3frac (problem 3.3.3)

Percentage Accurate: 69.8% → 98.6%

Time: 10.0s

Alternatives: 11

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: 69.8% 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: 98.6% 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 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around inf 99.6%

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

   1. associate-*r/ 99.6%

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

   2. metadata-eval 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 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 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around inf 98.9%

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

   1. div-inv 98.9%

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

   2. pow-flip 99.3%

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

   3. metadata-eval 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 3: 98.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around -inf 69.6%

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

   1. associate-*r/ 69.6%

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

7. Simplified 69.6%

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

   1. frac-add 69.7%

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

   2. div-inv 69.7%

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

   3. *-un-lft-identity 69.7%

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

   4. *-commutative 69.7%

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

9. Applied egg-rr 69.7%

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

11. Simplified 69.7%

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

12. Taylor expanded in x around -inf 99.1%

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

    1. mul-1-neg 99.1%

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

    2. distribute-neg-frac2 99.1%

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

    3. sub-neg 99.1%

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

    4. associate-*r/ 99.1%

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

    5. sub-neg 99.1%

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

    6. metadata-eval 99.1%

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

    7. distribute-lft-in 99.1%

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

    8. neg-mul-1 99.1%

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

    9. distribute-neg-frac 99.1%

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

    10. metadata-eval 99.1%

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

    11. metadata-eval 99.1%

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

    12. metadata-eval 99.1%

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

14. Simplified 99.1%

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

15. Final simplification 99.1%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around -inf 69.6%

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

   1. associate-*r/ 69.6%

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

7. Simplified 69.6%

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

   1. frac-add 69.7%

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

   2. div-inv 69.7%

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

   3. *-un-lft-identity 69.7%

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

   4. *-commutative 69.7%

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

9. Applied egg-rr 69.7%

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

11. Simplified 69.7%

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

12. Taylor expanded in x around inf 99.1%

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

    1. associate-*r/ 99.1%

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

    2. metadata-eval 99.1%

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

14. Simplified 99.1%

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

15. Final simplification 99.1%

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

Alternative 5: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around -inf 69.6%

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

   1. associate-*r/ 69.6%

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

7. Simplified 69.6%

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

   1. frac-add 69.7%

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

   2. div-inv 69.7%

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

   3. *-un-lft-identity 69.7%

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

   4. *-commutative 69.7%

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

9. Applied egg-rr 69.7%

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

11. Simplified 69.7%

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

12. Taylor expanded in x around inf 97.9%

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

13. Final simplification 97.9%

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

Alternative 6: 68.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around inf 68.5%

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

6. Final simplification 68.5%

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

Alternative 7: 72.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around inf 68.8%

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

   1. associate-*r/ 68.8%

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

   2. neg-mul-1 68.8%

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

   3. distribute-neg-in 68.8%

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

   4. metadata-eval 68.8%

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

   5. distribute-neg-frac 68.8%

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

   6. metadata-eval 68.8%

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

7. Simplified 68.8%

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

   1. frac-add 68.8%

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

   2. associate-/r* 68.8%

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

   3. *-un-lft-identity 68.8%

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

9. Applied egg-rr 68.8%

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

10. Taylor expanded in x around 0 72.8%

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

11. Final simplification 72.8%

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

Alternative 8: 53.5% 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 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around inf 68.5%

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

   1. frac-add 68.5%

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

   2. div-inv 68.5%

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

   3. *-un-lft-identity 68.5%

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

   4. *-commutative 68.5%

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

   5. neg-mul-1 68.5%

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

   6. +-commutative 68.5%

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

   7. distribute-neg-in 68.5%

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

   8. metadata-eval 68.5%

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

   9. *-commutative 68.5%

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

7. Applied egg-rr 68.5%

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

   1. associate-*r/ 68.5%

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

   2. *-rgt-identity 68.5%

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

   3. +-commutative 68.5%

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

   4. associate-+r+ 56.3%

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

   5. neg-mul-1 56.3%

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

   6. distribute-lft1-in 56.3%

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

   7. metadata-eval 56.3%

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

   8. +-commutative 56.3%

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

9. Simplified 56.3%

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

    1. div-inv 56.3%

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

    2. mul0-lft 56.3%

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

    3. metadata-eval 56.3%

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

    4. +-commutative 56.3%

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

11. Applied egg-rr 56.3%

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

    1. *-lft-identity 56.3%

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

13. Simplified 56.3%

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

14. Final simplification 56.3%

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

Alternative 9: 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 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around 0 5.3%

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

6. Final simplification 5.3%

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

Alternative 10: 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 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around inf 68.5%

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

6. Taylor expanded in x around 0 5.3%

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

7. Final simplification 5.3%

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

Alternative 11: 3.4% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 69.7%

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

   1. +-commutative 69.7%

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

   2. associate-+r- 69.6%

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

   3. sub-neg 69.6%

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

   4. remove-double-neg 69.6%

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

   5. neg-sub0 69.6%

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

   6. associate-+l- 69.6%

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

   7. neg-sub0 69.6%

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

   8. distribute-neg-frac2 69.6%

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

   9. distribute-frac-neg2 69.6%

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

   10. associate-+r+ 69.7%

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

   11. +-commutative 69.7%

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

   12. remove-double-neg 69.7%

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

   13. distribute-neg-frac2 69.7%

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

   14. sub0-neg 69.7%

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

   15. associate-+l- 69.7%

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

   16. neg-sub0 69.7%

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

3. Simplified 69.7%

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

5. Taylor expanded in x around 0 3.3%

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

6. Taylor expanded in x around inf 3.3%

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

7. Final simplification 3.3%

   \[\leadsto 1 \]
8. 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 2024112 
(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))))