Asymptote A

Percentage Accurate: 77.5% → 99.9%

Time: 6.3s

Alternatives: 6

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.5% 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

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (/ (/ 2.0 (+ x -1.0)) (- -1.0 x)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 / (x + (-1.0d0))) / ((-1.0d0) - x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[(2.0 / N[(x + -1.0), $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. neg-sub0 77.9%

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

   4. associate-+l- 77.9%

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

   5. sub-neg 77.9%

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

   6. associate--r+ 77.9%

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

   7. neg-sub0 77.9%

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

   8. distribute-neg-frac 77.9%

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

   9. metadata-eval 77.9%

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

   10. metadata-eval 77.9%

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

   11. metadata-eval 77.9%

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

   12. associate-/r* 77.9%

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

   13. metadata-eval 77.9%

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

   14. neg-mul-1 77.9%

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

   15. sub0-neg 77.9%

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

   16. associate-+l- 77.9%

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

   17. neg-sub0 77.9%

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

   18. +-commutative 77.9%

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

   19. unsub-neg 77.9%

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

   20. distribute-neg-frac 77.9%

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

   21. metadata-eval 77.9%

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

   22. metadata-eval 77.9%

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

   23. metadata-eval 77.9%

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

   24. associate-/r* 77.9%

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

   25. metadata-eval 77.9%

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

   26. neg-mul-1 77.9%

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

3. Simplified 77.9%

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

   1. frac-sub 78.4%

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

   2. *-rgt-identity 78.4%

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

   3. metadata-eval 78.4%

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

   4. div-inv 78.4%

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

   5. associate-/r* 78.4%

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

   6. *-un-lft-identity 78.4%

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

   7. metadata-eval 78.4%

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

   8. div-inv 78.4%

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

   9. associate--l- 81.3%

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

   10. div-inv 81.3%

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

   11. metadata-eval 81.3%

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

   12. *-rgt-identity 81.3%

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

   13. div-inv 81.3%

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

   14. metadata-eval 81.3%

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

   15. *-rgt-identity 81.3%

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

5. Applied egg-rr 81.3%

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

   1. div-sub 81.3%

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

   2. sub-neg 81.3%

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

7. Applied egg-rr 81.3%

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

   1. distribute-neg-frac 81.3%

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

   2. +-commutative 81.3%

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

   3. associate--r- 99.9%

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

   4. +-inverses 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. count-2 99.9%

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

   8. associate-*r/ 99.9%

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

   9. metadata-eval 99.9%

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

   10. metadata-eval 99.9%

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

   11. associate-/r* 99.9%

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

   12. neg-mul-1 99.9%

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

   13. neg-sub0 99.9%

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

   14. associate--r- 99.9%

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

   15. metadata-eval 99.9%

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

   16. +-commutative 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x + -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 0.76) 2.0 (/ (/ -2.0 x) (+ x -1.0))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / x) / (x + -1.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.76d0) then
        tmp = 2.0d0
    else
        tmp = ((-2.0d0) / x) / (x + (-1.0d0))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / x) / (x + -1.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.76:
		tmp = 2.0
	else:
		tmp = (-2.0 / x) / (x + -1.0)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.76)
		tmp = 2.0;
	else
		tmp = Float64(Float64(-2.0 / x) / Float64(x + -1.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.76)
		tmp = 2.0;
	else
		tmp = (-2.0 / x) / (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.76], 2.0, N[(N[(-2.0 / x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.76000000000000001

   1. Initial program 83.4%

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

      1. sub-neg 83.4%

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

      2. +-commutative 83.4%

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

      3. neg-sub0 83.4%

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

      4. associate-+l- 83.4%

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

      5. sub-neg 83.4%

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

      6. associate--r+ 83.4%

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

      7. neg-sub0 83.4%

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

      8. distribute-neg-frac 83.4%

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

      9. metadata-eval 83.4%

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

      10. metadata-eval 83.4%

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

      11. metadata-eval 83.4%

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

      12. associate-/r* 83.4%

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

      13. metadata-eval 83.4%

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

      14. neg-mul-1 83.4%

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

      15. sub0-neg 83.4%

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

      16. associate-+l- 83.4%

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

      17. neg-sub0 83.4%

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

      18. +-commutative 83.4%

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

      19. unsub-neg 83.4%

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

      20. distribute-neg-frac 83.4%

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

      21. metadata-eval 83.4%

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

      22. metadata-eval 83.4%

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

      23. metadata-eval 83.4%

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

      24. associate-/r* 83.4%

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

      25. metadata-eval 83.4%

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

      26. neg-mul-1 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in x around 0 62.0%

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

   if 0.76000000000000001 < x

   1. Initial program 62.9%

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

      1. sub-neg 62.9%

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

      2. +-commutative 62.9%

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

      3. neg-sub0 62.9%

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

      4. associate-+l- 62.9%

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

      5. sub-neg 62.9%

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

      6. associate--r+ 62.9%

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

      7. neg-sub0 62.9%

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

      8. distribute-neg-frac 62.9%

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

      9. metadata-eval 62.9%

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

      10. metadata-eval 62.9%

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

      11. metadata-eval 62.9%

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

      12. associate-/r* 62.9%

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

      13. metadata-eval 62.9%

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

      14. neg-mul-1 62.9%

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

      15. sub0-neg 62.9%

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

      16. associate-+l- 62.9%

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

      17. neg-sub0 62.9%

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

      18. +-commutative 62.9%

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

      19. unsub-neg 62.9%

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

      20. distribute-neg-frac 62.9%

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

      21. metadata-eval 62.9%

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

      22. metadata-eval 62.9%

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

      23. metadata-eval 62.9%

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

      24. associate-/r* 62.9%

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

      25. metadata-eval 62.9%

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

      26. neg-mul-1 62.9%

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

   3. Simplified 62.9%

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

      1. frac-sub 63.8%

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

      2. *-rgt-identity 63.8%

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

      3. metadata-eval 63.8%

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

      4. div-inv 63.8%

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

      5. associate-/r* 63.8%

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

      6. *-un-lft-identity 63.8%

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

      7. metadata-eval 63.8%

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

      8. div-inv 63.8%

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

      9. associate--l- 68.6%

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

      10. div-inv 68.6%

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

      11. metadata-eval 68.6%

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

      12. *-rgt-identity 68.6%

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

      13. div-inv 68.6%

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

      14. metadata-eval 68.6%

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

      15. *-rgt-identity 68.6%

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

   5. Applied egg-rr 68.6%

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

   6. Taylor expanded in x around inf 98.4%

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

      1. div-inv 98.4%

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

      2. *-un-lft-identity 98.4%

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

      3. times-frac 98.4%

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

      4. metadata-eval 98.4%

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

      5. frac-2neg 98.4%

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

      6. metadata-eval 98.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 60.3%

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

      9. sqr-neg 60.3%

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

      10. sqrt-unprod 60.3%

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

      11. add-sqr-sqrt 60.3%

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

      12. sub-neg 60.3%

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

      13. +-commutative 60.3%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 98.1%

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

      16. sqr-neg 98.1%

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

      17. sqrt-unprod 98.2%

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

      18. add-sqr-sqrt 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. associate-*r/ 98.4%

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

      2. associate-*r/ 98.4%

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

      3. metadata-eval 98.4%

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

   10. Simplified 98.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x}}{x + -1}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.2× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (/ 2.0 (* (+ x -1.0) (- -1.0 x))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / ((x + (-1.0d0)) * ((-1.0d0) - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(2.0 / N[(N[(x + -1.0), $MachinePrecision] * N[(-1.0 - x), $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. neg-sub0 77.9%

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

   4. associate-+l- 77.9%

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

   5. sub-neg 77.9%

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

   6. associate--r+ 77.9%

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

   7. neg-sub0 77.9%

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

   8. distribute-neg-frac 77.9%

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

   9. metadata-eval 77.9%

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

   10. metadata-eval 77.9%

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

   11. metadata-eval 77.9%

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

   12. associate-/r* 77.9%

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

   13. metadata-eval 77.9%

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

   14. neg-mul-1 77.9%

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

   15. sub0-neg 77.9%

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

   16. associate-+l- 77.9%

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

   17. neg-sub0 77.9%

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

   18. +-commutative 77.9%

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

   19. unsub-neg 77.9%

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

   20. distribute-neg-frac 77.9%

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

   21. metadata-eval 77.9%

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

   22. metadata-eval 77.9%

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

   23. metadata-eval 77.9%

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

   24. associate-/r* 77.9%

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

   25. metadata-eval 77.9%

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

   26. neg-mul-1 77.9%

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

3. Simplified 77.9%

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

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

5. Applied egg-rr 77.9%

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

   1. *-rgt-identity 77.9%

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

   2. cancel-sign-sub 77.9%

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

   3. distribute-neg-frac 77.9%

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

   4. metadata-eval 77.9%

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

   5. *-rgt-identity 77.9%

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

   6. *-inverses 77.9%

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

   7. associate-/r* 50.1%

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

   8. distribute-lft-neg-in 50.1%

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

   9. distribute-rgt-neg-in 50.1%

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

   10. *-commutative 50.1%

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

   11. *-lft-identity 50.1%

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

   12. *-inverses 50.1%

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

   13. times-frac 77.9%

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

   14. div-sub 78.4%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 52.8% accurate, 2.2× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 (/ -2.0 x)))

C

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

Fortran

NOTE: x should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

x = abs(x)
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

NOTE: x should be positive before calling this function
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 83.4%

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

      1. sub-neg 83.4%

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

      2. +-commutative 83.4%

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

      3. neg-sub0 83.4%

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

      4. associate-+l- 83.4%

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

      5. sub-neg 83.4%

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

      6. associate--r+ 83.4%

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

      7. neg-sub0 83.4%

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

      8. distribute-neg-frac 83.4%

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

      9. metadata-eval 83.4%

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

      10. metadata-eval 83.4%

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

      11. metadata-eval 83.4%

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

      12. associate-/r* 83.4%

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

      13. metadata-eval 83.4%

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

      14. neg-mul-1 83.4%

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

      15. sub0-neg 83.4%

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

      16. associate-+l- 83.4%

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

      17. neg-sub0 83.4%

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

      18. +-commutative 83.4%

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

      19. unsub-neg 83.4%

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

      20. distribute-neg-frac 83.4%

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

      21. metadata-eval 83.4%

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

      22. metadata-eval 83.4%

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

      23. metadata-eval 83.4%

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

      24. associate-/r* 83.4%

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

      25. metadata-eval 83.4%

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

      26. neg-mul-1 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in x around 0 62.0%

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

   if 1 < x

   1. Initial program 62.9%

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

      1. sub-neg 62.9%

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

      2. +-commutative 62.9%

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

      3. neg-sub0 62.9%

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

      4. associate-+l- 62.9%

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

      5. sub-neg 62.9%

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

      6. associate--r+ 62.9%

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

      7. neg-sub0 62.9%

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

      8. distribute-neg-frac 62.9%

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

      9. metadata-eval 62.9%

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

      10. metadata-eval 62.9%

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

      11. metadata-eval 62.9%

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

      12. associate-/r* 62.9%

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

      13. metadata-eval 62.9%

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

      14. neg-mul-1 62.9%

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

      15. sub0-neg 62.9%

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

      16. associate-+l- 62.9%

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

      17. neg-sub0 62.9%

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

      18. +-commutative 62.9%

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

      19. unsub-neg 62.9%

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

      20. distribute-neg-frac 62.9%

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

      21. metadata-eval 62.9%

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

      22. metadata-eval 62.9%

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

      23. metadata-eval 62.9%

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

      24. associate-/r* 62.9%

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

      25. metadata-eval 62.9%

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

      26. neg-mul-1 62.9%

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

   3. Simplified 62.9%

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

      1. frac-sub 63.8%

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

      2. *-rgt-identity 63.8%

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

      3. metadata-eval 63.8%

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

      4. div-inv 63.8%

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

      5. associate-/r* 63.8%

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

      6. *-un-lft-identity 63.8%

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

      7. metadata-eval 63.8%

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

      8. div-inv 63.8%

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

      9. associate--l- 68.6%

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

      10. div-inv 68.6%

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

      11. metadata-eval 68.6%

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

      12. *-rgt-identity 68.6%

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

      13. div-inv 68.6%

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

      14. metadata-eval 68.6%

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

      15. *-rgt-identity 68.6%

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

   5. Applied egg-rr 68.6%

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

   6. Taylor expanded in x around inf 98.4%

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

   7. Taylor expanded in x around 0 6.8%

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

4. Final simplification 47.4%

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

Alternative 5: 2.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ -2 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 -2.0)

C

x = abs(x);
double code(double x) {
	return -2.0;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return -2.0;
}

Python

x = abs(x)
def code(x):
	return -2.0

Julia

x = abs(x)
function code(x)
	return -2.0
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = -2.0;
end

Wolfram

NOTE: x should be positive before calling this function
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. neg-sub0 77.9%

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

   4. associate-+l- 77.9%

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

   5. sub-neg 77.9%

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

   6. associate--r+ 77.9%

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

   7. neg-sub0 77.9%

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

   8. distribute-neg-frac 77.9%

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

   9. metadata-eval 77.9%

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

   10. metadata-eval 77.9%

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

   11. metadata-eval 77.9%

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

   12. associate-/r* 77.9%

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

   13. metadata-eval 77.9%

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

   14. neg-mul-1 77.9%

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

   15. sub0-neg 77.9%

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

   16. associate-+l- 77.9%

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

   17. neg-sub0 77.9%

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

   18. +-commutative 77.9%

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

   19. unsub-neg 77.9%

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

   20. distribute-neg-frac 77.9%

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

   21. metadata-eval 77.9%

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

   22. metadata-eval 77.9%

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

   23. metadata-eval 77.9%

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

   24. associate-/r* 77.9%

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

   25. metadata-eval 77.9%

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

   26. neg-mul-1 77.9%

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

3. Simplified 77.9%

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

   1. frac-sub 78.4%

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

   2. *-rgt-identity 78.4%

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

   3. metadata-eval 78.4%

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

   4. div-inv 78.4%

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

   5. associate-/r* 78.4%

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

   6. *-un-lft-identity 78.4%

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

   7. metadata-eval 78.4%

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

   8. div-inv 78.4%

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

   9. associate--l- 81.3%

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

   10. div-inv 81.3%

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

   11. metadata-eval 81.3%

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

   12. *-rgt-identity 81.3%

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

   13. div-inv 81.3%

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

   14. metadata-eval 81.3%

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

   15. *-rgt-identity 81.3%

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

5. Applied egg-rr 81.3%

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

   1. div-sub 81.3%

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

   2. sub-neg 81.3%

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

7. Applied egg-rr 81.3%

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

   1. distribute-neg-frac 81.3%

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

   2. +-commutative 81.3%

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

   3. associate--r- 99.9%

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

   4. +-inverses 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. count-2 99.9%

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

   8. associate-*r/ 99.9%

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

   9. metadata-eval 99.9%

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

   10. metadata-eval 99.9%

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

   11. associate-/r* 99.9%

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

   12. neg-mul-1 99.9%

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

   13. neg-sub0 99.9%

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

   14. associate--r- 99.9%

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

   15. metadata-eval 99.9%

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

   16. +-commutative 99.9%

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

9. Simplified 99.9%

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

    1. associate-/r* 99.6%

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

    2. sub-neg 99.6%

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

    3. distribute-lft-out 99.7%

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

    4. frac-2neg 99.7%

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

    5. metadata-eval 99.7%

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

    6. div-inv 99.7%

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

    7. metadata-eval 99.7%

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

    8. distribute-lft-out 99.6%

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

    9. sub-neg 99.6%

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

    10. distribute-lft-neg-in 99.6%

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

11. Applied egg-rr 54.2%

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

12. Taylor expanded in x around 0 2.9%

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

13. Final simplification 2.9%

    \[\leadsto -2 \]

Alternative 6: 50.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 2 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 2.0)

C

x = abs(x);
double code(double x) {
	return 2.0;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 2.0;
}

Python

x = abs(x)
def code(x):
	return 2.0

Julia

x = abs(x)
function code(x)
	return 2.0
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = 2.0;
end

Wolfram

NOTE: x should be positive before calling this function
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. neg-sub0 77.9%

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

   4. associate-+l- 77.9%

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

   5. sub-neg 77.9%

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

   6. associate--r+ 77.9%

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

   7. neg-sub0 77.9%

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

   8. distribute-neg-frac 77.9%

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

   9. metadata-eval 77.9%

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

   10. metadata-eval 77.9%

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

   11. metadata-eval 77.9%

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

   12. associate-/r* 77.9%

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

   13. metadata-eval 77.9%

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

   14. neg-mul-1 77.9%

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

   15. sub0-neg 77.9%

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

   16. associate-+l- 77.9%

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

   17. neg-sub0 77.9%

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

   18. +-commutative 77.9%

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

   19. unsub-neg 77.9%

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

   20. distribute-neg-frac 77.9%

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

   21. metadata-eval 77.9%

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

   22. metadata-eval 77.9%

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

   23. metadata-eval 77.9%

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

   24. associate-/r* 77.9%

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

   25. metadata-eval 77.9%

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

   26. neg-mul-1 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in x around 0 46.3%

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

5. Final simplification 46.3%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2024024 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))