove-double-neg [<=] 18.1	\[ \frac{x - \left(\left(--1\right) + \color{blue}{\left(-\left(-x\right)\right)}\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      distribute-neg-in [<=] 18.1	\[ \frac{x - \color{blue}{\left(-\left(-1 + \left(-x\right)\right)\right)} \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      sub-neg [<=] 18.1	\[ \frac{x - \left(-\color{blue}{\left(-1 - x\right)}\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      sub-neg [=>] 18.1	\[ \frac{x - \left(-\color{blue}{\left(-1 + \left(-x\right)\right)}\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      distribute-neg-in [=>] 18.1	\[ \frac{x - \color{blue}{\left(\left(--1\right) + \left(-\left(-x\right)\right)\right)} \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      metadata-eval [=>] 18.1	\[ \frac{x - \left(\color{blue}{1} + \left(-\left(-x\right)\right)\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      remove-double-neg [=>] 18.1	\[ \frac{x - \left(1 + \color{blue}{x}\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      +-commutative [=>] 18.1	\[ \frac{x - \color{blue}{\left(x + 1\right)} \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      +-commutative [=>] 18.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{\color{blue}{-2 + x}}{x + -1}}{\left(1 + x\right) \cdot x} \]
      *-commutative [=>] 18.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{\color{blue}{x \cdot \left(1 + x\right)}} \]
      +-commutative [=>] 18.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Taylor expanded in x around inf 0.1

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

   if -6.2e15 < x < 2.3e8

   1. Initial program 1.4

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

   2. Simplified 1.4

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

      [Start] 1.4	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 1.4	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 1.4	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 1.4	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 1.4	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 1.4	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 1.4	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 1.4	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 1.4	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 1.4	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 1.4

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

   4. Simplified 1.4

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

      [Start] 1.4	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]
      *-commutative [=>] 1.4	\[ \frac{1}{1 + x} - \color{blue}{\frac{1}{x + -1} \cdot \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1}} \]
      /-rgt-identity [=>] 1.4	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{x}} \]
      *-commutative [=>] 1.4	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{-2 + \left(\color{blue}{x \cdot 2} - x\right)}{x} \]

   5. Taylor expanded in x around 0 1.4

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

   6. Simplified 1.4

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

      [Start] 1.4	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \left(1 - 2 \cdot \frac{1}{x}\right) \]
      associate-*r/ [=>] 1.4	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \left(1 - \color{blue}{\frac{2 \cdot 1}{x}}\right) \]
      metadata-eval [=>] 1.4	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \left(1 - \frac{\color{blue}{2}}{x}\right) \]

   7. Applied egg-rr 1.4

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

   8. Applied egg-rr 0.3

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

   if 2.3e8 < x

   1. Initial program 19.2

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

   2. Simplified 19.2

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

      [Start] 19.2	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 19.2	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.2	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 19.2	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 19.2	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 19.2	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 19.2	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 19.2	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 19.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 19.3

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

   4. Simplified 19.3

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

      [Start] 19.3	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]
      *-commutative [=>] 19.3	\[ \frac{1}{1 + x} - \color{blue}{\frac{1}{x + -1} \cdot \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1}} \]
      /-rgt-identity [=>] 19.3	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{x}} \]
      *-commutative [=>] 19.3	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{-2 + \left(\color{blue}{x \cdot 2} - x\right)}{x} \]

   5. Applied egg-rr 19.3

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

   6. Simplified 19.3

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

      [Start] 19.3	\[ \frac{x - \left(1 + x\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      metadata-eval [<=] 19.3	\[ \frac{x - \left(\color{blue}{\left(--1\right)} + x\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      remove-double-neg [<=] 19.3	\[ \frac{x - \left(\left(--1\right) + \color{blue}{\left(-\left(-x\right)\right)}\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      distribute-neg-in [<=] 19.3	\[ \frac{x - \color{blue}{\left(-\left(-1 + \left(-x\right)\right)\right)} \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      sub-neg [<=] 19.3	\[ \frac{x - \left(-\color{blue}{\left(-1 - x\right)}\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      sub-neg [=>] 19.3	\[ \frac{x - \left(-\color{blue}{\left(-1 + \left(-x\right)\right)}\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      distribute-neg-in [=>] 19.3	\[ \frac{x - \color{blue}{\left(\left(--1\right) + \left(-\left(-x\right)\right)\right)} \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      metadata-eval [=>] 19.3	\[ \frac{x - \left(\color{blue}{1} + \left(-\left(-x\right)\right)\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      remove-double-neg [=>] 19.3	\[ \frac{x - \left(1 + \color{blue}{x}\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      +-commutative [=>] 19.3	\[ \frac{x - \color{blue}{\left(x + 1\right)} \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      +-commutative [=>] 19.3	\[ \frac{x - \left(x + 1\right) \cdot \frac{\color{blue}{-2 + x}}{x + -1}}{\left(1 + x\right) \cdot x} \]
      *-commutative [=>] 19.3	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{\color{blue}{x \cdot \left(1 + x\right)}} \]
      +-commutative [=>] 19.3	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Taylor expanded in x around inf 0.1

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

   8. Simplified 0.1

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

      [Start] 0.1	\[ \frac{2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{x}}{x \cdot \left(x + 1\right)} \]
      associate-*r/ [=>] 0.1	\[ \frac{\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + 2 \cdot \frac{1}{x}}{x \cdot \left(x + 1\right)} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\color{blue}{2}}{{x}^{2}} + 2 \cdot \frac{1}{x}}{x \cdot \left(x + 1\right)} \]
      unpow2 [=>] 0.1	\[ \frac{\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{x}}{x \cdot \left(x + 1\right)} \]
      associate-*r/ [=>] 0.1	\[ \frac{\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{x}}}{x \cdot \left(x + 1\right)} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{x}}{x \cdot \left(x + 1\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 230000000:\\ \;\;\;\;\frac{\frac{-2}{1 - x} \cdot \left(\left(-1 - x\right) \cdot \left(x + -1\right)\right) + x \cdot \left(x + \left(2 - x\right)\right)}{x \cdot \left(\left(-1 - x\right) \cdot \left(x + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{2}{x \cdot x}}{x \cdot \left(x + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	3529

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-7} \lor \neg \left(t_0 \leq 2 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(x + 1\right) \cdot \left(x + -2\right)}{\left(x + 1\right) \cdot \left(x - x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{2}{x \cdot x}}{x \cdot \left(x + 1\right)}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	3017

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-7} \lor \neg \left(t_0 \leq 5 \cdot 10^{-21}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot \left(x + 1\right)}\\ \end{array} \]

Alternative 3
Error	0.6
Cost	3016

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t_0 + \left(\frac{-2}{x} - \frac{-1}{x + -1}\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.3
Cost	3016

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t_0 + \left(\frac{-2}{x} - \frac{-1}{x + -1}\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{2}{x \cdot x}}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	1.0
Cost	841

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

Alternative 6
Error	15.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.55\right):\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 7
Error	10.6
Cost	448

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

Alternative 8
Error	30.7
Cost	192

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

Alternative 9
Error	61.9
Cost	64

\[-1 \]

Alternative 10
Error	61.9
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023041 
(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))))