3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 98.8%

Time: 9.3s

Alternatives: 12

Speedup: 1.0×

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: 69.3% 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: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((2.0 / (x * x)) + (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 * x)) + (2.0d0 + (2.0d0 / (x ** 4.0d0)))) / (x ** 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(2.0 / N[(x * x), $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 74.3%

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

   1. +-commutative 74.3%

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

   2. associate-+r- 74.3%

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

   3. sub-neg 74.3%

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

   4. remove-double-neg 74.3%

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

   5. neg-sub0 74.3%

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

   6. associate-+l- 74.3%

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

   7. neg-sub0 74.3%

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

   8. distribute-neg-frac2 74.3%

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

   9. distribute-frac-neg2 74.3%

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

   10. associate-+r+ 74.3%

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

   11. +-commutative 74.3%

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

   12. remove-double-neg 74.3%

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

   13. distribute-neg-frac2 74.3%

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

   14. sub0-neg 74.3%

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

   15. associate-+l- 74.3%

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

   16. neg-sub0 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 99.0%

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

   1. associate-+r+ 99.0%

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

   2. +-commutative 99.0%

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

   3. associate-+l+ 99.0%

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

   4. associate-*r/ 99.0%

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

   5. metadata-eval 99.0%

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

7. Simplified 99.0%

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

   1. unpow2 99.0%

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

9. Applied egg-rr 99.0%

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

Alternative 2: 98.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. +-commutative 74.3%

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

   2. associate-+r- 74.3%

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

   3. sub-neg 74.3%

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

   4. remove-double-neg 74.3%

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

   5. neg-sub0 74.3%

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

   6. associate-+l- 74.3%

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

   7. neg-sub0 74.3%

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

   8. distribute-neg-frac2 74.3%

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

   9. distribute-frac-neg2 74.3%

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

   10. associate-+r+ 74.3%

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

   11. +-commutative 74.3%

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

   12. remove-double-neg 74.3%

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

   13. distribute-neg-frac2 74.3%

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

   14. sub0-neg 74.3%

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

   15. associate-+l- 74.3%

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

   16. neg-sub0 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 98.8%

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

   1. associate-*r/ 98.8%

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

   2. metadata-eval 98.8%

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

7. Simplified 98.8%

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

   1. unpow2 99.0%

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

9. Applied egg-rr 98.8%

   \[\leadsto \frac{2 + \frac{2}{\color{blue}{x \cdot x}}}{{x}^{3}} \]
10. 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 74.3%

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

   1. +-commutative 74.3%

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

   2. associate-+r- 74.3%

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

   3. sub-neg 74.3%

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

   4. remove-double-neg 74.3%

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

   5. neg-sub0 74.3%

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

   6. associate-+l- 74.3%

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

   7. neg-sub0 74.3%

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

   8. distribute-neg-frac2 74.3%

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

   9. distribute-frac-neg2 74.3%

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

   10. associate-+r+ 74.3%

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

   11. +-commutative 74.3%

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

   12. remove-double-neg 74.3%

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

   13. distribute-neg-frac2 74.3%

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

   14. sub0-neg 74.3%

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

   15. associate-+l- 74.3%

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

   16. neg-sub0 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 98.3%

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

   1. div-inv 98.3%

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

   2. pow-flip 98.7%

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

   3. metadata-eval 98.7%

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

7. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{2 \cdot {x}^{-3}} \]
8. Add Preprocessing

Alternative 4: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-1 - x\right)\\ \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(x + -1\right) \cdot \frac{x}{1 + \frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -1\right) \cdot \left(2 + x\right) + t\_0}{\left(x + -1\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (- -1.0 x))))
   (if (<= (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))) 2e-32)
     (/ (/ 1.0 x) (* (+ x -1.0) (/ x (+ 1.0 (/ 1.0 x)))))
     (/ (+ (* (+ x -1.0) (+ 2.0 x)) t_0) (* (+ x -1.0) t_0)))))

C

double code(double x) {
	double t_0 = x * (-1.0 - x);
	double tmp;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 2e-32) {
		tmp = (1.0 / x) / ((x + -1.0) * (x / (1.0 + (1.0 / x))));
	} else {
		tmp = (((x + -1.0) * (2.0 + x)) + t_0) / ((x + -1.0) * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) - x)
    if ((((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))) <= 2d-32) then
        tmp = (1.0d0 / x) / ((x + (-1.0d0)) * (x / (1.0d0 + (1.0d0 / x))))
    else
        tmp = (((x + (-1.0d0)) * (2.0d0 + x)) + t_0) / ((x + (-1.0d0)) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (-1.0 - x);
	double tmp;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 2e-32) {
		tmp = (1.0 / x) / ((x + -1.0) * (x / (1.0 + (1.0 / x))));
	} else {
		tmp = (((x + -1.0) * (2.0 + x)) + t_0) / ((x + -1.0) * t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (-1.0 - x)
	tmp = 0
	if (((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 2e-32:
		tmp = (1.0 / x) / ((x + -1.0) * (x / (1.0 + (1.0 / x))))
	else:
		tmp = (((x + -1.0) * (2.0 + x)) + t_0) / ((x + -1.0) * t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(-1.0 - x))
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0))) <= 2e-32)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(x + -1.0) * Float64(x / Float64(1.0 + Float64(1.0 / x)))));
	else
		tmp = Float64(Float64(Float64(Float64(x + -1.0) * Float64(2.0 + x)) + t_0) / Float64(Float64(x + -1.0) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (-1.0 - x);
	tmp = 0.0;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 2e-32)
		tmp = (1.0 / x) / ((x + -1.0) * (x / (1.0 + (1.0 / x))));
	else
		tmp = (((x + -1.0) * (2.0 + x)) + t_0) / ((x + -1.0) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 2e-32], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(x / N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x + -1.0), $MachinePrecision] * N[(2.0 + x), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64)))) < 2.00000000000000011e-32

   1. Initial program 74.2%

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

      1. +-commutative 74.2%

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

      2. associate-+r- 74.2%

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

      3. sub-neg 74.2%

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

      4. remove-double-neg 74.2%

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

      5. neg-sub0 74.2%

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

      6. associate-+l- 74.2%

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

      7. neg-sub0 74.2%

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

      8. distribute-neg-frac2 74.2%

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

      9. distribute-frac-neg2 74.2%

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

      10. associate-+r+ 74.2%

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

      11. +-commutative 74.2%

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

      12. remove-double-neg 74.2%

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

      13. distribute-neg-frac2 74.2%

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

      14. sub0-neg 74.2%

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

      15. associate-+l- 74.2%

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

      16. neg-sub0 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in x around inf 73.7%

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

      1. associate-*r/ 73.7%

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

      2. neg-mul-1 73.7%

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

      3. distribute-neg-in 73.7%

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

      4. metadata-eval 73.7%

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

      5. distribute-neg-frac 73.7%

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

      6. metadata-eval 73.7%

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

   7. Simplified 73.7%

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

      1. frac-2neg 73.7%

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

      2. metadata-eval 73.7%

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

      3. clear-num 73.7%

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

      4. frac-add 73.7%

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

   9. Applied egg-rr 73.7%

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

       1. *-rgt-identity 73.7%

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

       2. mul-1-neg 73.7%

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

       3. distribute-neg-out 73.7%

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

       4. +-commutative 73.7%

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

       5. distribute-neg-out 73.7%

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

       6. distribute-neg-in 73.7%

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

       7. metadata-eval 73.7%

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

       8. +-commutative 73.7%

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

       9. unsub-neg 73.7%

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

       10. distribute-neg-frac2 73.7%

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

       11. distribute-neg-in 73.7%

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

       12. metadata-eval 73.7%

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

       13. distribute-neg-frac 73.7%

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

       14. metadata-eval 73.7%

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

       15. distribute-lft-neg-out 73.7%

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

       16. distribute-rgt-neg-in 73.7%

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

       17. distribute-neg-frac2 73.7%

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

   11. Simplified 73.7%

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

   12. Taylor expanded in x around inf 77.2%

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

   if 2.00000000000000011e-32 < (+.f64 (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 2 binary64) x)) (/.f64 #s(literal 1 binary64) (-.f64 x #s(literal 1 binary64))))

   1. Initial program 87.3%

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

      1. +-commutative 87.3%

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

      2. associate-+r- 86.4%

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

      3. sub-neg 86.4%

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

      4. remove-double-neg 86.4%

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

      5. neg-sub0 86.4%

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

      6. associate-+l- 86.4%

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

      7. neg-sub0 86.4%

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

      8. distribute-neg-frac2 86.4%

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

      9. distribute-frac-neg2 86.4%

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

      10. associate-+r+ 87.3%

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

      11. +-commutative 87.3%

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

      12. remove-double-neg 87.3%

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

      13. distribute-neg-frac2 87.3%

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

      14. sub0-neg 87.3%

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

      15. associate-+l- 87.3%

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

      16. neg-sub0 87.3%

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

   3. Simplified 87.3%

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

      1. frac-sub 87.9%

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

      2. div-inv 87.9%

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

      3. *-rgt-identity 87.9%

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

      4. fmm-def 87.9%

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

   6. Applied egg-rr 87.9%

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

      1. fmm-undef 87.9%

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

   8. Simplified 87.9%

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

   9. Taylor expanded in x around 0 87.9%

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

       1. +-commutative 87.9%

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

       2. un-div-inv 87.9%

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

       3. frac-add 99.5%

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

       4. metadata-eval 99.5%

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

       5. div-inv 99.5%

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

       6. /-rgt-identity 99.5%

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

   11. Applied egg-rr 99.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(x + -1\right) \cdot \frac{x}{1 + \frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -1\right) \cdot \left(2 + x\right) + x \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \left(x \cdot \left(-1 - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. +-commutative 74.3%

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

   2. associate-+r- 74.3%

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

   3. sub-neg 74.3%

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

   4. remove-double-neg 74.3%

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

   5. neg-sub0 74.3%

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

   6. associate-+l- 74.3%

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

   7. neg-sub0 74.3%

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

   8. distribute-neg-frac2 74.3%

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

   9. distribute-frac-neg2 74.3%

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

   10. associate-+r+ 74.3%

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

   11. +-commutative 74.3%

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

   12. remove-double-neg 74.3%

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

   13. distribute-neg-frac2 74.3%

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

   14. sub0-neg 74.3%

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

   15. associate-+l- 74.3%

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

   16. neg-sub0 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 73.1%

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

   1. associate-*r/ 73.1%

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

   2. neg-mul-1 73.1%

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

   3. distribute-neg-in 73.1%

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

   4. metadata-eval 73.1%

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

   5. distribute-neg-frac 73.1%

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

   6. metadata-eval 73.1%

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

7. Simplified 73.1%

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

   1. frac-2neg 73.1%

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

   2. metadata-eval 73.1%

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

   3. clear-num 73.1%

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

   4. frac-add 73.1%

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

9. Applied egg-rr 73.1%

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

    1. *-rgt-identity 73.1%

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

    2. mul-1-neg 73.1%

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

    3. distribute-neg-out 73.1%

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

    4. +-commutative 73.1%

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

    5. distribute-neg-out 73.1%

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

    6. distribute-neg-in 73.1%

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

    7. metadata-eval 73.1%

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

    8. +-commutative 73.1%

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

    9. unsub-neg 73.1%

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

    10. distribute-neg-frac2 73.1%

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

    11. distribute-neg-in 73.1%

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

    12. metadata-eval 73.1%

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

    13. distribute-neg-frac 73.1%

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

    14. metadata-eval 73.1%

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

    15. distribute-lft-neg-out 73.1%

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

    16. distribute-rgt-neg-in 73.1%

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

    17. distribute-neg-frac2 73.1%

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

11. Simplified 73.1%

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

12. Taylor expanded in x around inf 76.5%

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

Alternative 6: 69.3% 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]

Derivation

1. Initial program 74.3%

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

3. Final simplification 74.3%

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

Alternative 7: 68.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. +-commutative 74.3%

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

   2. associate-+r- 74.3%

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

   3. sub-neg 74.3%

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

   4. remove-double-neg 74.3%

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

   5. neg-sub0 74.3%

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

   6. associate-+l- 74.3%

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

   7. neg-sub0 74.3%

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

   8. distribute-neg-frac2 74.3%

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

   9. distribute-frac-neg2 74.3%

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

   10. associate-+r+ 74.3%

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

   11. +-commutative 74.3%

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

   12. remove-double-neg 74.3%

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

   13. distribute-neg-frac2 74.3%

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

   14. sub0-neg 74.3%

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

   15. associate-+l- 74.3%

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

   16. neg-sub0 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 73.1%

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

   1. associate-*r/ 73.1%

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

   2. neg-mul-1 73.1%

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

   3. distribute-neg-in 73.1%

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

   4. metadata-eval 73.1%

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

   5. distribute-neg-frac 73.1%

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

   6. metadata-eval 73.1%

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

7. Simplified 73.1%

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

Alternative 8: 67.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. +-commutative 74.3%

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

   2. associate-+r- 74.3%

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

   3. sub-neg 74.3%

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

   4. remove-double-neg 74.3%

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

   5. neg-sub0 74.3%

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

   6. associate-+l- 74.3%

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

   7. neg-sub0 74.3%

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

   8. distribute-neg-frac2 74.3%

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

   9. distribute-frac-neg2 74.3%

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

   10. associate-+r+ 74.3%

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

   11. +-commutative 74.3%

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

   12. remove-double-neg 74.3%

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

   13. distribute-neg-frac2 74.3%

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

   14. sub0-neg 74.3%

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

   15. associate-+l- 74.3%

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

   16. neg-sub0 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 72.9%

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

   1. frac-2neg 72.9%

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

   2. metadata-eval 72.9%

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

   3. frac-add 72.9%

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

7. Applied egg-rr 72.9%

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

   1. *-lft-identity 72.9%

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

   2. *-rgt-identity 72.9%

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

   3. +-commutative 72.9%

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

   4. associate-+l+ 72.9%

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

   5. sub-neg 72.9%

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

9. Simplified 72.9%

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

10. Final simplification 72.9%

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

Alternative 9: 67.9% 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 74.3%

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

   1. +-commutative 74.3%

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

   2. associate-+r- 74.3%

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

   3. sub-neg 74.3%

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

   4. remove-double-neg 74.3%

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

   5. neg-sub0 74.3%

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

   6. associate-+l- 74.3%

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

   7. neg-sub0 74.3%

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

   8. distribute-neg-frac2 74.3%

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

   9. distribute-frac-neg2 74.3%

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

   10. associate-+r+ 74.3%

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

   11. +-commutative 74.3%

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

   12. remove-double-neg 74.3%

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

   13. distribute-neg-frac2 74.3%

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

   14. sub0-neg 74.3%

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

   15. associate-+l- 74.3%

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

   16. neg-sub0 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 72.9%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1}{x}} \]
6. Add Preprocessing

Alternative 10: 67.7% accurate, 2.1× speedup

Math

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

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(Float64(-1.0 / x) + Float64(1.0 / x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. +-commutative 74.3%

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

   2. associate-+r- 74.3%

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

   3. sub-neg 74.3%

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

   4. remove-double-neg 74.3%

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

   5. neg-sub0 74.3%

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

   6. associate-+l- 74.3%

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

   7. neg-sub0 74.3%

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

   8. distribute-neg-frac2 74.3%

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

   9. distribute-frac-neg2 74.3%

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

   10. associate-+r+ 74.3%

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

   11. +-commutative 74.3%

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

   12. remove-double-neg 74.3%

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

   13. distribute-neg-frac2 74.3%

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

   14. sub0-neg 74.3%

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

   15. associate-+l- 74.3%

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

   16. neg-sub0 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 72.9%

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

6. Taylor expanded in x around inf 72.6%

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

7. Final simplification 72.6%

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

Alternative 11: 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 74.3%

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

   1. +-commutative 74.3%

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

   2. associate-+r- 74.3%

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

   3. sub-neg 74.3%

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

   4. remove-double-neg 74.3%

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

   5. neg-sub0 74.3%

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

   6. associate-+l- 74.3%

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

   7. neg-sub0 74.3%

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

   8. distribute-neg-frac2 74.3%

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

   9. distribute-frac-neg2 74.3%

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

   10. associate-+r+ 74.3%

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

   11. +-commutative 74.3%

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

   12. remove-double-neg 74.3%

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

   13. distribute-neg-frac2 74.3%

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

   14. sub0-neg 74.3%

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

   15. associate-+l- 74.3%

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

   16. neg-sub0 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 72.9%

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

6. Taylor expanded in x around 0 5.2%

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

Alternative 12: 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 74.3%

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

   1. +-commutative 74.3%

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

   2. associate-+r- 74.3%

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

   3. sub-neg 74.3%

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

   4. remove-double-neg 74.3%

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

   5. neg-sub0 74.3%

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

   6. associate-+l- 74.3%

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

   7. neg-sub0 74.3%

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

   8. distribute-neg-frac2 74.3%

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

   9. distribute-frac-neg2 74.3%

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

   10. associate-+r+ 74.3%

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

   11. +-commutative 74.3%

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

   12. remove-double-neg 74.3%

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

   13. distribute-neg-frac2 74.3%

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

   14. sub0-neg 74.3%

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

   15. associate-+l- 74.3%

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

   16. neg-sub0 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around 0 5.2%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
6. Add Preprocessing

Developer Target 1: 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 2024180 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

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