Asymptote A

Percentage Accurate: 77.6% → 99.9%

Time: 5.0s

Alternatives: 7

Speedup: 1.6×

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: 77.6% 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

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

FPCore

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

C

x = abs(x);
double code(double x) {
	return (2.0 / (1.0 - x)) / (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 / (1.0d0 - x)) / (1.0d0 + x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. sub-neg 78.1%

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

   2. +-commutative 78.1%

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

   3. neg-sub0 78.1%

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

   4. associate-+l- 78.1%

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

   5. sub-neg 78.1%

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

   6. associate--r+ 78.1%

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

   7. neg-sub0 78.1%

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

   8. distribute-neg-frac 78.1%

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

   9. metadata-eval 78.1%

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

   10. metadata-eval 78.1%

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

   11. metadata-eval 78.1%

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

   12. associate-/r* 78.1%

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

   13. metadata-eval 78.1%

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

   14. neg-mul-1 78.1%

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

   15. sub0-neg 78.1%

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

   16. associate-+l- 78.1%

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

   17. neg-sub0 78.1%

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

   18. +-commutative 78.1%

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

   19. unsub-neg 78.1%

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

   20. distribute-neg-frac 78.1%

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

   21. metadata-eval 78.1%

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

   22. metadata-eval 78.1%

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

   23. metadata-eval 78.1%

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

   24. associate-/r* 78.1%

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

   25. metadata-eval 78.1%

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

   26. neg-mul-1 78.1%

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

3. Simplified 78.1%

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

   1. frac-sub 78.8%

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

   2. *-rgt-identity 78.8%

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

   3. metadata-eval 78.8%

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

   4. div-inv 78.8%

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

   5. associate-/r* 78.7%

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

   6. *-un-lft-identity 78.7%

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

   7. metadata-eval 78.7%

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

   8. div-inv 78.7%

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

   9. associate--l- 81.7%

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

   10. div-inv 81.7%

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

   11. metadata-eval 81.7%

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

   12. *-rgt-identity 81.7%

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

   13. div-inv 81.7%

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

   14. metadata-eval 81.7%

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

   15. *-rgt-identity 81.7%

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

5. Applied egg-rr 81.7%

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

   1. frac-2neg 81.7%

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

   2. div-inv 81.7%

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

   3. distribute-neg-frac 81.7%

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

   4. associate--r+ 78.8%

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

   5. associate--r- 78.8%

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

   6. sub-neg 78.8%

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

   7. neg-mul-1 78.8%

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

   8. distribute-neg-in 78.8%

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

   9. metadata-eval 78.8%

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

   10. *-commutative 78.8%

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

   11. sub-neg 78.8%

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

   12. *-commutative 78.8%

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

   13. neg-mul-1 78.8%

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

7. Applied egg-rr 78.8%

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

   1. associate-*r/ 78.7%

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

   2. *-rgt-identity 78.7%

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

   3. neg-mul-1 78.7%

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

   4. associate--l- 78.7%

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

   5. +-commutative 78.7%

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

   6. associate-+l- 81.7%

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

   7. associate--l+ 99.9%

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

   8. +-inverses 99.9%

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

   9. metadata-eval 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

   12. sub-neg 99.9%

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

   13. remove-double-neg 99.9%

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

   14. +-commutative 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. sub-neg 78.1%

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

   2. +-commutative 78.1%

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

   3. neg-sub0 78.1%

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

   4. associate-+l- 78.1%

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

   5. sub-neg 78.1%

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

   6. associate--r+ 78.1%

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

   7. neg-sub0 78.1%

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

   8. distribute-neg-frac 78.1%

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

   9. metadata-eval 78.1%

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

   10. metadata-eval 78.1%

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

   11. metadata-eval 78.1%

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

   12. associate-/r* 78.1%

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

   13. metadata-eval 78.1%

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

   14. neg-mul-1 78.1%

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

   15. sub0-neg 78.1%

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

   16. associate-+l- 78.1%

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

   17. neg-sub0 78.1%

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

   18. +-commutative 78.1%

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

   19. unsub-neg 78.1%

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

   20. distribute-neg-frac 78.1%

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

   21. metadata-eval 78.1%

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

   22. metadata-eval 78.1%

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

   23. metadata-eval 78.1%

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

   24. associate-/r* 78.1%

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

   25. metadata-eval 78.1%

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

   26. neg-mul-1 78.1%

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

3. Simplified 78.1%

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

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. metadata-eval 78.1%

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

5. Applied egg-rr 78.1%

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

   1. *-rgt-identity 78.1%

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

   2. cancel-sign-sub 78.1%

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

   3. distribute-neg-frac 78.1%

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

   4. metadata-eval 78.1%

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

   5. *-rgt-identity 78.1%

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

   6. *-inverses 78.1%

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

   7. associate-/r* 57.3%

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

   8. distribute-lft-neg-in 57.3%

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

   9. distribute-rgt-neg-in 57.3%

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

   10. *-commutative 57.3%

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

   11. *-lft-identity 57.3%

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

   12. *-inverses 57.3%

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

   13. times-frac 78.1%

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

   14. div-sub 78.8%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 98.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 84.2%

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

      1. sub-neg 84.2%

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

      2. +-commutative 84.2%

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

      3. neg-sub0 84.2%

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

      4. associate-+l- 84.2%

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

      5. sub-neg 84.2%

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

      6. associate--r+ 84.2%

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

      7. neg-sub0 84.2%

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

      8. distribute-neg-frac 84.2%

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

      9. metadata-eval 84.2%

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

      10. metadata-eval 84.2%

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

      11. metadata-eval 84.2%

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

      12. associate-/r* 84.2%

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

      13. metadata-eval 84.2%

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

      14. neg-mul-1 84.2%

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

      15. sub0-neg 84.2%

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

      16. associate-+l- 84.2%

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

      17. neg-sub0 84.2%

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

      18. +-commutative 84.2%

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

      19. unsub-neg 84.2%

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

      20. distribute-neg-frac 84.2%

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

      21. metadata-eval 84.2%

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

      22. metadata-eval 84.2%

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

      23. metadata-eval 84.2%

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

      24. associate-/r* 84.2%

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

      25. metadata-eval 84.2%

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

      26. neg-mul-1 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in x around 0 69.5%

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

   if 1 < x

   1. Initial program 58.2%

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

      1. sub-neg 58.2%

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

      2. +-commutative 58.2%

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

      3. neg-sub0 58.2%

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

      4. associate-+l- 58.2%

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

      5. sub-neg 58.2%

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

      6. associate--r+ 58.2%

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

      7. neg-sub0 58.2%

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

      8. distribute-neg-frac 58.2%

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

      9. metadata-eval 58.2%

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

      10. metadata-eval 58.2%

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

      11. metadata-eval 58.2%

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

      12. associate-/r* 58.2%

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

      13. metadata-eval 58.2%

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

      14. neg-mul-1 58.2%

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

      15. sub0-neg 58.2%

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

      16. associate-+l- 58.2%

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

      17. neg-sub0 58.2%

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

      18. +-commutative 58.2%

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

      19. unsub-neg 58.2%

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

      20. distribute-neg-frac 58.2%

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

      21. metadata-eval 58.2%

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

      22. metadata-eval 58.2%

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

      23. metadata-eval 58.2%

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

      24. associate-/r* 58.2%

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

      25. metadata-eval 58.2%

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

      26. neg-mul-1 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in x around inf 98.9%

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

      1. unpow2 98.9%

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

   6. Applied egg-rr 98.9%

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

4. Final simplification 76.4%

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

Alternative 4: 98.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 84.2%

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

      1. sub-neg 84.2%

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

      2. +-commutative 84.2%

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

      3. neg-sub0 84.2%

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

      4. associate-+l- 84.2%

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

      5. sub-neg 84.2%

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

      6. associate--r+ 84.2%

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

      7. neg-sub0 84.2%

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

      8. distribute-neg-frac 84.2%

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

      9. metadata-eval 84.2%

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

      10. metadata-eval 84.2%

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

      11. metadata-eval 84.2%

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

      12. associate-/r* 84.2%

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

      13. metadata-eval 84.2%

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

      14. neg-mul-1 84.2%

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

      15. sub0-neg 84.2%

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

      16. associate-+l- 84.2%

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

      17. neg-sub0 84.2%

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

      18. +-commutative 84.2%

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

      19. unsub-neg 84.2%

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

      20. distribute-neg-frac 84.2%

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

      21. metadata-eval 84.2%

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

      22. metadata-eval 84.2%

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

      23. metadata-eval 84.2%

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

      24. associate-/r* 84.2%

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

      25. metadata-eval 84.2%

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

      26. neg-mul-1 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in x around 0 69.5%

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

   if 1 < x

   1. Initial program 58.2%

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

      1. sub-neg 58.2%

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

      2. +-commutative 58.2%

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

      3. neg-sub0 58.2%

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

      4. associate-+l- 58.2%

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

      5. sub-neg 58.2%

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

      6. associate--r+ 58.2%

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

      7. neg-sub0 58.2%

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

      8. distribute-neg-frac 58.2%

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

      9. metadata-eval 58.2%

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

      10. metadata-eval 58.2%

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

      11. metadata-eval 58.2%

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

      12. associate-/r* 58.2%

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

      13. metadata-eval 58.2%

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

      14. neg-mul-1 58.2%

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

      15. sub0-neg 58.2%

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

      16. associate-+l- 58.2%

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

      17. neg-sub0 58.2%

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

      18. +-commutative 58.2%

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

      19. unsub-neg 58.2%

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

      20. distribute-neg-frac 58.2%

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

      21. metadata-eval 58.2%

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

      22. metadata-eval 58.2%

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

      23. metadata-eval 58.2%

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

      24. associate-/r* 58.2%

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

      25. metadata-eval 58.2%

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

      26. neg-mul-1 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in x around inf 98.9%

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

      1. expm1-log1p-u 98.9%

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

      2. expm1-udef 57.7%

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

      3. div-inv 57.7%

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

      4. pow-flip 57.7%

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

      5. metadata-eval 57.7%

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

   6. Applied egg-rr 57.7%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

   8. Simplified 99.6%

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

      1. metadata-eval 99.6%

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

      2. pow-flip 98.9%

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

      3. pow2 98.9%

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

      4. associate-/r/ 98.9%

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

      5. clear-num 98.9%

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

      6. associate-/r* 99.4%

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

   10. Applied egg-rr 99.4%

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

4. Final simplification 76.5%

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

Alternative 5: 52.4% 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 84.2%

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

      1. sub-neg 84.2%

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

      2. +-commutative 84.2%

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

      3. neg-sub0 84.2%

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

      4. associate-+l- 84.2%

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

      5. sub-neg 84.2%

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

      6. associate--r+ 84.2%

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

      7. neg-sub0 84.2%

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

      8. distribute-neg-frac 84.2%

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

      9. metadata-eval 84.2%

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

      10. metadata-eval 84.2%

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

      11. metadata-eval 84.2%

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

      12. associate-/r* 84.2%

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

      13. metadata-eval 84.2%

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

      14. neg-mul-1 84.2%

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

      15. sub0-neg 84.2%

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

      16. associate-+l- 84.2%

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

      17. neg-sub0 84.2%

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

      18. +-commutative 84.2%

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

      19. unsub-neg 84.2%

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

      20. distribute-neg-frac 84.2%

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

      21. metadata-eval 84.2%

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

      22. metadata-eval 84.2%

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

      23. metadata-eval 84.2%

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

      24. associate-/r* 84.2%

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

      25. metadata-eval 84.2%

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

      26. neg-mul-1 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in x around 0 69.5%

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

   if 1 < x

   1. Initial program 58.2%

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

      1. sub-neg 58.2%

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

      2. +-commutative 58.2%

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

      3. neg-sub0 58.2%

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

      4. associate-+l- 58.2%

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

      5. sub-neg 58.2%

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

      6. associate--r+ 58.2%

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

      7. neg-sub0 58.2%

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

      8. distribute-neg-frac 58.2%

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

      9. metadata-eval 58.2%

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

      10. metadata-eval 58.2%

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

      11. metadata-eval 58.2%

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

      12. associate-/r* 58.2%

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

      13. metadata-eval 58.2%

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

      14. neg-mul-1 58.2%

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

      15. sub0-neg 58.2%

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

      16. associate-+l- 58.2%

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

      17. neg-sub0 58.2%

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

      18. +-commutative 58.2%

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

      19. unsub-neg 58.2%

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

      20. distribute-neg-frac 58.2%

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

      21. metadata-eval 58.2%

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

      22. metadata-eval 58.2%

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

      23. metadata-eval 58.2%

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

      24. associate-/r* 58.2%

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

      25. metadata-eval 58.2%

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

      26. neg-mul-1 58.2%

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

   3. Simplified 58.2%

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

      1. frac-sub 58.7%

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

      2. *-rgt-identity 58.7%

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

      3. metadata-eval 58.7%

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

      4. div-inv 58.7%

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

      5. associate-/r* 58.7%

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

      6. *-un-lft-identity 58.7%

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

      7. metadata-eval 58.7%

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

      8. div-inv 58.7%

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

      9. associate--l- 64.3%

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

      10. div-inv 64.3%

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

      11. metadata-eval 64.3%

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

      12. *-rgt-identity 64.3%

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

      13. div-inv 64.3%

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

      14. metadata-eval 64.3%

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

      15. *-rgt-identity 64.3%

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

   5. Applied egg-rr 64.3%

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

   6. Taylor expanded in x around inf 98.9%

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

   7. Taylor expanded in x around 0 7.2%

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

4. Final simplification 54.9%

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

Alternative 6: 10.6% 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 78.1%

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

   1. sub-neg 78.1%

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

   2. +-commutative 78.1%

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

   3. neg-sub0 78.1%

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

   4. associate-+l- 78.1%

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

   5. sub-neg 78.1%

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

   6. associate--r+ 78.1%

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

   7. neg-sub0 78.1%

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

   8. distribute-neg-frac 78.1%

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

   9. metadata-eval 78.1%

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

   10. metadata-eval 78.1%

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

   11. metadata-eval 78.1%

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

   12. associate-/r* 78.1%

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

   13. metadata-eval 78.1%

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

   14. neg-mul-1 78.1%

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

   15. sub0-neg 78.1%

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

   16. associate-+l- 78.1%

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

   17. neg-sub0 78.1%

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

   18. +-commutative 78.1%

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

   19. unsub-neg 78.1%

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

   20. distribute-neg-frac 78.1%

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

   21. metadata-eval 78.1%

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

   22. metadata-eval 78.1%

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

   23. metadata-eval 78.1%

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

   24. associate-/r* 78.1%

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

   25. metadata-eval 78.1%

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

   26. neg-mul-1 78.1%

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

3. Simplified 78.1%

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

4. Taylor expanded in x around 0 53.3%

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

5. Taylor expanded in x around inf 11.2%

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

6. Final simplification 11.2%

   \[\leadsto 1 \]

Alternative 7: 50.2% 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 78.1%

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

   1. sub-neg 78.1%

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

   2. +-commutative 78.1%

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

   3. neg-sub0 78.1%

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

   4. associate-+l- 78.1%

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

   5. sub-neg 78.1%

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

   6. associate--r+ 78.1%

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

   7. neg-sub0 78.1%

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

   8. distribute-neg-frac 78.1%

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

   9. metadata-eval 78.1%

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

   10. metadata-eval 78.1%

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

   11. metadata-eval 78.1%

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

   12. associate-/r* 78.1%

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

   13. metadata-eval 78.1%

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

   14. neg-mul-1 78.1%

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

   15. sub0-neg 78.1%

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

   16. associate-+l- 78.1%

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

   17. neg-sub0 78.1%

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

   18. +-commutative 78.1%

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

   19. unsub-neg 78.1%

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

   20. distribute-neg-frac 78.1%

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

   21. metadata-eval 78.1%

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

   22. metadata-eval 78.1%

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

   23. metadata-eval 78.1%

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

   24. associate-/r* 78.1%

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

   25. metadata-eval 78.1%

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

   26. neg-mul-1 78.1%

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

3. Simplified 78.1%

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

4. Taylor expanded in x around 0 53.9%

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

5. Final simplification 53.9%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))