3frac (problem 3.3.3)

Percentage Accurate: 69.2% → 98.7%

Time: 10.8s

Alternatives: 10

Speedup: 1.7×

Specification

\[\left|x\right| > 1\]

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: 69.2% 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: 98.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ 2.0 (pow x 2.0)) (+ 2.0 (/ 2.0 (pow x 4.0)))) (pow x 3.0)))

C

double code(double x) {
	return ((2.0 / pow(x, 2.0)) + (2.0 + (2.0 / pow(x, 4.0)))) / pow(x, 3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.0d0 / (x ** 2.0d0)) + (2.0d0 + (2.0d0 / (x ** 4.0d0)))) / (x ** 3.0d0)
end function

Java

public static double code(double x) {
	return ((2.0 / Math.pow(x, 2.0)) + (2.0 + (2.0 / Math.pow(x, 4.0)))) / Math.pow(x, 3.0);
}

Python

def code(x):
	return ((2.0 / math.pow(x, 2.0)) + (2.0 + (2.0 / math.pow(x, 4.0)))) / math.pow(x, 3.0)

Julia

function code(x)
	return Float64(Float64(Float64(2.0 / (x ^ 2.0)) + Float64(2.0 + Float64(2.0 / (x ^ 4.0)))) / (x ^ 3.0))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 / (x ^ 2.0)) + (2.0 + (2.0 / (x ^ 4.0)))) / (x ^ 3.0);
end

Wolfram

code[x_] := N[(N[(N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(2.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

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

   1. +-commutative 69.5%

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

   2. associate-+r- 69.5%

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

   3. sub-neg 69.5%

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

   4. remove-double-neg 69.5%

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

   5. neg-sub0 69.5%

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

   6. associate-+l- 69.5%

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

   7. neg-sub0 69.5%

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

   8. distribute-neg-frac2 69.5%

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

   9. distribute-frac-neg2 69.5%

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

   10. associate-+r+ 69.5%

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

   11. +-commutative 69.5%

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

   12. remove-double-neg 69.5%

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

   13. distribute-neg-frac2 69.5%

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

   14. sub0-neg 69.5%

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

   15. associate-+l- 69.5%

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

   16. neg-sub0 69.5%

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

3. Simplified 69.5%

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

5. Taylor expanded in x around inf 99.6%

   \[\leadsto \color{blue}{\frac{2 + \left(2 \cdot \frac{1}{{x}^{2}} + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
6. Step-by-step derivation

   1. associate-+r+ 99.6%

      \[\leadsto \frac{\color{blue}{\left(2 + 2 \cdot \frac{1}{{x}^{2}}\right) + \frac{2}{{x}^{4}}}}{{x}^{3}} \]

   2. +-commutative 99.6%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{{x}^{2}} + 2\right)} + \frac{2}{{x}^{4}}}{{x}^{3}} \]

   3. associate-+l+ 99.6%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}}{{x}^{3}} \]

   4. associate-*r/ 99.6%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\frac{\color{blue}{2}}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]

8. Final simplification 99.6%

   \[\leadsto \frac{\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}} \]
9. Add Preprocessing

Alternative 2: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \frac{2}{{x}^{2}}}{{x}^{3}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ 2.0 (/ 2.0 (pow x 2.0))) (pow x 3.0)))

C

double code(double x) {
	return (2.0 + (2.0 / pow(x, 2.0))) / pow(x, 3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 + (2.0d0 / (x ** 2.0d0))) / (x ** 3.0d0)
end function

Java

public static double code(double x) {
	return (2.0 + (2.0 / Math.pow(x, 2.0))) / Math.pow(x, 3.0);
}

Python

def code(x):
	return (2.0 + (2.0 / math.pow(x, 2.0))) / math.pow(x, 3.0)

Julia

function code(x)
	return Float64(Float64(2.0 + Float64(2.0 / (x ^ 2.0))) / (x ^ 3.0))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 + (2.0 / (x ^ 2.0))) / (x ^ 3.0);
end

Wolfram

code[x_] := N[(N[(2.0 + N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

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

   1. +-commutative 69.5%

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

   2. associate-+r- 69.5%

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

   3. sub-neg 69.5%

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

   4. remove-double-neg 69.5%

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

   5. neg-sub0 69.5%

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

   6. associate-+l- 69.5%

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

   7. neg-sub0 69.5%

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

   8. distribute-neg-frac2 69.5%

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

   9. distribute-frac-neg2 69.5%

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

   10. associate-+r+ 69.5%

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

   11. +-commutative 69.5%

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

   12. remove-double-neg 69.5%

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

   13. distribute-neg-frac2 69.5%

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

   14. sub0-neg 69.5%

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

   15. associate-+l- 69.5%

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

   16. neg-sub0 69.5%

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

3. Simplified 69.5%

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

5. Taylor expanded in x around inf 99.3%

   \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{3}}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.3%

      \[\leadsto \frac{2 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}}{{x}^{3}} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{2 + \frac{\color{blue}{2}}{{x}^{2}}}{{x}^{3}} \]

7. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{2 + \frac{2}{{x}^{2}}}{{x}^{3}}} \]

8. Final simplification 99.3%

   \[\leadsto \frac{2 + \frac{2}{{x}^{2}}}{{x}^{3}} \]
9. Add Preprocessing

Alternative 3: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (pow x -3.0)))

C

double code(double x) {
	return 2.0 * pow(x, -3.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * (x ** (-3.0d0))
end function

Java

public static double code(double x) {
	return 2.0 * Math.pow(x, -3.0);
}

Python

def code(x):
	return 2.0 * math.pow(x, -3.0)

Julia

function code(x)
	return Float64(2.0 * (x ^ -3.0))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * (x ^ -3.0);
end

Wolfram

code[x_] := N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

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

   1. +-commutative 69.5%

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

   2. associate-+r- 69.5%

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

   3. sub-neg 69.5%

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

   4. remove-double-neg 69.5%

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

   5. neg-sub0 69.5%

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

   6. associate-+l- 69.5%

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

   7. neg-sub0 69.5%

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

   8. distribute-neg-frac2 69.5%

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

   9. distribute-frac-neg2 69.5%

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

   10. associate-+r+ 69.5%

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

   11. +-commutative 69.5%

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

   12. remove-double-neg 69.5%

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

   13. distribute-neg-frac2 69.5%

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

   14. sub0-neg 69.5%

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

   15. associate-+l- 69.5%

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

   16. neg-sub0 69.5%

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

3. Simplified 69.5%

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

5. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
6. Step-by-step derivation

   1. div-inv 98.8%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}}} \]

   2. pow-flip 99.2%

      \[\leadsto 2 \cdot \color{blue}{{x}^{\left(-3\right)}} \]

   3. metadata-eval 99.2%

      \[\leadsto 2 \cdot {x}^{\color{blue}{-3}} \]

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{2 \cdot {x}^{-3}} \]

8. Final simplification 99.2%

   \[\leadsto 2 \cdot {x}^{-3} \]
9. Add Preprocessing

Alternative 4: 67.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. +-commutative 69.5%

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

   2. associate-+r- 69.5%

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

   3. sub-neg 69.5%

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

   4. remove-double-neg 69.5%

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

   5. neg-sub0 69.5%

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

   6. associate-+l- 69.5%

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

   7. neg-sub0 69.5%

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

   8. distribute-neg-frac2 69.5%

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

   9. distribute-frac-neg2 69.5%

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

   10. associate-+r+ 69.5%

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

   11. +-commutative 69.5%

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

   12. remove-double-neg 69.5%

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

   13. distribute-neg-frac2 69.5%

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

   14. sub0-neg 69.5%

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

   15. associate-+l- 69.5%

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

   16. neg-sub0 69.5%

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

3. Simplified 69.5%

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

5. Taylor expanded in x around inf 68.4%

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

6. Final simplification 68.4%

   \[\leadsto \frac{1}{x + -1} + \frac{-1}{x} \]
7. Add Preprocessing

Alternative 5: 72.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. +-commutative 69.5%

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

   2. associate-+r- 69.5%

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

   3. sub-neg 69.5%

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

   4. remove-double-neg 69.5%

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

   5. neg-sub0 69.5%

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

   6. associate-+l- 69.5%

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

   7. neg-sub0 69.5%

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

   8. distribute-neg-frac2 69.5%

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

   9. distribute-frac-neg2 69.5%

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

   10. associate-+r+ 69.5%

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

   11. +-commutative 69.5%

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

   12. remove-double-neg 69.5%

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

   13. distribute-neg-frac2 69.5%

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

   14. sub0-neg 69.5%

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

   15. associate-+l- 69.5%

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

   16. neg-sub0 69.5%

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

3. Simplified 69.5%

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

5. Taylor expanded in x around inf 68.6%

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

   1. associate-*r/ 68.6%

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

   2. neg-mul-1 68.6%

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

   3. distribute-neg-in 68.6%

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

   4. metadata-eval 68.6%

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

   5. distribute-neg-frac 68.6%

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

   6. metadata-eval 68.6%

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

7. Simplified 68.6%

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

   1. frac-add 68.7%

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

   2. *-un-lft-identity 68.7%

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

9. Applied egg-rr 68.7%

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

10. Taylor expanded in x around 0 72.7%

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

11. Final simplification 72.7%

    \[\leadsto \frac{\frac{1}{x}}{x \cdot \left(x + -1\right)} \]
12. Add Preprocessing

Alternative 6: 52.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. +-commutative 69.5%

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

   2. associate-+r- 69.5%

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

   3. sub-neg 69.5%

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

   4. remove-double-neg 69.5%

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

   5. neg-sub0 69.5%

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

   6. associate-+l- 69.5%

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

   7. neg-sub0 69.5%

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

   8. distribute-neg-frac2 69.5%

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

   9. distribute-frac-neg2 69.5%

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

   10. associate-+r+ 69.5%

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

   11. +-commutative 69.5%

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

   12. remove-double-neg 69.5%

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

   13. distribute-neg-frac2 69.5%

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

   14. sub0-neg 69.5%

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

   15. associate-+l- 69.5%

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

   16. neg-sub0 69.5%

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

3. Simplified 69.5%

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

5. Taylor expanded in x around inf 68.4%

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

   1. clear-num 68.4%

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

   2. frac-add 68.4%

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

   3. *-un-lft-identity 68.4%

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

   4. div-inv 68.4%

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

   5. metadata-eval 68.4%

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

   6. metadata-eval 68.4%

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

   7. div-inv 68.4%

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

   8. /-rgt-identity 68.4%

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

   9. div-inv 68.4%

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

   10. metadata-eval 68.4%

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

7. Applied egg-rr 68.4%

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

   1. +-commutative 68.4%

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

   2. +-commutative 68.4%

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

   3. associate-+l+ 53.9%

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

   4. *-commutative 53.9%

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

   5. mul-1-neg 53.9%

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

   6. sub-neg 53.9%

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

   7. +-inverses 53.9%

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

   8. metadata-eval 53.9%

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

   9. *-commutative 53.9%

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

   10. +-commutative 53.9%

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

   11. distribute-lft-in 53.9%

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

   12. associate-*l* 53.9%

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

   13. metadata-eval 53.9%

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

   14. *-commutative 53.9%

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

   15. distribute-rgt-in 53.9%

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

   16. *-commutative 53.9%

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

   17. mul-1-neg 53.9%

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

   18. unsub-neg 53.9%

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

9. Simplified 53.9%

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

10. Final simplification 53.9%

    \[\leadsto \frac{-1}{x \cdot \left(1 - x\right)} \]
11. Add Preprocessing

Alternative 7: 52.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. +-commutative 69.5%

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

   2. associate-+r- 69.5%

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

   3. sub-neg 69.5%

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

   4. remove-double-neg 69.5%

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

   5. neg-sub0 69.5%

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

   6. associate-+l- 69.5%

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

   7. neg-sub0 69.5%

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

   8. distribute-neg-frac2 69.5%

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

   9. distribute-frac-neg2 69.5%

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

   10. associate-+r+ 69.5%

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

   11. +-commutative 69.5%

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

   12. remove-double-neg 69.5%

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

   13. distribute-neg-frac2 69.5%

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

   14. sub0-neg 69.5%

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

   15. associate-+l- 69.5%

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

   16. neg-sub0 69.5%

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

3. Simplified 69.5%

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

5. Taylor expanded in x around inf 68.4%

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

   1. frac-2neg 68.4%

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

   2. metadata-eval 68.4%

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

   3. add-sqr-sqrt 21.6%

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

   4. sqrt-unprod 14.8%

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

   5. sqr-neg 14.8%

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

   6. unpow2 14.8%

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

   7. sqrt-pow1 6.5%

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

   8. metadata-eval 6.5%

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

   9. pow1 6.5%

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

   10. *-un-lft-identity 6.5%

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

   11. metadata-eval 6.5%

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

   12. cancel-sign-sub-inv 6.5%

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

   13. div-inv 6.5%

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

   14. frac-2neg 6.5%

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

   15. metadata-eval 6.5%

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

   16. clear-num 6.5%

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

   17. frac-sub 53.6%

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

7. Applied egg-rr 68.4%

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

   1. *-commutative 68.4%

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

   2. *-rgt-identity 68.4%

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

   3. associate-*l* 68.4%

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

   4. metadata-eval 68.4%

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

   5. *-rgt-identity 68.4%

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

   6. associate--r+ 53.9%

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

   7. +-inverses 53.9%

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

   8. metadata-eval 53.9%

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

   9. *-commutative 53.9%

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

   10. associate-*l* 53.9%

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

   11. associate-/r* 52.4%

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

   12. +-commutative 52.4%

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

   13. distribute-lft-in 52.4%

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

   14. metadata-eval 52.4%

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

   15. mul-1-neg 52.4%

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

   16. unsub-neg 52.4%

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

9. Simplified 52.4%

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

    1. div-inv 52.4%

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

    2. *-un-lft-identity 52.4%

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

    3. times-frac 52.4%

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

    4. metadata-eval 52.4%

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

11. Applied egg-rr 52.4%

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

    1. associate-/l/ 53.9%

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

    2. mul-1-neg 53.9%

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

    3. *-commutative 53.9%

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

    4. distribute-frac-neg2 53.9%

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

    5. distribute-rgt-neg-in 53.9%

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

    6. neg-sub0 53.9%

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

    7. associate--r- 53.9%

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

    8. metadata-eval 53.9%

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

13. Simplified 53.9%

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

14. Final simplification 53.9%

    \[\leadsto \frac{1}{x \cdot \left(x + 1\right)} \]
15. Add Preprocessing

Alternative 8: 5.1% 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 69.5%

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

   1. +-commutative 69.5%

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

   2. associate-+r- 69.5%

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

   3. sub-neg 69.5%

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

   4. remove-double-neg 69.5%

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

   5. neg-sub0 69.5%

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

   6. associate-+l- 69.5%

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

   7. neg-sub0 69.5%

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

   8. distribute-neg-frac2 69.5%

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

   9. distribute-frac-neg2 69.5%

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

   10. associate-+r+ 69.5%

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

   11. +-commutative 69.5%

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

   12. remove-double-neg 69.5%

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

   13. distribute-neg-frac2 69.5%

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

   14. sub0-neg 69.5%

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

   15. associate-+l- 69.5%

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

   16. neg-sub0 69.5%

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

3. Simplified 69.5%

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

5. Taylor expanded in x around 0 5.2%

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

6. Final simplification 5.2%

   \[\leadsto \frac{-2}{x} \]
7. Add Preprocessing

Alternative 9: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. +-commutative 69.5%

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

   2. associate-+r- 69.5%

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

   3. sub-neg 69.5%

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

   4. remove-double-neg 69.5%

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

   5. neg-sub0 69.5%

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

   6. associate-+l- 69.5%

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

   7. neg-sub0 69.5%

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

   8. distribute-neg-frac2 69.5%

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

   9. distribute-frac-neg2 69.5%

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

   10. associate-+r+ 69.5%

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

   11. +-commutative 69.5%

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

   12. remove-double-neg 69.5%

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

   13. distribute-neg-frac2 69.5%

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

   14. sub0-neg 69.5%

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

   15. associate-+l- 69.5%

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

   16. neg-sub0 69.5%

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

3. Simplified 69.5%

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

5. Taylor expanded in x around inf 68.4%

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

6. Taylor expanded in x around 0 5.2%

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

7. Final simplification 5.2%

   \[\leadsto \frac{-1}{x} \]
8. Add Preprocessing

Alternative 10: 6.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. +-commutative 69.5%

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

   2. associate-+r- 69.5%

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

   3. sub-neg 69.5%

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

   4. remove-double-neg 69.5%

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

   5. neg-sub0 69.5%

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

   6. associate-+l- 69.5%

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

   7. neg-sub0 69.5%

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

   8. distribute-neg-frac2 69.5%

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

   9. distribute-frac-neg2 69.5%

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

   10. associate-+r+ 69.5%

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

   11. +-commutative 69.5%

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

   12. remove-double-neg 69.5%

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

   13. distribute-neg-frac2 69.5%

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

   14. sub0-neg 69.5%

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

   15. associate-+l- 69.5%

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

   16. neg-sub0 69.5%

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

3. Simplified 69.5%

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

5. Taylor expanded in x around inf 68.4%

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

   1. frac-2neg 68.4%

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

   2. metadata-eval 68.4%

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

   3. add-sqr-sqrt 21.6%

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

   4. sqrt-unprod 14.8%

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

   5. sqr-neg 14.8%

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

   6. unpow2 14.8%

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

   7. sqrt-pow1 6.5%

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

   8. metadata-eval 6.5%

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

   9. pow1 6.5%

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

   10. *-un-lft-identity 6.5%

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

   11. metadata-eval 6.5%

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

   12. cancel-sign-sub-inv 6.5%

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

   13. div-inv 6.5%

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

   14. frac-2neg 6.5%

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

   15. metadata-eval 6.5%

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

   16. clear-num 6.5%

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

   17. frac-sub 53.6%

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

7. Applied egg-rr 68.4%

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

   1. *-commutative 68.4%

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

   2. *-rgt-identity 68.4%

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

   3. associate-*l* 68.4%

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

   4. metadata-eval 68.4%

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

   5. *-rgt-identity 68.4%

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

   6. associate--r+ 53.9%

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

   7. +-inverses 53.9%

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

   8. metadata-eval 53.9%

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

   9. *-commutative 53.9%

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

   10. associate-*l* 53.9%

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

   11. associate-/r* 52.4%

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

   12. +-commutative 52.4%

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

   13. distribute-lft-in 52.4%

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

   14. metadata-eval 52.4%

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

   15. mul-1-neg 52.4%

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

   16. unsub-neg 52.4%

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

9. Simplified 52.4%

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

10. Taylor expanded in x around 0 6.5%

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

11. Final simplification 6.5%

    \[\leadsto \frac{1}{x} \]
12. Add Preprocessing

Developer target: 99.1% 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 2024077 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (/ 2.0 (* x (- (* x x) 1.0)))

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