Asymptote A

Percentage Accurate: 77.2% → 99.9%

Time: 5.7s

Alternatives: 3

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} - \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} - \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{1 - x}}{1 + x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 2.0 (- 1.0 x)) (+ 1.0 x)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(2.0 / Float64(1.0 - x)) / Float64(1.0 + x))
end

MATLAB

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

Wolfram

code[x_] := N[(N[(2.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. sub-neg 78.6%

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

   2. +-commutative 78.6%

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

   3. neg-sub0 78.6%

      \[\leadsto \color{blue}{\left(0 - \frac{1}{x - 1}\right)} + \frac{1}{x + 1} \]

   4. associate-+l- 78.6%

      \[\leadsto \color{blue}{0 - \left(\frac{1}{x - 1} - \frac{1}{x + 1}\right)} \]

   5. sub-neg 78.6%

      \[\leadsto 0 - \color{blue}{\left(\frac{1}{x - 1} + \left(-\frac{1}{x + 1}\right)\right)} \]

   6. associate--r+ 78.6%

      \[\leadsto \color{blue}{\left(0 - \frac{1}{x - 1}\right) - \left(-\frac{1}{x + 1}\right)} \]

   7. neg-sub0 78.6%

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

   8. distribute-neg-frac 78.6%

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

   9. metadata-eval 78.6%

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

   10. metadata-eval 78.6%

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

   11. metadata-eval 78.6%

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

   12. associate-/r* 78.6%

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

   13. metadata-eval 78.6%

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

   14. neg-mul-1 78.6%

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

   15. sub0-neg 78.6%

       \[\leadsto \frac{1}{\color{blue}{0 - \left(x - 1\right)}} - \left(-\frac{1}{x + 1}\right) \]

   16. associate-+l- 78.6%

       \[\leadsto \frac{1}{\color{blue}{\left(0 - x\right) + 1}} - \left(-\frac{1}{x + 1}\right) \]

   17. neg-sub0 78.6%

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

   18. +-commutative 78.6%

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

   19. unsub-neg 78.6%

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

   20. distribute-neg-frac 78.6%

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

   21. metadata-eval 78.6%

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

   22. metadata-eval 78.6%

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

   23. metadata-eval 78.6%

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

   24. associate-/r* 78.6%

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

   25. metadata-eval 78.6%

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

   26. neg-mul-1 78.6%

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

3. Simplified 78.6%

   \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
4. Step-by-step derivation

   1. sub-neg 78.6%

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

   2. distribute-neg-frac 78.6%

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

   3. metadata-eval 78.6%

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

5. Applied egg-rr 78.6%

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

7. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. flip-- 99.6%

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

   3. associate-*l/ 90.7%

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

   4. metadata-eval 90.7%

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

   5. pow2 90.7%

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

   6. +-commutative 90.7%

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

9. Applied egg-rr 90.7%

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

    1. associate-/r/ 90.6%

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

    2. *-commutative 90.6%

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

    3. associate-/r* 92.0%

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

    4. metadata-eval 92.0%

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

    5. associate-/r/ 99.6%

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

    6. metadata-eval 99.6%

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

    7. unpow2 99.6%

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

    8. +-commutative 99.6%

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

    9. flip-- 99.9%

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

    10. frac-2neg 99.9%

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

    11. div-inv 99.8%

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

    12. metadata-eval 99.8%

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

    13. distribute-neg-frac 99.8%

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

    14. metadata-eval 99.8%

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

    15. sub-neg 99.8%

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

    16. neg-mul-1 99.8%

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

    17. distribute-neg-in 99.8%

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

    18. metadata-eval 99.8%

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

    19. *-commutative 99.8%

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

11. Applied egg-rr 99.8%

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

    1. associate-*r/ 99.9%

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

    2. *-rgt-identity 99.9%

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

    3. sub-neg 99.9%

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

    4. remove-double-neg 99.9%

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

    5. +-commutative 99.9%

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

13. Simplified 99.9%

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

14. Final simplification 99.9%

    \[\leadsto \frac{\frac{2}{1 - x}}{1 + x} \]

Alternative 2: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 (* (- 1.0 x) (- -1.0 x))))

C

double code(double x) {
	return -2.0 / ((1.0 - x) * (-1.0 - x));
}

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(-2.0 / Float64(Float64(1.0 - x) * Float64(-1.0 - x)))
end

MATLAB

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

Wolfram

code[x_] := N[(-2.0 / N[(N[(1.0 - x), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. sub-neg 78.6%

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

   2. +-commutative 78.6%

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

   3. neg-sub0 78.6%

      \[\leadsto \color{blue}{\left(0 - \frac{1}{x - 1}\right)} + \frac{1}{x + 1} \]

   4. associate-+l- 78.6%

      \[\leadsto \color{blue}{0 - \left(\frac{1}{x - 1} - \frac{1}{x + 1}\right)} \]

   5. sub-neg 78.6%

      \[\leadsto 0 - \color{blue}{\left(\frac{1}{x - 1} + \left(-\frac{1}{x + 1}\right)\right)} \]

   6. associate--r+ 78.6%

      \[\leadsto \color{blue}{\left(0 - \frac{1}{x - 1}\right) - \left(-\frac{1}{x + 1}\right)} \]

   7. neg-sub0 78.6%

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

   8. distribute-neg-frac 78.6%

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

   9. metadata-eval 78.6%

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

   10. metadata-eval 78.6%

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

   11. metadata-eval 78.6%

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

   12. associate-/r* 78.6%

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

   13. metadata-eval 78.6%

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

   14. neg-mul-1 78.6%

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

   15. sub0-neg 78.6%

       \[\leadsto \frac{1}{\color{blue}{0 - \left(x - 1\right)}} - \left(-\frac{1}{x + 1}\right) \]

   16. associate-+l- 78.6%

       \[\leadsto \frac{1}{\color{blue}{\left(0 - x\right) + 1}} - \left(-\frac{1}{x + 1}\right) \]

   17. neg-sub0 78.6%

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

   18. +-commutative 78.6%

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

   19. unsub-neg 78.6%

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

   20. distribute-neg-frac 78.6%

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

   21. metadata-eval 78.6%

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

   22. metadata-eval 78.6%

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

   23. metadata-eval 78.6%

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

   24. associate-/r* 78.6%

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

   25. metadata-eval 78.6%

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

   26. neg-mul-1 78.6%

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

3. Simplified 78.6%

   \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
4. Step-by-step derivation

   1. sub-neg 78.6%

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

   2. distribute-neg-frac 78.6%

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

   3. metadata-eval 78.6%

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

5. Applied egg-rr 78.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 50.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 78.6%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. sub-neg 78.6%

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

   2. +-commutative 78.6%

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

   3. neg-sub0 78.6%

      \[\leadsto \color{blue}{\left(0 - \frac{1}{x - 1}\right)} + \frac{1}{x + 1} \]

   4. associate-+l- 78.6%

      \[\leadsto \color{blue}{0 - \left(\frac{1}{x - 1} - \frac{1}{x + 1}\right)} \]

   5. sub-neg 78.6%

      \[\leadsto 0 - \color{blue}{\left(\frac{1}{x - 1} + \left(-\frac{1}{x + 1}\right)\right)} \]

   6. associate--r+ 78.6%

      \[\leadsto \color{blue}{\left(0 - \frac{1}{x - 1}\right) - \left(-\frac{1}{x + 1}\right)} \]

   7. neg-sub0 78.6%

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

   8. distribute-neg-frac 78.6%

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

   9. metadata-eval 78.6%

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

   10. metadata-eval 78.6%

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

   11. metadata-eval 78.6%

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

   12. associate-/r* 78.6%

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

   13. metadata-eval 78.6%

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

   14. neg-mul-1 78.6%

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

   15. sub0-neg 78.6%

       \[\leadsto \frac{1}{\color{blue}{0 - \left(x - 1\right)}} - \left(-\frac{1}{x + 1}\right) \]

   16. associate-+l- 78.6%

       \[\leadsto \frac{1}{\color{blue}{\left(0 - x\right) + 1}} - \left(-\frac{1}{x + 1}\right) \]

   17. neg-sub0 78.6%

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

   18. +-commutative 78.6%

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

   19. unsub-neg 78.6%

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

   20. distribute-neg-frac 78.6%

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

   21. metadata-eval 78.6%

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

   22. metadata-eval 78.6%

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

   23. metadata-eval 78.6%

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

   24. associate-/r* 78.6%

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

   25. metadata-eval 78.6%

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

   26. neg-mul-1 78.6%

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

3. Simplified 78.6%

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

4. Taylor expanded in x around 0 48.4%

   \[\leadsto \color{blue}{2} \]

5. Final simplification 48.4%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))