Asymptote A

Percentage Accurate: 77.4% → 99.9%

Time: 6.3s

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: 77.4% 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 70.2%

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

   1. sub-neg 70.2%

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

   2. +-commutative 70.2%

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

   3. neg-sub0 70.2%

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

   4. associate-+l- 70.2%

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

   5. sub-neg 70.2%

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

   6. associate--r+ 70.2%

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

   7. neg-sub0 70.2%

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

   8. distribute-neg-frac 70.2%

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

   9. metadata-eval 70.2%

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

   10. metadata-eval 70.2%

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

   11. metadata-eval 70.2%

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

   12. associate-/r* 70.2%

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

   13. metadata-eval 70.2%

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

   14. neg-mul-1 70.2%

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

   15. sub0-neg 70.2%

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

   16. associate-+l- 70.2%

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

   17. neg-sub0 70.2%

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

   18. +-commutative 70.2%

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

   19. unsub-neg 70.2%

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

   20. distribute-neg-frac 70.2%

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

   21. metadata-eval 70.2%

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

   22. metadata-eval 70.2%

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

   23. metadata-eval 70.2%

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

   24. associate-/r* 70.2%

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

   25. metadata-eval 70.2%

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

   26. neg-mul-1 70.2%

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

3. Simplified 70.2%

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

   1. frac-sub 71.1%

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

   2. *-rgt-identity 71.1%

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

   3. metadata-eval 71.1%

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

   4. div-inv 71.1%

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

   5. associate-/r* 71.1%

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

   6. *-un-lft-identity 71.1%

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

   7. metadata-eval 71.1%

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

   8. div-inv 71.1%

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

   9. associate--l- 75.1%

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

   10. div-inv 75.1%

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

   11. metadata-eval 75.1%

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

   12. *-rgt-identity 75.1%

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

   13. div-inv 75.1%

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

   14. metadata-eval 75.1%

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

   15. *-rgt-identity 75.1%

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

5. Applied egg-rr 75.1%

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

   1. div-sub 75.1%

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

   2. sub-neg 75.1%

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

7. Applied egg-rr 75.1%

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

   1. metadata-eval 75.1%

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

   2. associate-*r/ 75.1%

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

   3. neg-mul-1 75.1%

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

   4. +-commutative 75.1%

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

   5. associate--r- 99.9%

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

   6. +-inverses 99.9%

      \[\leadsto \frac{-1 \cdot \frac{1}{1 - x} + -1 \cdot \frac{1 - \color{blue}{0}}{1 - x}}{-1 - x} \]

   7. metadata-eval 99.9%

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

   8. distribute-rgt-out 99.9%

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

   9. metadata-eval 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

   12. +-inverses 99.9%

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

   13. associate--l+ 75.1%

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

   14. associate-+l- 71.1%

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

   15. +-commutative 71.1%

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

   16. associate--l- 71.1%

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

   17. associate-*l/ 71.1%

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

   18. associate-*r/ 71.1%

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

   19. metadata-eval 71.1%

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

   20. times-frac 71.1%

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

   21. neg-mul-1 71.1%

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

   22. neg-mul-1 71.1%

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

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: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = 2.0 / (x * (-1.0 - 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 * ((-1.0d0) - 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 * (-1.0 - x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 2.0
	else:
		tmp = 2.0 / (x * (-1.0 - 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 / Float64(x * Float64(-1.0 - 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 * (-1.0 - 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 / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 76.5%

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

      1. sub-neg 76.5%

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

      2. +-commutative 76.5%

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

      3. neg-sub0 76.5%

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

      4. associate-+l- 76.5%

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

      5. sub-neg 76.5%

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

      6. associate--r+ 76.5%

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

      7. neg-sub0 76.5%

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

      8. distribute-neg-frac 76.5%

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

      9. metadata-eval 76.5%

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

      10. metadata-eval 76.5%

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

      11. metadata-eval 76.5%

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

      12. associate-/r* 76.5%

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

      13. metadata-eval 76.5%

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

      14. neg-mul-1 76.5%

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

      15. sub0-neg 76.5%

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

      16. associate-+l- 76.5%

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

      17. neg-sub0 76.5%

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

      18. +-commutative 76.5%

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

      19. unsub-neg 76.5%

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

      20. distribute-neg-frac 76.5%

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

      21. metadata-eval 76.5%

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

      22. metadata-eval 76.5%

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

      23. metadata-eval 76.5%

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

      24. associate-/r* 76.5%

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

      25. metadata-eval 76.5%

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

      26. neg-mul-1 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in x around 0 59.8%

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

   if 1 < x

   1. Initial program 47.0%

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

      1. sub-neg 47.0%

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

      2. +-commutative 47.0%

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

      3. neg-sub0 47.0%

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

      4. associate-+l- 47.0%

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

      5. sub-neg 47.0%

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

      6. associate--r+ 47.0%

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

      7. neg-sub0 47.0%

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

      8. distribute-neg-frac 47.0%

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

      9. metadata-eval 47.0%

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

      10. metadata-eval 47.0%

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

      11. metadata-eval 47.0%

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

      12. associate-/r* 47.0%

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

      13. metadata-eval 47.0%

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

      14. neg-mul-1 47.0%

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

      15. sub0-neg 47.0%

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

      16. associate-+l- 47.0%

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

      17. neg-sub0 47.0%

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

      18. +-commutative 47.0%

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

      19. unsub-neg 47.0%

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

      20. distribute-neg-frac 47.0%

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

      21. metadata-eval 47.0%

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

      22. metadata-eval 47.0%

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

      23. metadata-eval 47.0%

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

      24. associate-/r* 47.0%

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

      25. metadata-eval 47.0%

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

      26. neg-mul-1 47.0%

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

   3. Simplified 47.0%

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

      1. frac-sub 48.7%

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

      2. *-rgt-identity 48.7%

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

      3. metadata-eval 48.7%

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

      4. div-inv 48.7%

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

      5. associate-/r* 48.8%

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

      6. *-un-lft-identity 48.8%

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

      7. metadata-eval 48.8%

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

      8. div-inv 48.8%

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

      9. associate--l- 55.8%

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

      10. div-inv 55.8%

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

      11. metadata-eval 55.8%

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

      12. *-rgt-identity 55.8%

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

      13. div-inv 55.8%

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

      14. metadata-eval 55.8%

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

      15. *-rgt-identity 55.8%

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

   5. Applied egg-rr 55.8%

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

   6. Taylor expanded in x around inf 97.3%

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

      1. expm1-log1p-u 97.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{x}}{-1 - x}\right)\right)} \]

      2. expm1-udef 44.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{x}}{-1 - x}\right)} - 1} \]

      3. associate-/l/ 44.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{\left(-1 - x\right) \cdot x}}\right)} - 1 \]

      4. *-commutative 44.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{x \cdot \left(-1 - x\right)}}\right)} - 1 \]

   8. Applied egg-rr 44.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{x \cdot \left(-1 - x\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 95.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{x \cdot \left(-1 - x\right)}\right)\right)} \]

      2. expm1-log1p 95.9%

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

   10. Simplified 95.9%

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

4. Final simplification 67.6%

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

Alternative 3: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = (2.0 / x) / (-1.0 - 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) / ((-1.0d0) - 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) / (-1.0 - x);
	}
	return tmp;
}

Python

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

Julia

x = abs(x)
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

x = abs(x)
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

NOTE: x should be positive before calling this function
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 76.5%

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

      1. sub-neg 76.5%

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

      2. +-commutative 76.5%

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

      3. neg-sub0 76.5%

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

      4. associate-+l- 76.5%

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

      5. sub-neg 76.5%

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

      6. associate--r+ 76.5%

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

      7. neg-sub0 76.5%

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

      8. distribute-neg-frac 76.5%

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

      9. metadata-eval 76.5%

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

      10. metadata-eval 76.5%

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

      11. metadata-eval 76.5%

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

      12. associate-/r* 76.5%

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

      13. metadata-eval 76.5%

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

      14. neg-mul-1 76.5%

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

      15. sub0-neg 76.5%

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

      16. associate-+l- 76.5%

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

      17. neg-sub0 76.5%

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

      18. +-commutative 76.5%

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

      19. unsub-neg 76.5%

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

      20. distribute-neg-frac 76.5%

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

      21. metadata-eval 76.5%

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

      22. metadata-eval 76.5%

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

      23. metadata-eval 76.5%

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

      24. associate-/r* 76.5%

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

      25. metadata-eval 76.5%

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

      26. neg-mul-1 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in x around 0 59.8%

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

   if 1 < x

   1. Initial program 47.0%

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

      1. sub-neg 47.0%

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

      2. +-commutative 47.0%

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

      3. neg-sub0 47.0%

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

      4. associate-+l- 47.0%

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

      5. sub-neg 47.0%

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

      6. associate--r+ 47.0%

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

      7. neg-sub0 47.0%

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

      8. distribute-neg-frac 47.0%

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

      9. metadata-eval 47.0%

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

      10. metadata-eval 47.0%

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

      11. metadata-eval 47.0%

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

      12. associate-/r* 47.0%

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

      13. metadata-eval 47.0%

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

      14. neg-mul-1 47.0%

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

      15. sub0-neg 47.0%

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

      16. associate-+l- 47.0%

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

      17. neg-sub0 47.0%

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

      18. +-commutative 47.0%

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

      19. unsub-neg 47.0%

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

      20. distribute-neg-frac 47.0%

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

      21. metadata-eval 47.0%

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

      22. metadata-eval 47.0%

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

      23. metadata-eval 47.0%

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

      24. associate-/r* 47.0%

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

      25. metadata-eval 47.0%

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

      26. neg-mul-1 47.0%

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

   3. Simplified 47.0%

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

      1. frac-sub 48.7%

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

      2. *-rgt-identity 48.7%

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

      3. metadata-eval 48.7%

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

      4. div-inv 48.7%

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

      5. associate-/r* 48.8%

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

      6. *-un-lft-identity 48.8%

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

      7. metadata-eval 48.8%

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

      8. div-inv 48.8%

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

      9. associate--l- 55.8%

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

      10. div-inv 55.8%

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

      11. metadata-eval 55.8%

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

      12. *-rgt-identity 55.8%

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

      13. div-inv 55.8%

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

      14. metadata-eval 55.8%

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

      15. *-rgt-identity 55.8%

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

   5. Applied egg-rr 55.8%

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

   6. Taylor expanded in x around inf 97.3%

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

4. Final simplification 67.9%

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

Alternative 4: 99.5% 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 70.2%

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

   1. sub-neg 70.2%

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

   2. +-commutative 70.2%

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

   3. neg-sub0 70.2%

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

   4. associate-+l- 70.2%

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

   5. sub-neg 70.2%

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

   6. associate--r+ 70.2%

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

   7. neg-sub0 70.2%

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

   8. distribute-neg-frac 70.2%

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

   9. metadata-eval 70.2%

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

   10. metadata-eval 70.2%

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

   11. metadata-eval 70.2%

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

   12. associate-/r* 70.2%

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

   13. metadata-eval 70.2%

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

   14. neg-mul-1 70.2%

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

   15. sub0-neg 70.2%

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

   16. associate-+l- 70.2%

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

   17. neg-sub0 70.2%

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

   18. +-commutative 70.2%

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

   19. unsub-neg 70.2%

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

   20. distribute-neg-frac 70.2%

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

   21. metadata-eval 70.2%

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

   22. metadata-eval 70.2%

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

   23. metadata-eval 70.2%

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

   24. associate-/r* 70.2%

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

   25. metadata-eval 70.2%

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

   26. neg-mul-1 70.2%

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

3. Simplified 70.2%

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

   1. sub-neg 70.2%

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

   2. distribute-neg-frac 70.2%

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

   3. metadata-eval 70.2%

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

5. Applied egg-rr 70.2%

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

   1. *-rgt-identity 70.2%

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

   2. cancel-sign-sub 70.2%

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

   3. distribute-neg-frac 70.2%

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

   4. metadata-eval 70.2%

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

   5. *-rgt-identity 70.2%

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

   6. *-inverses 70.2%

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

   7. associate-/r* 51.6%

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

   8. distribute-lft-neg-in 51.6%

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

   9. distribute-rgt-neg-in 51.6%

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

   10. *-commutative 51.6%

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

   11. *-lft-identity 51.6%

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

   12. *-inverses 51.6%

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

   13. times-frac 70.1%

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

   14. div-sub 71.1%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 5: 53.2% 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 76.5%

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

      1. sub-neg 76.5%

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

      2. +-commutative 76.5%

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

      3. neg-sub0 76.5%

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

      4. associate-+l- 76.5%

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

      5. sub-neg 76.5%

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

      6. associate--r+ 76.5%

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

      7. neg-sub0 76.5%

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

      8. distribute-neg-frac 76.5%

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

      9. metadata-eval 76.5%

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

      10. metadata-eval 76.5%

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

      11. metadata-eval 76.5%

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

      12. associate-/r* 76.5%

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

      13. metadata-eval 76.5%

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

      14. neg-mul-1 76.5%

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

      15. sub0-neg 76.5%

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

      16. associate-+l- 76.5%

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

      17. neg-sub0 76.5%

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

      18. +-commutative 76.5%

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

      19. unsub-neg 76.5%

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

      20. distribute-neg-frac 76.5%

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

      21. metadata-eval 76.5%

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

      22. metadata-eval 76.5%

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

      23. metadata-eval 76.5%

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

      24. associate-/r* 76.5%

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

      25. metadata-eval 76.5%

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

      26. neg-mul-1 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in x around 0 59.8%

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

   if 1 < x

   1. Initial program 47.0%

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

      1. sub-neg 47.0%

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

      2. +-commutative 47.0%

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

      3. neg-sub0 47.0%

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

      4. associate-+l- 47.0%

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

      5. sub-neg 47.0%

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

      6. associate--r+ 47.0%

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

      7. neg-sub0 47.0%

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

      8. distribute-neg-frac 47.0%

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

      9. metadata-eval 47.0%

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

      10. metadata-eval 47.0%

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

      11. metadata-eval 47.0%

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

      12. associate-/r* 47.0%

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

      13. metadata-eval 47.0%

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

      14. neg-mul-1 47.0%

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

      15. sub0-neg 47.0%

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

      16. associate-+l- 47.0%

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

      17. neg-sub0 47.0%

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

      18. +-commutative 47.0%

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

      19. unsub-neg 47.0%

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

      20. distribute-neg-frac 47.0%

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

      21. metadata-eval 47.0%

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

      22. metadata-eval 47.0%

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

      23. metadata-eval 47.0%

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

      24. associate-/r* 47.0%

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

      25. metadata-eval 47.0%

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

      26. neg-mul-1 47.0%

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

   3. Simplified 47.0%

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

      1. frac-sub 48.7%

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

      2. *-rgt-identity 48.7%

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

      3. metadata-eval 48.7%

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

      4. div-inv 48.7%

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

      5. associate-/r* 48.8%

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

      6. *-un-lft-identity 48.8%

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

      7. metadata-eval 48.8%

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

      8. div-inv 48.8%

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

      9. associate--l- 55.8%

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

      10. div-inv 55.8%

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

      11. metadata-eval 55.8%

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

      12. *-rgt-identity 55.8%

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

      13. div-inv 55.8%

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

      14. metadata-eval 55.8%

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

      15. *-rgt-identity 55.8%

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

   5. Applied egg-rr 55.8%

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

   6. Taylor expanded in x around inf 97.3%

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

   7. Taylor expanded in x around 0 7.1%

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

4. Final simplification 48.5%

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

Alternative 6: 10.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 1 \end{array} \]

FPCore

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

C

x = abs(x);
double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

x = abs(x)
def code(x):
	return 1.0

Julia

x = abs(x)
function code(x)
	return 1.0
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 1.0

Derivation

1. Initial program 70.2%

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

   1. sub-neg 70.2%

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

   2. +-commutative 70.2%

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

   3. neg-sub0 70.2%

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

   4. associate-+l- 70.2%

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

   5. sub-neg 70.2%

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

   6. associate--r+ 70.2%

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

   7. neg-sub0 70.2%

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

   8. distribute-neg-frac 70.2%

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

   9. metadata-eval 70.2%

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

   10. metadata-eval 70.2%

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

   11. metadata-eval 70.2%

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

   12. associate-/r* 70.2%

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

   13. metadata-eval 70.2%

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

   14. neg-mul-1 70.2%

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

   15. sub0-neg 70.2%

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

   16. associate-+l- 70.2%

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

   17. neg-sub0 70.2%

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

   18. +-commutative 70.2%

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

   19. unsub-neg 70.2%

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

   20. distribute-neg-frac 70.2%

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

   21. metadata-eval 70.2%

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

   22. metadata-eval 70.2%

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

   23. metadata-eval 70.2%

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

   24. associate-/r* 70.2%

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

   25. metadata-eval 70.2%

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

   26. neg-mul-1 70.2%

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

3. Simplified 70.2%

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

4. Taylor expanded in x around 0 47.3%

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

5. Taylor expanded in x around inf 10.2%

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

6. Final simplification 10.2%

   \[\leadsto 1 \]

Alternative 7: 51.1% 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 70.2%

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

   1. sub-neg 70.2%

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

   2. +-commutative 70.2%

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

   3. neg-sub0 70.2%

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

   4. associate-+l- 70.2%

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

   5. sub-neg 70.2%

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

   6. associate--r+ 70.2%

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

   7. neg-sub0 70.2%

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

   8. distribute-neg-frac 70.2%

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

   9. metadata-eval 70.2%

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

   10. metadata-eval 70.2%

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

   11. metadata-eval 70.2%

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

   12. associate-/r* 70.2%

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

   13. metadata-eval 70.2%

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

   14. neg-mul-1 70.2%

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

   15. sub0-neg 70.2%

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

   16. associate-+l- 70.2%

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

   17. neg-sub0 70.2%

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

   18. +-commutative 70.2%

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

   19. unsub-neg 70.2%

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

   20. distribute-neg-frac 70.2%

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

   21. metadata-eval 70.2%

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

   22. metadata-eval 70.2%

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

   23. metadata-eval 70.2%

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

   24. associate-/r* 70.2%

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

   25. metadata-eval 70.2%

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

   26. neg-mul-1 70.2%

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

3. Simplified 70.2%

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

4. Taylor expanded in x around 0 47.5%

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

5. Final simplification 47.5%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023308 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))