3frac (problem 3.3.3)

Percentage Accurate: 9.9% → 99.4%

Time: 8.1s

Alternatives: 5

Speedup: 1.7×

Specification

\[\left|x\right| > 1 \land \left|x\right| < 10^{+100}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 9.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.7%

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

   1. sub-neg 10.7%

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

   2. distribute-neg-frac 10.7%

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

   3. metadata-eval 10.7%

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

   4. metadata-eval 10.7%

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

   5. metadata-eval 10.7%

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

   6. associate-/r* 10.7%

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

   7. metadata-eval 10.7%

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

   8. neg-mul-1 10.7%

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

   9. +-commutative 10.7%

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

   10. associate-+l+ 10.6%

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

   11. +-commutative 10.6%

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

   12. neg-mul-1 10.6%

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

   13. metadata-eval 10.6%

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

   14. associate-/r* 10.6%

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

   15. metadata-eval 10.6%

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

   16. metadata-eval 10.6%

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

   17. +-commutative 10.6%

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

   18. +-commutative 10.6%

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

3. Simplified 10.6%

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

   1. frac-2neg 10.6%

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

   2. metadata-eval 10.6%

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

   3. frac-add 10.2%

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

   4. *-un-lft-identity 10.2%

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

   5. *-commutative 10.2%

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

   6. +-commutative 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. +-commutative 10.2%

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

5. Applied egg-rr 10.2%

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

   1. associate-/r* 10.6%

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

   2. neg-sub0 10.6%

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

   3. associate--r- 10.6%

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

   4. metadata-eval 10.6%

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

   5. +-commutative 10.6%

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

   6. neg-sub0 10.6%

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

   7. associate--r- 10.6%

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

   8. metadata-eval 10.6%

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

   9. +-commutative 10.6%

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

   10. associate-/r* 10.2%

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

   11. associate-+l+ 10.2%

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

   12. +-commutative 10.2%

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

   13. associate-+r+ 10.2%

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

   14. metadata-eval 10.2%

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

7. Simplified 10.2%

   \[\leadsto \frac{-2}{x} + \color{blue}{\frac{x + \left(0 + x\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}} \]
8. Step-by-step derivation

   1. +-commutative 10.2%

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

   2. clear-num 10.3%

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

   3. frac-add 10.3%

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

   4. *-un-lft-identity 10.3%

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

   5. +-lft-identity 10.3%

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

   6. +-lft-identity 10.3%

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

9. Applied egg-rr 10.3%

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

10. Taylor expanded in x around 0 99.5%

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

    1. expm1-log1p-u 99.5%

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

    2. expm1-udef 8.8%

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

12. Applied egg-rr 8.8%

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

    1. expm1-def 99.2%

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

    2. expm1-log1p 99.2%

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

    3. associate-*r/ 99.2%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.7%

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

   1. sub-neg 10.7%

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

   2. distribute-neg-frac 10.7%

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

   3. metadata-eval 10.7%

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

   4. metadata-eval 10.7%

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

   5. metadata-eval 10.7%

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

   6. associate-/r* 10.7%

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

   7. metadata-eval 10.7%

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

   8. neg-mul-1 10.7%

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

   9. +-commutative 10.7%

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

   10. associate-+l+ 10.6%

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

   11. +-commutative 10.6%

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

   12. neg-mul-1 10.6%

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

   13. metadata-eval 10.6%

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

   14. associate-/r* 10.6%

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

   15. metadata-eval 10.6%

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

   16. metadata-eval 10.6%

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

   17. +-commutative 10.6%

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

   18. +-commutative 10.6%

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

3. Simplified 10.6%

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

   1. frac-2neg 10.6%

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

   2. metadata-eval 10.6%

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

   3. frac-add 10.2%

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

   4. *-un-lft-identity 10.2%

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

   5. *-commutative 10.2%

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

   6. +-commutative 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. +-commutative 10.2%

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

5. Applied egg-rr 10.2%

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

   1. associate-/r* 10.6%

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

   2. neg-sub0 10.6%

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

   3. associate--r- 10.6%

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

   4. metadata-eval 10.6%

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

   5. +-commutative 10.6%

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

   6. neg-sub0 10.6%

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

   7. associate--r- 10.6%

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

   8. metadata-eval 10.6%

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

   9. +-commutative 10.6%

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

   10. associate-/r* 10.2%

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

   11. associate-+l+ 10.2%

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

   12. +-commutative 10.2%

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

   13. associate-+r+ 10.2%

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

   14. metadata-eval 10.2%

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

7. Simplified 10.2%

   \[\leadsto \frac{-2}{x} + \color{blue}{\frac{x + \left(0 + x\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}} \]
8. Step-by-step derivation

   1. +-commutative 10.2%

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

   2. frac-add 13.8%

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

   3. +-lft-identity 13.8%

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

9. Applied egg-rr 13.8%

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

10. Taylor expanded in x around 0 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 96.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{x \cdot \left(x \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.0 x) (* x (* x 0.5))))

C

double code(double x) {
	return (1.0 / x) / (x * (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) / (x * (x * 0.5d0))
end function

Java

public static double code(double x) {
	return (1.0 / x) / (x * (x * 0.5));
}

Python

def code(x):
	return (1.0 / x) / (x * (x * 0.5))

Julia

function code(x)
	return Float64(Float64(1.0 / x) / Float64(x * Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) / (x * (x * 0.5));
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] / N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.7%

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

   1. sub-neg 10.7%

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

   2. distribute-neg-frac 10.7%

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

   3. metadata-eval 10.7%

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

   4. metadata-eval 10.7%

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

   5. metadata-eval 10.7%

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

   6. associate-/r* 10.7%

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

   7. metadata-eval 10.7%

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

   8. neg-mul-1 10.7%

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

   9. +-commutative 10.7%

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

   10. associate-+l+ 10.6%

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

   11. +-commutative 10.6%

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

   12. neg-mul-1 10.6%

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

   13. metadata-eval 10.6%

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

   14. associate-/r* 10.6%

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

   15. metadata-eval 10.6%

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

   16. metadata-eval 10.6%

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

   17. +-commutative 10.6%

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

   18. +-commutative 10.6%

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

3. Simplified 10.6%

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

   1. frac-2neg 10.6%

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

   2. metadata-eval 10.6%

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

   3. frac-add 10.2%

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

   4. *-un-lft-identity 10.2%

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

   5. *-commutative 10.2%

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

   6. +-commutative 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. +-commutative 10.2%

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

5. Applied egg-rr 10.2%

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

   1. associate-/r* 10.6%

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

   2. neg-sub0 10.6%

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

   3. associate--r- 10.6%

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

   4. metadata-eval 10.6%

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

   5. +-commutative 10.6%

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

   6. neg-sub0 10.6%

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

   7. associate--r- 10.6%

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

   8. metadata-eval 10.6%

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

   9. +-commutative 10.6%

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

   10. associate-/r* 10.2%

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

   11. associate-+l+ 10.2%

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

   12. +-commutative 10.2%

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

   13. associate-+r+ 10.2%

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

   14. metadata-eval 10.2%

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

7. Simplified 10.2%

   \[\leadsto \frac{-2}{x} + \color{blue}{\frac{x + \left(0 + x\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}} \]
8. Step-by-step derivation

   1. +-commutative 10.2%

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

   2. clear-num 10.3%

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

   3. frac-add 10.3%

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

   4. *-un-lft-identity 10.3%

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

   5. +-lft-identity 10.3%

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

   6. +-lft-identity 10.3%

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

9. Applied egg-rr 10.3%

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

10. Taylor expanded in x around 0 99.5%

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

11. Taylor expanded in x around inf 95.8%

    \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\left(0.5 \cdot x\right)} \cdot x} \]
12. Step-by-step derivation

    1. *-commutative 95.8%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\left(x \cdot 0.5\right)} \cdot x} \]

13. Simplified 95.8%

    \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\left(x \cdot 0.5\right)} \cdot x} \]

14. Final simplification 95.8%

    \[\leadsto \frac{\frac{1}{x}}{x \cdot \left(x \cdot 0.5\right)} \]

Alternative 4: 2.5% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ -2.0 x))

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

function code(x)
	return Float64(-2.0 / x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.7%

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

   1. sub-neg 10.7%

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

   2. distribute-neg-frac 10.7%

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

   3. metadata-eval 10.7%

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

   4. metadata-eval 10.7%

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

   5. metadata-eval 10.7%

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

   6. associate-/r* 10.7%

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

   7. metadata-eval 10.7%

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

   8. neg-mul-1 10.7%

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

   9. +-commutative 10.7%

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

   10. associate-+l+ 10.6%

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

   11. +-commutative 10.6%

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

   12. neg-mul-1 10.6%

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

   13. metadata-eval 10.6%

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

   14. associate-/r* 10.6%

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

   15. metadata-eval 10.6%

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

   16. metadata-eval 10.6%

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

   17. +-commutative 10.6%

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

   18. +-commutative 10.6%

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

3. Simplified 10.6%

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

4. Taylor expanded in x around 0 2.6%

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

5. Final simplification 2.6%

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

Alternative 5: 6.3% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 2.0 x))

C

double code(double x) {
	return 2.0 / x;
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 / x;
}

Python

def code(x):
	return 2.0 / x

Julia

function code(x)
	return Float64(2.0 / x)
end

MATLAB

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

Wolfram

code[x_] := N[(2.0 / x), $MachinePrecision]

Derivation

1. Initial program 10.7%

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

   1. sub-neg 10.7%

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

   2. distribute-neg-frac 10.7%

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

   3. metadata-eval 10.7%

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

   4. metadata-eval 10.7%

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

   5. metadata-eval 10.7%

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

   6. associate-/r* 10.7%

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

   7. metadata-eval 10.7%

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

   8. neg-mul-1 10.7%

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

   9. +-commutative 10.7%

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

   10. associate-+l+ 10.6%

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

   11. +-commutative 10.6%

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

   12. neg-mul-1 10.6%

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

   13. metadata-eval 10.6%

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

   14. associate-/r* 10.6%

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

   15. metadata-eval 10.6%

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

   16. metadata-eval 10.6%

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

   17. +-commutative 10.6%

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

   18. +-commutative 10.6%

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

3. Simplified 10.6%

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

   1. frac-2neg 10.6%

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

   2. metadata-eval 10.6%

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

   3. frac-add 10.2%

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

   4. *-un-lft-identity 10.2%

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

   5. *-commutative 10.2%

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

   6. +-commutative 10.2%

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

   7. *-un-lft-identity 10.2%

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

   8. +-commutative 10.2%

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

5. Applied egg-rr 10.2%

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

   1. associate-/r* 10.6%

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

   2. neg-sub0 10.6%

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

   3. associate--r- 10.6%

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

   4. metadata-eval 10.6%

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

   5. +-commutative 10.6%

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

   6. neg-sub0 10.6%

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

   7. associate--r- 10.6%

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

   8. metadata-eval 10.6%

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

   9. +-commutative 10.6%

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

   10. associate-/r* 10.2%

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

   11. associate-+l+ 10.2%

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

   12. +-commutative 10.2%

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

   13. associate-+r+ 10.2%

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

   14. metadata-eval 10.2%

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

7. Simplified 10.2%

   \[\leadsto \frac{-2}{x} + \color{blue}{\frac{x + \left(0 + x\right)}{\left(x + 1\right) \cdot \left(x + -1\right)}} \]
8. Step-by-step derivation

   1. +-commutative 10.2%

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

   2. associate-/r* 10.6%

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

   3. frac-2neg 10.6%

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

   4. metadata-eval 10.6%

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

   5. frac-add 10.5%

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

   6. +-lft-identity 10.5%

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

   7. add-sqr-sqrt 4.2%

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

   8. sqrt-unprod 5.8%

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

   9. sqr-neg 5.8%

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

   10. sqrt-unprod 1.3%

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

   11. add-sqr-sqrt 2.6%

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

   12. add-sqr-sqrt 1.3%

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

   13. sqrt-unprod 4.5%

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

   14. sqr-neg 4.5%

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

   15. sqrt-unprod 3.2%

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

   16. add-sqr-sqrt 6.3%

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

9. Applied egg-rr 6.3%

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

10. Taylor expanded in x around 0 6.3%

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

11. Taylor expanded in x around 0 6.3%

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

12. Final simplification 6.3%

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

Developer target: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023333 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (and (> (fabs x) 1.0) (< (fabs x) 1e+100))

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))