Asymptote A

Percentage Accurate: 77.1% → 99.9%

Time: 6.5s

Alternatives: 7

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: 77.1% 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 81.0%

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

   1. sub-neg 81.0%

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

   2. +-commutative 81.0%

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

   3. distribute-neg-frac 81.0%

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

   4. metadata-eval 81.0%

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

   5. metadata-eval 81.0%

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

   6. metadata-eval 81.0%

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

   7. associate-/r* 81.0%

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

   8. metadata-eval 81.0%

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

   9. neg-mul-1 81.0%

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

   10. sub0-neg 81.0%

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

   11. associate-+l- 81.0%

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

   12. neg-sub0 81.0%

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

   13. remove-double-neg 81.0%

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

   14. distribute-neg-in 81.0%

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

   15. sub-neg 81.0%

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

   16. mul-1-neg 81.0%

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

   17. metadata-eval 81.0%

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

   18. associate-/r* 81.0%

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

   19. metadata-eval 81.0%

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

   20. metadata-eval 81.0%

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

   21. metadata-eval 81.0%

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

   22. metadata-eval 81.0%

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

   23. associate-*l/ 81.0%

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

3. Simplified 81.0%

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

   1. sub-neg 81.0%

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

   2. distribute-neg-frac 81.0%

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

   3. metadata-eval 81.0%

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

5. Applied egg-rr 81.0%

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

   1. *-rgt-identity 81.0%

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

   2. cancel-sign-sub 81.0%

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

   3. distribute-neg-frac 81.0%

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

   4. metadata-eval 81.0%

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

   5. *-rgt-identity 81.0%

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

   6. *-inverses 81.0%

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

   7. associate-/r* 56.5%

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

   8. distribute-lft-neg-in 56.5%

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

   9. distribute-rgt-neg-in 56.5%

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

   10. *-commutative 56.5%

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

   11. *-lft-identity 56.5%

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

   12. *-inverses 56.5%

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

   13. times-frac 81.0%

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

   14. div-sub 81.7%

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

7. Simplified 99.7%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.8%

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

   3. +-commutative 99.8%

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

9. Applied egg-rr 99.8%

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

    1. un-div-inv 99.9%

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

    2. frac-2neg 99.9%

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

    3. add-sqr-sqrt 23.7%

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

    4. sqrt-unprod 38.4%

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

    5. sqr-neg 38.4%

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

    6. sqrt-unprod 14.9%

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

    7. add-sqr-sqrt 27.9%

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

    8. neg-sub0 27.9%

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

    9. associate--r- 27.9%

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

    10. metadata-eval 27.9%

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

    11. flip-+ 27.9%

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

    12. distribute-neg-frac 27.9%

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

    13. add-sqr-sqrt 13.0%

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

    14. sqrt-unprod 62.8%

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

    15. sqr-neg 62.8%

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

    16. sqrt-unprod 76.1%

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

    17. add-sqr-sqrt 99.7%

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

    18. frac-2neg 99.7%

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

    19. flip-+ 99.9%

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

11. Applied egg-rr 99.9%

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

    1. distribute-neg-frac 99.9%

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

    2. metadata-eval 99.9%

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

    3. div-inv 99.9%

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

    4. *-un-lft-identity 99.9%

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

    5. times-frac 99.9%

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

    6. metadata-eval 99.9%

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

    7. frac-2neg 99.9%

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

    8. metadata-eval 99.9%

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

    9. distribute-neg-in 99.9%

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

    10. metadata-eval 99.9%

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

    11. sub-neg 99.9%

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

    12. +-commutative 99.9%

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

13. Applied egg-rr 99.9%

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

    1. associate-*r/ 99.9%

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

    2. associate-*r/ 99.9%

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

    3. metadata-eval 99.9%

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

15. Simplified 99.9%

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

16. Final simplification 99.9%

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

Alternative 2: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x + 1}}{x}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / (x + 1.0)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 2.0d0
    else
        tmp = ((-2.0d0) / (x + 1.0d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / (x + 1.0)) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 2.0
	else:
		tmp = (-2.0 / (x + 1.0)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = Float64(Float64(-2.0 / Float64(x + 1.0)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = (-2.0 / (x + 1.0)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 86.4%

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

      1. sub-neg 86.4%

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

      2. +-commutative 86.4%

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

      3. distribute-neg-frac 86.4%

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

      4. metadata-eval 86.4%

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

      5. metadata-eval 86.4%

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

      6. metadata-eval 86.4%

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

      7. associate-/r* 86.4%

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

      8. metadata-eval 86.4%

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

      9. neg-mul-1 86.4%

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

      10. sub0-neg 86.4%

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

      11. associate-+l- 86.4%

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

      12. neg-sub0 86.4%

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

      13. remove-double-neg 86.4%

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

      14. distribute-neg-in 86.4%

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

      15. sub-neg 86.4%

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

      16. mul-1-neg 86.4%

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

      17. metadata-eval 86.4%

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

      18. associate-/r* 86.4%

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

      19. metadata-eval 86.4%

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

      20. metadata-eval 86.4%

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

      21. metadata-eval 86.4%

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

      22. metadata-eval 86.4%

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

      23. associate-*l/ 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x around 0 68.9%

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

   if 1 < x

   1. Initial program 63.6%

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

      1. sub-neg 63.6%

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

      2. +-commutative 63.6%

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

      3. distribute-neg-frac 63.6%

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

      4. metadata-eval 63.6%

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

      5. metadata-eval 63.6%

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

      6. metadata-eval 63.6%

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

      7. associate-/r* 63.6%

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

      8. metadata-eval 63.6%

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

      9. neg-mul-1 63.6%

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

      10. sub0-neg 63.6%

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

      11. associate-+l- 63.6%

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

      12. neg-sub0 63.6%

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

      13. remove-double-neg 63.6%

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

      14. distribute-neg-in 63.6%

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

      15. sub-neg 63.6%

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

      16. mul-1-neg 63.6%

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

      17. metadata-eval 63.6%

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

      18. associate-/r* 63.6%

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

      19. metadata-eval 63.6%

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

      20. metadata-eval 63.6%

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

      21. metadata-eval 63.6%

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

      22. metadata-eval 63.6%

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

      23. associate-*l/ 63.6%

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

   3. Simplified 63.6%

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

      1. frac-sub 66.0%

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

      2. *-rgt-identity 66.0%

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

      3. metadata-eval 66.0%

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

      4. div-inv 66.0%

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

      5. associate-/r* 66.0%

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

      6. *-un-lft-identity 66.0%

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

      7. metadata-eval 66.0%

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

      8. div-inv 66.0%

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

      9. associate--l- 70.6%

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

      10. div-inv 70.6%

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

      11. metadata-eval 70.6%

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

      12. *-rgt-identity 70.6%

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

      13. div-inv 70.6%

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

      14. metadata-eval 70.6%

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

      15. *-rgt-identity 70.6%

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

   5. Applied egg-rr 70.6%

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

   6. Taylor expanded in x around inf 97.4%

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

      1. div-inv 97.3%

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

      2. div-inv 97.3%

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

      3. associate-*l* 97.3%

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

      4. frac-2neg 97.3%

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

      5. metadata-eval 97.3%

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

      6. neg-sub0 97.3%

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

      7. associate--r- 97.3%

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

      8. metadata-eval 97.3%

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

   8. Applied egg-rr 97.3%

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

      1. associate-*r* 97.3%

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

      2. associate-*r/ 97.3%

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

      3. metadata-eval 97.3%

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

      4. associate-*l/ 97.4%

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

      5. associate-*r/ 97.4%

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

      6. metadata-eval 97.4%

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

      7. +-commutative 97.4%

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

   10. Simplified 97.4%

       \[\leadsto \color{blue}{\frac{\frac{-2}{x + 1}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x + 1}}{x}\\ \end{array} \]

Alternative 3: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{-1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 (/ (/ 2.0 x) (- -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = (2.0 / x) / (-1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 2.0d0
    else
        tmp = (2.0d0 / x) / ((-1.0d0) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = (2.0 / x) / (-1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 2.0
	else:
		tmp = (2.0 / x) / (-1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = Float64(Float64(2.0 / x) / Float64(-1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = (2.0 / x) / (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 86.4%

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

      1. sub-neg 86.4%

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

      2. +-commutative 86.4%

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

      3. distribute-neg-frac 86.4%

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

      4. metadata-eval 86.4%

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

      5. metadata-eval 86.4%

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

      6. metadata-eval 86.4%

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

      7. associate-/r* 86.4%

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

      8. metadata-eval 86.4%

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

      9. neg-mul-1 86.4%

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

      10. sub0-neg 86.4%

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

      11. associate-+l- 86.4%

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

      12. neg-sub0 86.4%

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

      13. remove-double-neg 86.4%

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

      14. distribute-neg-in 86.4%

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

      15. sub-neg 86.4%

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

      16. mul-1-neg 86.4%

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

      17. metadata-eval 86.4%

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

      18. associate-/r* 86.4%

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

      19. metadata-eval 86.4%

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

      20. metadata-eval 86.4%

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

      21. metadata-eval 86.4%

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

      22. metadata-eval 86.4%

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

      23. associate-*l/ 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x around 0 68.9%

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

   if 1 < x

   1. Initial program 63.6%

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

      1. sub-neg 63.6%

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

      2. +-commutative 63.6%

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

      3. distribute-neg-frac 63.6%

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

      4. metadata-eval 63.6%

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

      5. metadata-eval 63.6%

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

      6. metadata-eval 63.6%

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

      7. associate-/r* 63.6%

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

      8. metadata-eval 63.6%

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

      9. neg-mul-1 63.6%

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

      10. sub0-neg 63.6%

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

      11. associate-+l- 63.6%

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

      12. neg-sub0 63.6%

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

      13. remove-double-neg 63.6%

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

      14. distribute-neg-in 63.6%

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

      15. sub-neg 63.6%

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

      16. mul-1-neg 63.6%

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

      17. metadata-eval 63.6%

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

      18. associate-/r* 63.6%

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

      19. metadata-eval 63.6%

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

      20. metadata-eval 63.6%

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

      21. metadata-eval 63.6%

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

      22. metadata-eval 63.6%

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

      23. associate-*l/ 63.6%

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

   3. Simplified 63.6%

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

      1. frac-sub 66.0%

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

      2. *-rgt-identity 66.0%

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

      3. metadata-eval 66.0%

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

      4. div-inv 66.0%

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

      5. associate-/r* 66.0%

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

      6. *-un-lft-identity 66.0%

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

      7. metadata-eval 66.0%

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

      8. div-inv 66.0%

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

      9. associate--l- 70.6%

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

      10. div-inv 70.6%

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

      11. metadata-eval 70.6%

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

      12. *-rgt-identity 70.6%

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

      13. div-inv 70.6%

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

      14. metadata-eval 70.6%

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

      15. *-rgt-identity 70.6%

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

   5. Applied egg-rr 70.6%

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

   6. Taylor expanded in x around inf 97.4%

      \[\leadsto \frac{\color{blue}{\frac{2}{x}}}{-1 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{-1 - x}\\ \end{array} \]

Alternative 4: 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 81.0%

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

   1. sub-neg 81.0%

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

   2. +-commutative 81.0%

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

   3. distribute-neg-frac 81.0%

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

   4. metadata-eval 81.0%

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

   5. metadata-eval 81.0%

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

   6. metadata-eval 81.0%

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

   7. associate-/r* 81.0%

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

   8. metadata-eval 81.0%

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

   9. neg-mul-1 81.0%

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

   10. sub0-neg 81.0%

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

   11. associate-+l- 81.0%

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

   12. neg-sub0 81.0%

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

   13. remove-double-neg 81.0%

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

   14. distribute-neg-in 81.0%

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

   15. sub-neg 81.0%

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

   16. mul-1-neg 81.0%

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

   17. metadata-eval 81.0%

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

   18. associate-/r* 81.0%

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

   19. metadata-eval 81.0%

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

   20. metadata-eval 81.0%

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

   21. metadata-eval 81.0%

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

   22. metadata-eval 81.0%

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

   23. associate-*l/ 81.0%

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

3. Simplified 81.0%

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

   1. sub-neg 81.0%

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

   2. distribute-neg-frac 81.0%

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

   3. metadata-eval 81.0%

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

5. Applied egg-rr 81.0%

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

   1. *-rgt-identity 81.0%

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

   2. cancel-sign-sub 81.0%

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

   3. distribute-neg-frac 81.0%

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

   4. metadata-eval 81.0%

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

   5. *-rgt-identity 81.0%

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

   6. *-inverses 81.0%

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

   7. associate-/r* 56.5%

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

   8. distribute-lft-neg-in 56.5%

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

   9. distribute-rgt-neg-in 56.5%

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

   10. *-commutative 56.5%

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

   11. *-lft-identity 56.5%

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

   12. *-inverses 56.5%

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

   13. times-frac 81.0%

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

   14. div-sub 81.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 5: 52.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 (/ -2.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 2.0d0
    else
        tmp = (-2.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 2.0
	else:
		tmp = -2.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = Float64(-2.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = -2.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 86.4%

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

      1. sub-neg 86.4%

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

      2. +-commutative 86.4%

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

      3. distribute-neg-frac 86.4%

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

      4. metadata-eval 86.4%

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

      5. metadata-eval 86.4%

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

      6. metadata-eval 86.4%

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

      7. associate-/r* 86.4%

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

      8. metadata-eval 86.4%

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

      9. neg-mul-1 86.4%

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

      10. sub0-neg 86.4%

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

      11. associate-+l- 86.4%

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

      12. neg-sub0 86.4%

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

      13. remove-double-neg 86.4%

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

      14. distribute-neg-in 86.4%

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

      15. sub-neg 86.4%

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

      16. mul-1-neg 86.4%

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

      17. metadata-eval 86.4%

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

      18. associate-/r* 86.4%

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

      19. metadata-eval 86.4%

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

      20. metadata-eval 86.4%

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

      21. metadata-eval 86.4%

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

      22. metadata-eval 86.4%

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

      23. associate-*l/ 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x around 0 68.9%

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

   if 1 < x

   1. Initial program 63.6%

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

      1. sub-neg 63.6%

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

      2. +-commutative 63.6%

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

      3. distribute-neg-frac 63.6%

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

      4. metadata-eval 63.6%

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

      5. metadata-eval 63.6%

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

      6. metadata-eval 63.6%

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

      7. associate-/r* 63.6%

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

      8. metadata-eval 63.6%

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

      9. neg-mul-1 63.6%

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

      10. sub0-neg 63.6%

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

      11. associate-+l- 63.6%

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

      12. neg-sub0 63.6%

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

      13. remove-double-neg 63.6%

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

      14. distribute-neg-in 63.6%

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

      15. sub-neg 63.6%

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

      16. mul-1-neg 63.6%

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

      17. metadata-eval 63.6%

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

      18. associate-/r* 63.6%

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

      19. metadata-eval 63.6%

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

      20. metadata-eval 63.6%

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

      21. metadata-eval 63.6%

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

      22. metadata-eval 63.6%

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

      23. associate-*l/ 63.6%

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

   3. Simplified 63.6%

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

      1. frac-sub 66.0%

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

      2. *-rgt-identity 66.0%

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

      3. metadata-eval 66.0%

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

      4. div-inv 66.0%

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

      5. associate-/r* 66.0%

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

      6. *-un-lft-identity 66.0%

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

      7. metadata-eval 66.0%

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

      8. div-inv 66.0%

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

      9. associate--l- 70.6%

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

      10. div-inv 70.6%

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

      11. metadata-eval 70.6%

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

      12. *-rgt-identity 70.6%

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

      13. div-inv 70.6%

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

      14. metadata-eval 70.6%

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

      15. *-rgt-identity 70.6%

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

   5. Applied egg-rr 70.6%

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

   6. Taylor expanded in x around inf 97.4%

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

      1. div-inv 97.3%

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

      2. div-inv 97.3%

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

      3. associate-*l* 97.3%

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

      4. frac-2neg 97.3%

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

      5. metadata-eval 97.3%

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

      6. neg-sub0 97.3%

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

      7. associate--r- 97.3%

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

      8. metadata-eval 97.3%

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

   8. Applied egg-rr 97.3%

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

      1. associate-*r* 97.3%

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

      2. associate-*r/ 97.3%

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

      3. metadata-eval 97.3%

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

      4. associate-*l/ 97.4%

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

      5. associate-*r/ 97.4%

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

      6. metadata-eval 97.4%

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

      7. +-commutative 97.4%

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

   10. Simplified 97.4%

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

   11. Taylor expanded in x around 0 7.3%

       \[\leadsto \color{blue}{\frac{-2}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 6: 52.0% accurate, 2.2× speedup

Math

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

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 81.0%

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

   1. sub-neg 81.0%

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

   2. +-commutative 81.0%

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

   3. distribute-neg-frac 81.0%

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

   4. metadata-eval 81.0%

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

   5. metadata-eval 81.0%

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

   6. metadata-eval 81.0%

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

   7. associate-/r* 81.0%

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

   8. metadata-eval 81.0%

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

   9. neg-mul-1 81.0%

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

   10. sub0-neg 81.0%

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

   11. associate-+l- 81.0%

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

   12. neg-sub0 81.0%

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

   13. remove-double-neg 81.0%

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

   14. distribute-neg-in 81.0%

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

   15. sub-neg 81.0%

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

   16. mul-1-neg 81.0%

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

   17. metadata-eval 81.0%

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

   18. associate-/r* 81.0%

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

   19. metadata-eval 81.0%

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

   20. metadata-eval 81.0%

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

   21. metadata-eval 81.0%

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

   22. metadata-eval 81.0%

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

   23. associate-*l/ 81.0%

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

3. Simplified 81.0%

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

   1. frac-sub 81.7%

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

   2. *-rgt-identity 81.7%

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

   3. metadata-eval 81.7%

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

   4. div-inv 81.7%

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

   5. associate-/r* 81.7%

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

   6. *-un-lft-identity 81.7%

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

   7. metadata-eval 81.7%

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

   8. div-inv 81.7%

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

   9. associate--l- 84.2%

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

   10. div-inv 84.2%

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

   11. metadata-eval 84.2%

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

   12. *-rgt-identity 84.2%

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

   13. div-inv 84.2%

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

   14. metadata-eval 84.2%

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

   15. *-rgt-identity 84.2%

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

5. Applied egg-rr 84.2%

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

6. Taylor expanded in x around 0 54.1%

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

7. Final simplification 54.1%

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

Alternative 7: 50.9% 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 81.0%

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

   1. sub-neg 81.0%

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

   2. +-commutative 81.0%

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

   3. distribute-neg-frac 81.0%

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

   4. metadata-eval 81.0%

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

   5. metadata-eval 81.0%

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

   6. metadata-eval 81.0%

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

   7. associate-/r* 81.0%

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

   8. metadata-eval 81.0%

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

   9. neg-mul-1 81.0%

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

   10. sub0-neg 81.0%

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

   11. associate-+l- 81.0%

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

   12. neg-sub0 81.0%

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

   13. remove-double-neg 81.0%

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

   14. distribute-neg-in 81.0%

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

   15. sub-neg 81.0%

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

   16. mul-1-neg 81.0%

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

   17. metadata-eval 81.0%

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

   18. associate-/r* 81.0%

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

   19. metadata-eval 81.0%

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

   20. metadata-eval 81.0%

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

   21. metadata-eval 81.0%

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

   22. metadata-eval 81.0%

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

   23. associate-*l/ 81.0%

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

3. Simplified 81.0%

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

4. Taylor expanded in x around 0 53.2%

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

5. Final simplification 53.2%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))