Asymptote A

Percentage Accurate: 78.0% → 99.9%

Time: 5.9s

Alternatives: 5

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 77.9%

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

   1. sub-neg 77.9%

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

   2. +-commutative 77.9%

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

   3. distribute-neg-frac2 77.9%

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

   4. neg-sub0 77.9%

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

   5. associate-+l- 77.9%

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

   6. neg-sub0 77.9%

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

   7. remove-double-neg 77.9%

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

   8. distribute-neg-in 77.9%

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

   9. sub-neg 77.9%

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

   10. distribute-neg-frac2 77.9%

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

   11. sub-neg 77.9%

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

   12. +-commutative 77.9%

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

   13. unsub-neg 77.9%

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

   14. sub-neg 77.9%

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

   15. +-commutative 77.9%

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

   16. unsub-neg 77.9%

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

   17. metadata-eval 77.9%

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

3. Simplified 77.9%

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

   1. sub-neg 77.9%

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

   2. distribute-neg-frac 77.9%

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

   3. metadata-eval 77.9%

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

6. Applied egg-rr 77.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. sub-neg 77.9%

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

   2. +-commutative 77.9%

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

   3. distribute-neg-frac2 77.9%

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

   4. neg-sub0 77.9%

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

   5. associate-+l- 77.9%

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

   6. neg-sub0 77.9%

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

   7. remove-double-neg 77.9%

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

   8. distribute-neg-in 77.9%

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

   9. sub-neg 77.9%

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

   10. distribute-neg-frac2 77.9%

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

   11. sub-neg 77.9%

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

   12. +-commutative 77.9%

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

   13. unsub-neg 77.9%

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

   14. sub-neg 77.9%

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

   15. +-commutative 77.9%

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

   16. unsub-neg 77.9%

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

   17. metadata-eval 77.9%

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

3. Simplified 77.9%

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

   1. sub-neg 77.9%

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

   2. distribute-neg-frac 77.9%

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

   3. metadata-eval 77.9%

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

6. Applied egg-rr 77.9%

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

8. Simplified 99.9%

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

   1. flip-- 99.6%

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

   2. associate-/r/ 93.2%

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

   3. metadata-eval 93.2%

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

   4. pow2 93.2%

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

   5. +-commutative 93.2%

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

10. Applied egg-rr 93.2%

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

    1. +-commutative 93.2%

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

    2. metadata-eval 93.2%

       \[\leadsto \frac{\frac{-2}{1 - x}}{1 - {x}^{2}} \cdot \left(\color{blue}{\left(0 - 1\right)} + x\right) \]

    3. associate--r- 93.2%

       \[\leadsto \frac{\frac{-2}{1 - x}}{1 - {x}^{2}} \cdot \color{blue}{\left(0 - \left(1 - x\right)\right)} \]

    4. neg-sub0 93.2%

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

    5. associate-*l/ 99.6%

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

    6. associate-/l* 99.5%

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

    7. metadata-eval 99.5%

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

    8. distribute-neg-frac 99.5%

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

    9. distribute-neg-frac2 99.5%

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

    10. neg-sub0 99.5%

        \[\leadsto \frac{2}{\color{blue}{0 - \left(1 - x\right)}} \cdot \frac{-\left(1 - x\right)}{1 - {x}^{2}} \]

    11. associate--r- 99.5%

        \[\leadsto \frac{2}{\color{blue}{\left(0 - 1\right) + x}} \cdot \frac{-\left(1 - x\right)}{1 - {x}^{2}} \]

    12. metadata-eval 99.5%

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

    13. +-commutative 99.5%

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

    14. neg-sub0 99.5%

        \[\leadsto \frac{2}{x + -1} \cdot \frac{\color{blue}{0 - \left(1 - x\right)}}{1 - {x}^{2}} \]

    15. associate--r- 99.5%

        \[\leadsto \frac{2}{x + -1} \cdot \frac{\color{blue}{\left(0 - 1\right) + x}}{1 - {x}^{2}} \]

    16. metadata-eval 99.5%

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

    17. +-commutative 99.5%

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

12. Simplified 99.5%

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

    1. *-commutative 99.5%

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

    2. clear-num 99.5%

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

    3. metadata-eval 99.5%

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

    4. unpow2 99.5%

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

    5. +-commutative 99.5%

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

    6. flip-- 99.8%

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

    7. frac-times 99.6%

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

    8. metadata-eval 99.6%

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

    9. +-commutative 99.6%

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

14. Applied egg-rr 99.6%

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

    1. +-commutative 99.6%

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

16. Simplified 99.6%

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

17. Final simplification 99.6%

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

Alternative 3: 52.2% accurate, 1.8× 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(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 77.9%

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

   1. sub-neg 77.9%

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

   2. +-commutative 77.9%

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

   3. distribute-neg-frac2 77.9%

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

   4. neg-sub0 77.9%

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

   5. associate-+l- 77.9%

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

   6. neg-sub0 77.9%

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

   7. remove-double-neg 77.9%

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

   8. distribute-neg-in 77.9%

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

   9. sub-neg 77.9%

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

   10. distribute-neg-frac2 77.9%

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

   11. sub-neg 77.9%

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

   12. +-commutative 77.9%

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

   13. unsub-neg 77.9%

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

   14. sub-neg 77.9%

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

   15. +-commutative 77.9%

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

   16. unsub-neg 77.9%

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

   17. metadata-eval 77.9%

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

3. Simplified 77.9%

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

   1. sub-neg 77.9%

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

   2. distribute-neg-frac 77.9%

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

   3. metadata-eval 77.9%

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

6. Applied egg-rr 77.9%

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

8. Simplified 99.9%

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

   1. flip-- 99.6%

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

   2. associate-/r/ 93.2%

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

   3. metadata-eval 93.2%

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

   4. pow2 93.2%

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

   5. +-commutative 93.2%

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

10. Applied egg-rr 93.2%

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

    1. +-commutative 93.2%

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

    2. metadata-eval 93.2%

       \[\leadsto \frac{\frac{-2}{1 - x}}{1 - {x}^{2}} \cdot \left(\color{blue}{\left(0 - 1\right)} + x\right) \]

    3. associate--r- 93.2%

       \[\leadsto \frac{\frac{-2}{1 - x}}{1 - {x}^{2}} \cdot \color{blue}{\left(0 - \left(1 - x\right)\right)} \]

    4. neg-sub0 93.2%

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

    5. associate-*l/ 99.6%

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

    6. associate-/l* 99.5%

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

    7. metadata-eval 99.5%

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

    8. distribute-neg-frac 99.5%

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

    9. distribute-neg-frac2 99.5%

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

    10. neg-sub0 99.5%

        \[\leadsto \frac{2}{\color{blue}{0 - \left(1 - x\right)}} \cdot \frac{-\left(1 - x\right)}{1 - {x}^{2}} \]

    11. associate--r- 99.5%

        \[\leadsto \frac{2}{\color{blue}{\left(0 - 1\right) + x}} \cdot \frac{-\left(1 - x\right)}{1 - {x}^{2}} \]

    12. metadata-eval 99.5%

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

    13. +-commutative 99.5%

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

    14. neg-sub0 99.5%

        \[\leadsto \frac{2}{x + -1} \cdot \frac{\color{blue}{0 - \left(1 - x\right)}}{1 - {x}^{2}} \]

    15. associate--r- 99.5%

        \[\leadsto \frac{2}{x + -1} \cdot \frac{\color{blue}{\left(0 - 1\right) + x}}{1 - {x}^{2}} \]

    16. metadata-eval 99.5%

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

    17. +-commutative 99.5%

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

12. Simplified 99.5%

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

13. Taylor expanded in x around 0 51.9%

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

14. Final simplification 51.9%

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

Alternative 4: 10.8% 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 77.9%

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

   1. sub-neg 77.9%

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

   2. +-commutative 77.9%

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

   3. distribute-neg-frac2 77.9%

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

   4. neg-sub0 77.9%

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

   5. associate-+l- 77.9%

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

   6. neg-sub0 77.9%

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

   7. remove-double-neg 77.9%

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

   8. distribute-neg-in 77.9%

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

   9. sub-neg 77.9%

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

   10. distribute-neg-frac2 77.9%

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

   11. sub-neg 77.9%

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

   12. +-commutative 77.9%

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

   13. unsub-neg 77.9%

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

   14. sub-neg 77.9%

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

   15. +-commutative 77.9%

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

   16. unsub-neg 77.9%

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

   17. metadata-eval 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in x around 0 50.3%

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

6. Taylor expanded in x around inf 10.7%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 10.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 5: 51.2% 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 77.9%

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

   1. sub-neg 77.9%

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

   2. +-commutative 77.9%

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

   3. distribute-neg-frac2 77.9%

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

   4. neg-sub0 77.9%

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

   5. associate-+l- 77.9%

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

   6. neg-sub0 77.9%

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

   7. remove-double-neg 77.9%

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

   8. distribute-neg-in 77.9%

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

   9. sub-neg 77.9%

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

   10. distribute-neg-frac2 77.9%

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

   11. sub-neg 77.9%

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

   12. +-commutative 77.9%

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

   13. unsub-neg 77.9%

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

   14. sub-neg 77.9%

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

   15. +-commutative 77.9%

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

   16. unsub-neg 77.9%

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

   17. metadata-eval 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in x around 0 51.0%

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

6. Final simplification 51.0%

   \[\leadsto 2 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))