Asymptote A

Percentage Accurate: 78.0% → 99.9%

Time: 5.5s

Alternatives: 4

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: 78.0% 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 76.0%

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

   1. sub-neg 76.0%

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

   2. +-commutative 76.0%

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

   3. distribute-neg-frac2 76.0%

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

   4. neg-sub0 76.0%

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

   5. associate-+l- 76.0%

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

   6. neg-sub0 76.0%

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

   7. remove-double-neg 76.0%

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

   8. distribute-neg-in 76.0%

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

   9. sub-neg 76.0%

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

   10. distribute-neg-frac2 76.0%

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

   11. sub-neg 76.0%

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

   12. +-commutative 76.0%

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

   13. unsub-neg 76.0%

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

   14. sub-neg 76.0%

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

   15. +-commutative 76.0%

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

   16. unsub-neg 76.0%

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

   17. metadata-eval 76.0%

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

3. Simplified 76.0%

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

   1. frac-sub 77.3%

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

   2. *-rgt-identity 77.3%

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

   3. metadata-eval 77.3%

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

   4. div-inv 77.3%

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

   5. associate-/r* 77.3%

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

   6. metadata-eval 77.3%

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

   7. div-inv 77.3%

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

   8. *-un-lft-identity 77.3%

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

   9. associate--l- 80.4%

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

   10. div-inv 80.4%

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

   11. metadata-eval 80.4%

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

   12. *-rgt-identity 80.4%

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

   13. div-inv 80.4%

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

   14. metadata-eval 80.4%

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

   15. *-rgt-identity 80.4%

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

6. Applied egg-rr 80.4%

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

   1. flip3-- 57.3%

      \[\leadsto \frac{\frac{-1 - \left(x + \left(1 - x\right)\right)}{1 - x}}{\color{blue}{\frac{{-1}^{3} - {x}^{3}}{-1 \cdot -1 + \left(x \cdot x + -1 \cdot x\right)}}} \]

   2. associate-/r/ 56.9%

      \[\leadsto \color{blue}{\frac{\frac{-1 - \left(x + \left(1 - x\right)\right)}{1 - x}}{{-1}^{3} - {x}^{3}} \cdot \left(-1 \cdot -1 + \left(x \cdot x + -1 \cdot x\right)\right)} \]

   3. +-commutative 56.9%

      \[\leadsto \frac{\frac{-1 - \color{blue}{\left(\left(1 - x\right) + x\right)}}{1 - x}}{{-1}^{3} - {x}^{3}} \cdot \left(-1 \cdot -1 + \left(x \cdot x + -1 \cdot x\right)\right) \]

   4. associate-+l- 65.8%

      \[\leadsto \frac{\frac{-1 - \color{blue}{\left(1 - \left(x - x\right)\right)}}{1 - x}}{{-1}^{3} - {x}^{3}} \cdot \left(-1 \cdot -1 + \left(x \cdot x + -1 \cdot x\right)\right) \]

   5. metadata-eval 65.8%

      \[\leadsto \frac{\frac{-1 - \left(1 - \left(x - x\right)\right)}{1 - x}}{\color{blue}{-1} - {x}^{3}} \cdot \left(-1 \cdot -1 + \left(x \cdot x + -1 \cdot x\right)\right) \]

   6. metadata-eval 65.8%

      \[\leadsto \frac{\frac{-1 - \left(1 - \left(x - x\right)\right)}{1 - x}}{-1 - {x}^{3}} \cdot \left(\color{blue}{1} + \left(x \cdot x + -1 \cdot x\right)\right) \]

   7. distribute-rgt-out 65.8%

      \[\leadsto \frac{\frac{-1 - \left(1 - \left(x - x\right)\right)}{1 - x}}{-1 - {x}^{3}} \cdot \left(1 + \color{blue}{x \cdot \left(x + -1\right)}\right) \]

   8. +-commutative 65.8%

      \[\leadsto \frac{\frac{-1 - \left(1 - \left(x - x\right)\right)}{1 - x}}{-1 - {x}^{3}} \cdot \left(1 + x \cdot \color{blue}{\left(-1 + x\right)}\right) \]

8. Applied egg-rr 65.8%

   \[\leadsto \color{blue}{\frac{\frac{-1 - \left(1 - \left(x - x\right)\right)}{1 - x}}{-1 - {x}^{3}} \cdot \left(1 + x \cdot \left(-1 + x\right)\right)} \]
9. Step-by-step derivation

   1. associate-/l/ 65.2%

      \[\leadsto \color{blue}{\frac{-1 - \left(1 - \left(x - x\right)\right)}{\left(-1 - {x}^{3}\right) \cdot \left(1 - x\right)}} \cdot \left(1 + x \cdot \left(-1 + x\right)\right) \]

   2. +-inverses 65.2%

      \[\leadsto \frac{-1 - \left(1 - \color{blue}{0}\right)}{\left(-1 - {x}^{3}\right) \cdot \left(1 - x\right)} \cdot \left(1 + x \cdot \left(-1 + x\right)\right) \]

   3. metadata-eval 65.2%

      \[\leadsto \frac{-1 - \color{blue}{1}}{\left(-1 - {x}^{3}\right) \cdot \left(1 - x\right)} \cdot \left(1 + x \cdot \left(-1 + x\right)\right) \]

   4. metadata-eval 65.2%

      \[\leadsto \frac{\color{blue}{-2}}{\left(-1 - {x}^{3}\right) \cdot \left(1 - x\right)} \cdot \left(1 + x \cdot \left(-1 + x\right)\right) \]

   5. +-commutative 65.2%

      \[\leadsto \frac{-2}{\left(-1 - {x}^{3}\right) \cdot \left(1 - x\right)} \cdot \left(1 + x \cdot \color{blue}{\left(x + -1\right)}\right) \]

10. Simplified 65.2%

    \[\leadsto \color{blue}{\frac{-2}{\left(-1 - {x}^{3}\right) \cdot \left(1 - x\right)} \cdot \left(1 + x \cdot \left(x + -1\right)\right)} \]
11. Step-by-step derivation

    1. *-commutative 65.2%

       \[\leadsto \frac{-2}{\color{blue}{\left(1 - x\right) \cdot \left(-1 - {x}^{3}\right)}} \cdot \left(1 + x \cdot \left(x + -1\right)\right) \]

    2. associate-/r* 65.8%

       \[\leadsto \color{blue}{\frac{\frac{-2}{1 - x}}{-1 - {x}^{3}}} \cdot \left(1 + x \cdot \left(x + -1\right)\right) \]

    3. metadata-eval 65.8%

       \[\leadsto \frac{\frac{\color{blue}{-1 - 1}}{1 - x}}{-1 - {x}^{3}} \cdot \left(1 + x \cdot \left(x + -1\right)\right) \]

    4. associate-/r/ 68.9%

       \[\leadsto \color{blue}{\frac{\frac{-1 - 1}{1 - x}}{\frac{-1 - {x}^{3}}{1 + x \cdot \left(x + -1\right)}}} \]

    5. metadata-eval 68.9%

       \[\leadsto \frac{\frac{-1 - 1}{1 - x}}{\frac{\color{blue}{{-1}^{3}} - {x}^{3}}{1 + x \cdot \left(x + -1\right)}} \]

    6. metadata-eval 68.9%

       \[\leadsto \frac{\frac{-1 - 1}{1 - x}}{\frac{{-1}^{3} - {x}^{3}}{\color{blue}{-1 \cdot -1} + x \cdot \left(x + -1\right)}} \]

    7. distribute-rgt-in 68.9%

       \[\leadsto \frac{\frac{-1 - 1}{1 - x}}{\frac{{-1}^{3} - {x}^{3}}{-1 \cdot -1 + \color{blue}{\left(x \cdot x + -1 \cdot x\right)}}} \]

    8. flip3-- 99.9%

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

    9. metadata-eval 99.9%

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

    10. associate-/l/ 99.0%

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

    11. *-commutative 99.0%

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

12. Applied egg-rr 99.0%

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

    1. associate-/l/ 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

    \[\leadsto \frac{\frac{-2}{-1 - x}}{1 - x} \]
16. Add Preprocessing

Alternative 2: 99.4% 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 76.0%

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

   1. sub-neg 76.0%

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

   2. +-commutative 76.0%

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

   3. distribute-neg-frac2 76.0%

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

   4. neg-sub0 76.0%

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

   5. associate-+l- 76.0%

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

   6. neg-sub0 76.0%

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

   7. remove-double-neg 76.0%

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

   8. distribute-neg-in 76.0%

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

   9. sub-neg 76.0%

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

   10. distribute-neg-frac2 76.0%

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

   11. sub-neg 76.0%

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

   12. +-commutative 76.0%

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

   13. unsub-neg 76.0%

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

   14. sub-neg 76.0%

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

   15. +-commutative 76.0%

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

   16. unsub-neg 76.0%

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

   17. metadata-eval 76.0%

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

3. Simplified 76.0%

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

   1. sub-neg 76.0%

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

   2. distribute-neg-frac 76.0%

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

   3. metadata-eval 76.0%

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

6. Applied egg-rr 76.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

   \[\leadsto \frac{2}{\left(-1 - x\right) \cdot \left(-1 + x\right)} \]
10. Add Preprocessing

Alternative 3: 50.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. sub-neg 76.0%

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

   2. +-commutative 76.0%

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

   3. distribute-neg-frac2 76.0%

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

   4. neg-sub0 76.0%

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

   5. associate-+l- 76.0%

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

   6. neg-sub0 76.0%

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

   7. remove-double-neg 76.0%

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

   8. distribute-neg-in 76.0%

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

   9. sub-neg 76.0%

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

   10. distribute-neg-frac2 76.0%

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

   11. sub-neg 76.0%

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

   12. +-commutative 76.0%

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

   13. unsub-neg 76.0%

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

   14. sub-neg 76.0%

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

   15. +-commutative 76.0%

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

   16. unsub-neg 76.0%

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

   17. metadata-eval 76.0%

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

3. Simplified 76.0%

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

   1. frac-sub 77.3%

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

   2. *-rgt-identity 77.3%

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

   3. metadata-eval 77.3%

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

   4. div-inv 77.3%

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

   5. associate-/r* 77.3%

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

   6. metadata-eval 77.3%

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

   7. div-inv 77.3%

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

   8. *-un-lft-identity 77.3%

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

   9. associate--l- 80.4%

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

   10. div-inv 80.4%

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

   11. metadata-eval 80.4%

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

   12. *-rgt-identity 80.4%

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

   13. div-inv 80.4%

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

   14. metadata-eval 80.4%

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

   15. *-rgt-identity 80.4%

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

6. Applied egg-rr 80.4%

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

7. Taylor expanded in x around 0 99.9%

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

   1. flip-- 98.9%

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

   2. +-commutative 98.9%

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

   3. associate-/r/ 91.3%

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

   4. metadata-eval 91.3%

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

   5. metadata-eval 91.3%

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

   6. pow2 91.3%

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

   7. +-commutative 91.3%

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

9. Applied egg-rr 91.3%

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

    1. *-commutative 91.3%

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

    2. +-commutative 91.3%

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

    3. associate-/l/ 91.1%

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

11. Simplified 91.1%

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

12. Taylor expanded in x around 0 52.3%

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

13. Final simplification 52.3%

    \[\leadsto -2 \cdot \left(-1 + x\right) \]
14. Add Preprocessing

Alternative 4: 51.5% 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 76.0%

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

   1. sub-neg 76.0%

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

   2. +-commutative 76.0%

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

   3. distribute-neg-frac2 76.0%

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

   4. neg-sub0 76.0%

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

   5. associate-+l- 76.0%

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

   6. neg-sub0 76.0%

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

   7. remove-double-neg 76.0%

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

   8. distribute-neg-in 76.0%

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

   9. sub-neg 76.0%

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

   10. distribute-neg-frac2 76.0%

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

   11. sub-neg 76.0%

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

   12. +-commutative 76.0%

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

   13. unsub-neg 76.0%

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

   14. sub-neg 76.0%

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

   15. +-commutative 76.0%

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

   16. unsub-neg 76.0%

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

   17. metadata-eval 76.0%

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

3. Simplified 76.0%

   \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.5%

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

6. Final simplification 52.5%

   \[\leadsto 2 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))