Asymptote A

Percentage Accurate: 77.8% → 99.9%

Time: 6.6s

Alternatives: 6

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: 77.8% 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 79.5%

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

   1. sub-neg 79.5%

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

   2. +-commutative 79.5%

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

   3. distribute-neg-frac2 79.5%

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

   4. neg-sub0 79.5%

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

   5. associate-+l- 79.5%

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

   6. neg-sub0 79.5%

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

   7. remove-double-neg 79.5%

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

   8. distribute-neg-in 79.5%

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

   9. sub-neg 79.5%

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

   10. distribute-neg-frac2 79.5%

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

   11. sub-neg 79.5%

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

   12. +-commutative 79.5%

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

   13. unsub-neg 79.5%

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

   14. sub-neg 79.5%

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

   15. +-commutative 79.5%

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

   16. unsub-neg 79.5%

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

   17. metadata-eval 79.5%

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

3. Simplified 79.5%

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

   1. frac-sub 80.0%

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

   2. *-rgt-identity 80.0%

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

   3. metadata-eval 80.0%

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

   4. div-inv 80.0%

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

   5. associate-/r* 80.0%

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

   6. metadata-eval 80.0%

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

   7. div-inv 80.0%

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

   8. *-un-lft-identity 80.0%

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

   9. associate--l- 82.7%

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

   10. div-inv 82.7%

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

   11. metadata-eval 82.7%

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

   12. *-rgt-identity 82.7%

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

   13. div-inv 82.7%

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

   14. metadata-eval 82.7%

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

   15. *-rgt-identity 82.7%

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

6. Applied egg-rr 82.7%

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

   1. +-commutative 82.7%

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

   2. associate-+l- 99.9%

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

8. Applied egg-rr 99.9%

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

   1. +-inverses 99.9%

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

   2. metadata-eval 99.9%

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

   3. metadata-eval 99.9%

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

   4. *-un-lft-identity 99.9%

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

10. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{1 \cdot \frac{\frac{-2}{1 - x}}{-1 - x}} \]
11. Step-by-step derivation

    1. *-lft-identity 99.9%

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

12. Simplified 99.9%

    \[\leadsto \color{blue}{\frac{\frac{-2}{1 - x}}{-1 - x}} \]
13. Add Preprocessing

Alternative 2: 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 79.5%

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

   1. sub-neg 79.5%

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

   2. +-commutative 79.5%

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

   3. distribute-neg-frac2 79.5%

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

   4. neg-sub0 79.5%

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

   5. associate-+l- 79.5%

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

   6. neg-sub0 79.5%

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

   7. remove-double-neg 79.5%

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

   8. distribute-neg-in 79.5%

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

   9. sub-neg 79.5%

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

   10. distribute-neg-frac2 79.5%

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

   11. sub-neg 79.5%

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

   12. +-commutative 79.5%

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

   13. unsub-neg 79.5%

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

   14. sub-neg 79.5%

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

   15. +-commutative 79.5%

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

   16. unsub-neg 79.5%

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

   17. metadata-eval 79.5%

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

3. Simplified 79.5%

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

   1. sub-neg 79.5%

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

   2. distribute-neg-frac 79.5%

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

   3. metadata-eval 79.5%

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

6. Applied egg-rr 79.5%

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

   1. metadata-eval 79.5%

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

   2. distribute-neg-frac 79.5%

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

   3. unsub-neg 79.5%

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

   4. *-rgt-identity 79.5%

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

   5. *-inverses 79.5%

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

   6. associate-/r* 57.4%

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

   7. *-commutative 57.4%

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

   8. *-lft-identity 57.4%

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

   9. associate-/r* 79.5%

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

   10. associate-*r/ 79.5%

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

   11. associate-*l/ 79.5%

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

   12. distribute-lft-out-- 79.5%

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

   13. *-inverses 79.5%

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

   14. div-sub 80.0%

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

   15. associate--r+ 82.7%

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

   16. *-commutative 82.7%

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

   17. times-frac 82.7%

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

8. Simplified 99.7%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.8%

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

10. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{\frac{-2}{1 - x} \cdot \frac{1}{-1 - x}} \]
11. Step-by-step derivation

    1. metadata-eval 99.8%

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

    2. metadata-eval 99.8%

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

    3. +-inverses 99.8%

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

    4. associate-*l/ 99.9%

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

    5. un-div-inv 99.9%

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

    6. +-inverses 99.9%

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

    7. metadata-eval 99.9%

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

    8. metadata-eval 99.9%

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

12. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{\frac{\frac{-2}{-1 - x}}{1 - x}} \]
13. Add Preprocessing

Alternative 3: 99.5% 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 79.5%

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

   1. sub-neg 79.5%

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

   2. +-commutative 79.5%

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

   3. distribute-neg-frac2 79.5%

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

   4. neg-sub0 79.5%

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

   5. associate-+l- 79.5%

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

   6. neg-sub0 79.5%

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

   7. remove-double-neg 79.5%

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

   8. distribute-neg-in 79.5%

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

   9. sub-neg 79.5%

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

   10. distribute-neg-frac2 79.5%

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

   11. sub-neg 79.5%

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

   12. +-commutative 79.5%

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

   13. unsub-neg 79.5%

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

   14. sub-neg 79.5%

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

   15. +-commutative 79.5%

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

   16. unsub-neg 79.5%

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

   17. metadata-eval 79.5%

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

3. Simplified 79.5%

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

   1. sub-neg 79.5%

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

   2. distribute-neg-frac 79.5%

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

   3. metadata-eval 79.5%

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

6. Applied egg-rr 79.5%

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

   1. metadata-eval 79.5%

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

   2. distribute-neg-frac 79.5%

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

   3. unsub-neg 79.5%

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

   4. *-rgt-identity 79.5%

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

   5. *-inverses 79.5%

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

   6. associate-/r* 57.4%

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

   7. *-commutative 57.4%

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

   8. *-lft-identity 57.4%

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

   9. associate-/r* 79.5%

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

   10. associate-*r/ 79.5%

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

   11. associate-*l/ 79.5%

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

   12. distribute-lft-out-- 79.5%

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

   13. *-inverses 79.5%

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

   14. div-sub 80.0%

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

   15. associate--r+ 82.7%

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

   16. *-commutative 82.7%

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

   17. times-frac 82.7%

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

8. Simplified 99.7%

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

Alternative 4: 52.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. sub-neg 79.5%

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

   2. +-commutative 79.5%

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

   3. distribute-neg-frac2 79.5%

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

   4. neg-sub0 79.5%

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

   5. associate-+l- 79.5%

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

   6. neg-sub0 79.5%

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

   7. remove-double-neg 79.5%

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

   8. distribute-neg-in 79.5%

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

   9. sub-neg 79.5%

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

   10. distribute-neg-frac2 79.5%

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

   11. sub-neg 79.5%

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

   12. +-commutative 79.5%

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

   13. unsub-neg 79.5%

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

   14. sub-neg 79.5%

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

   15. +-commutative 79.5%

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

   16. unsub-neg 79.5%

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

   17. metadata-eval 79.5%

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

3. Simplified 79.5%

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

   1. sub-neg 79.5%

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

   2. distribute-neg-frac 79.5%

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

   3. metadata-eval 79.5%

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

6. Applied egg-rr 79.5%

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

   1. metadata-eval 79.5%

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

   2. distribute-neg-frac 79.5%

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

   3. unsub-neg 79.5%

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

   4. *-rgt-identity 79.5%

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

   5. *-inverses 79.5%

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

   6. associate-/r* 57.4%

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

   7. *-commutative 57.4%

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

   8. *-lft-identity 57.4%

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

   9. associate-/r* 79.5%

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

   10. associate-*r/ 79.5%

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

   11. associate-*l/ 79.5%

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

   12. distribute-lft-out-- 79.5%

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

   13. *-inverses 79.5%

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

   14. div-sub 80.0%

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

   15. associate--r+ 82.7%

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

   16. *-commutative 82.7%

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

   17. times-frac 82.7%

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

8. Simplified 99.7%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.8%

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

10. Applied egg-rr 99.8%

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

11. Taylor expanded in x around 0 54.9%

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

12. Final simplification 54.9%

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

Alternative 5: 51.9% 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 79.5%

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

   1. sub-neg 79.5%

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

   2. +-commutative 79.5%

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

   3. distribute-neg-frac2 79.5%

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

   4. neg-sub0 79.5%

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

   5. associate-+l- 79.5%

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

   6. neg-sub0 79.5%

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

   7. remove-double-neg 79.5%

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

   8. distribute-neg-in 79.5%

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

   9. sub-neg 79.5%

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

   10. distribute-neg-frac2 79.5%

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

   11. sub-neg 79.5%

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

   12. +-commutative 79.5%

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

   13. unsub-neg 79.5%

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

   14. sub-neg 79.5%

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

   15. +-commutative 79.5%

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

   16. unsub-neg 79.5%

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

   17. metadata-eval 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in x around 0 54.0%

   \[\leadsto \color{blue}{2} \]
6. Add Preprocessing

Alternative 6: 10.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 79.5%

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

   1. sub-neg 79.5%

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

   2. +-commutative 79.5%

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

   3. distribute-neg-frac2 79.5%

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

   4. neg-sub0 79.5%

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

   5. associate-+l- 79.5%

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

   6. neg-sub0 79.5%

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

   7. remove-double-neg 79.5%

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

   8. distribute-neg-in 79.5%

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

   9. sub-neg 79.5%

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

   10. distribute-neg-frac2 79.5%

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

   11. sub-neg 79.5%

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

   12. +-commutative 79.5%

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

   13. unsub-neg 79.5%

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

   14. sub-neg 79.5%

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

   15. +-commutative 79.5%

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

   16. unsub-neg 79.5%

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

   17. metadata-eval 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in x around 0 53.5%

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

6. Taylor expanded in x around inf 11.2%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))