Asymptote A

Percentage Accurate: 78.0% → 99.9%

Time: 7.1s

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 76.1%

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

   1. sub-neg 76.1%

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

   2. +-commutative 76.1%

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

   3. distribute-neg-frac2 76.1%

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

   4. neg-sub0 76.1%

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

   5. associate-+l- 76.1%

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

   6. neg-sub0 76.1%

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

   7. remove-double-neg 76.1%

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

   8. distribute-neg-in 76.1%

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

   9. sub-neg 76.1%

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

   10. distribute-neg-frac2 76.1%

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

   11. sub-neg 76.1%

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

   12. +-commutative 76.1%

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

   13. unsub-neg 76.1%

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

   14. sub-neg 76.1%

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

   15. +-commutative 76.1%

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

   16. unsub-neg 76.1%

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

   17. metadata-eval 76.1%

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

3. Simplified 76.1%

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

   1. frac-sub 76.2%

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

   2. *-rgt-identity 76.2%

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

   3. metadata-eval 76.2%

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

   4. div-inv 76.2%

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

   5. associate-/r* 76.3%

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

   6. metadata-eval 76.3%

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

   7. div-inv 76.3%

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

   8. *-un-lft-identity 76.3%

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

   9. associate--l- 79.7%

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

   10. div-inv 79.7%

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

   11. metadata-eval 79.7%

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

   12. *-rgt-identity 79.7%

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

   13. div-inv 79.7%

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

   14. metadata-eval 79.7%

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

   15. *-rgt-identity 79.7%

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

6. Applied egg-rr 79.7%

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

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. *-un-lft-identity 99.9%

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

   2. add-sqr-sqrt 24.5%

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

   3. times-frac 24.5%

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

9. Applied egg-rr 52.6%

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

    1. frac-times 52.6%

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

    2. *-un-lft-identity 52.6%

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

    3. frac-2neg 52.6%

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

    4. metadata-eval 52.6%

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

    5. distribute-neg-in 52.6%

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

    6. metadata-eval 52.6%

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

    7. *-rgt-identity 52.6%

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

    8. sub-neg 52.6%

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

    9. *-rgt-identity 52.6%

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

    10. hypot-undefine 52.6%

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

    11. metadata-eval 52.6%

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

    12. add-sqr-sqrt 52.6%

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

    13. hypot-undefine 52.6%

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

    14. metadata-eval 52.6%

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

    15. add-sqr-sqrt 75.3%

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

    16. add-sqr-sqrt 99.9%

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

    17. +-commutative 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

    \[\leadsto \frac{\frac{2}{1 - x}}{1 + x} \]
13. 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 76.1%

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

   1. sub-neg 76.1%

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

   2. +-commutative 76.1%

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

   3. distribute-neg-frac2 76.1%

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

   4. neg-sub0 76.1%

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

   5. associate-+l- 76.1%

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

   6. neg-sub0 76.1%

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

   7. remove-double-neg 76.1%

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

   8. distribute-neg-in 76.1%

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

   9. sub-neg 76.1%

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

   10. distribute-neg-frac2 76.1%

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

   11. sub-neg 76.1%

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

   12. +-commutative 76.1%

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

   13. unsub-neg 76.1%

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

   14. sub-neg 76.1%

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

   15. +-commutative 76.1%

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

   16. unsub-neg 76.1%

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

   17. metadata-eval 76.1%

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

3. Simplified 76.1%

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

   1. sub-neg 76.1%

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

   2. distribute-neg-frac 76.1%

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

   3. metadata-eval 76.1%

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

6. Applied egg-rr 76.1%

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

   1. metadata-eval 76.1%

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

   2. distribute-neg-frac 76.1%

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

   3. unsub-neg 76.1%

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

   4. *-rgt-identity 76.1%

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

   5. *-inverses 76.1%

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

   6. associate-/r* 58.0%

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

   7. *-commutative 58.0%

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

   8. *-lft-identity 58.0%

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

   9. associate-/r* 76.1%

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

   10. associate-*r/ 76.1%

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

   11. associate-*l/ 76.1%

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

   12. distribute-lft-out-- 76.1%

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

   13. *-inverses 76.1%

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

   14. div-sub 76.2%

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

   15. associate--r+ 79.7%

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

   16. *-commutative 79.7%

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

   17. times-frac 79.6%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 3: 2.9% 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 76.1%

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

   1. sub-neg 76.1%

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

   2. +-commutative 76.1%

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

   3. distribute-neg-frac2 76.1%

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

   4. neg-sub0 76.1%

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

   5. associate-+l- 76.1%

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

   6. neg-sub0 76.1%

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

   7. remove-double-neg 76.1%

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

   8. distribute-neg-in 76.1%

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

   9. sub-neg 76.1%

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

   10. distribute-neg-frac2 76.1%

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

   11. sub-neg 76.1%

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

   12. +-commutative 76.1%

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

   13. unsub-neg 76.1%

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

   14. sub-neg 76.1%

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

   15. +-commutative 76.1%

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

   16. unsub-neg 76.1%

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

   17. metadata-eval 76.1%

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

3. Simplified 76.1%

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

   1. sub-neg 76.1%

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

   2. distribute-neg-frac 76.1%

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

   3. metadata-eval 76.1%

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

6. Applied egg-rr 76.1%

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

   1. metadata-eval 76.1%

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

   2. distribute-neg-frac 76.1%

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

   3. unsub-neg 76.1%

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

   4. *-rgt-identity 76.1%

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

   5. *-inverses 76.1%

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

   6. associate-/r* 58.0%

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

   7. *-commutative 58.0%

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

   8. *-lft-identity 58.0%

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

   9. associate-/r* 76.1%

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

   10. associate-*r/ 76.1%

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

   11. associate-*l/ 76.1%

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

   12. distribute-lft-out-- 76.1%

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

   13. *-inverses 76.1%

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

   14. div-sub 76.2%

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

   15. associate--r+ 79.7%

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

   16. *-commutative 79.7%

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

   17. times-frac 79.6%

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

8. Simplified 99.9%

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

   1. associate-/r* 99.9%

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

   2. flip3-- 74.2%

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

   3. associate-/r/ 69.4%

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

   4. metadata-eval 69.4%

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

   5. metadata-eval 69.4%

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

   6. distribute-rgt-out 69.4%

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

10. Applied egg-rr 69.4%

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

    1. metadata-eval 69.4%

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

    2. associate-*l/ 74.2%

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

    3. associate-/l* 74.2%

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

    4. clear-num 74.2%

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

    5. metadata-eval 74.2%

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

    6. metadata-eval 74.2%

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

    7. distribute-rgt-in 74.2%

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

    8. flip3-- 99.8%

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

    9. div-inv 99.9%

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

12. Applied egg-rr 45.5%

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

13. Taylor expanded in x around 0 2.7%

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

14. Final simplification 2.7%

    \[\leadsto -2 \]
15. Add Preprocessing

Alternative 4: 52.1% 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 76.1%

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

   1. sub-neg 76.1%

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

   2. +-commutative 76.1%

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

   3. distribute-neg-frac2 76.1%

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

   4. neg-sub0 76.1%

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

   5. associate-+l- 76.1%

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

   6. neg-sub0 76.1%

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

   7. remove-double-neg 76.1%

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

   8. distribute-neg-in 76.1%

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

   9. sub-neg 76.1%

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

   10. distribute-neg-frac2 76.1%

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

   11. sub-neg 76.1%

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

   12. +-commutative 76.1%

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

   13. unsub-neg 76.1%

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

   14. sub-neg 76.1%

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

   15. +-commutative 76.1%

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

   16. unsub-neg 76.1%

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

   17. metadata-eval 76.1%

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

3. Simplified 76.1%

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

5. Taylor expanded in x around 0 56.0%

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

6. Final simplification 56.0%

   \[\leadsto 2 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))