3frac (problem 3.3.3)

Percentage Accurate: 68.8% → 99.6%

Time: 12.1s

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: 68.8% 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: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left({x}^{-5} + \left({x}^{-3} + {x}^{-7}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 2.0 (+ (pow x -5.0) (+ (pow x -3.0) (pow x -7.0)))))

C

double code(double x) {
	return 2.0 * (pow(x, -5.0) + (pow(x, -3.0) + pow(x, -7.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * ((x ** (-5.0d0)) + ((x ** (-3.0d0)) + (x ** (-7.0d0))))
end function

Java

public static double code(double x) {
	return 2.0 * (Math.pow(x, -5.0) + (Math.pow(x, -3.0) + Math.pow(x, -7.0)));
}

Python

def code(x):
	return 2.0 * (math.pow(x, -5.0) + (math.pow(x, -3.0) + math.pow(x, -7.0)))

Julia

function code(x)
	return Float64(2.0 * Float64((x ^ -5.0) + Float64((x ^ -3.0) + (x ^ -7.0))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * ((x ^ -5.0) + ((x ^ -3.0) + (x ^ -7.0)));
end

Wolfram

code[x_] := N[(2.0 * N[(N[Power[x, -5.0], $MachinePrecision] + N[(N[Power[x, -3.0], $MachinePrecision] + N[Power[x, -7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

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

   1. associate-+l- 66.8%

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

   2. sub-neg 66.8%

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

   3. +-commutative 66.8%

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

   4. neg-sub0 66.8%

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

   5. associate-+l- 66.8%

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

   6. neg-sub0 66.8%

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

   7. distribute-neg-frac 66.8%

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

   8. metadata-eval 66.8%

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

   9. sub-neg 66.8%

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

   10. metadata-eval 66.8%

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

3. Simplified 66.8%

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

5. Taylor expanded in x around inf 98.7%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 98.7%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right) \]

   2. metadata-eval 98.7%

      \[\leadsto \frac{\color{blue}{2}}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right) \]

   3. +-commutative 98.7%

      \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{7}}\right)} \]

   4. associate-*r/ 98.7%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{\frac{2 \cdot 1}{{x}^{3}}} + 2 \cdot \frac{1}{{x}^{7}}\right) \]

   5. metadata-eval 98.7%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{\color{blue}{2}}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{7}}\right) \]

   6. associate-*r/ 98.7%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \color{blue}{\frac{2 \cdot 1}{{x}^{7}}}\right) \]

   7. metadata-eval 98.7%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \frac{\color{blue}{2}}{{x}^{7}}\right) \]

7. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 98.7%

      \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right)\right)} \]

   2. expm1-udef 66.3%

      \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right)} - 1\right)} \]

   3. div-inv 66.3%

      \[\leadsto \frac{2}{{x}^{5}} + \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}} + \frac{2}{{x}^{7}}\right)} - 1\right) \]

   4. div-inv 66.3%

      \[\leadsto \frac{2}{{x}^{5}} + \left(e^{\mathsf{log1p}\left(2 \cdot \frac{1}{{x}^{3}} + \color{blue}{2 \cdot \frac{1}{{x}^{7}}}\right)} - 1\right) \]

   5. distribute-lft-out 66.3%

      \[\leadsto \frac{2}{{x}^{5}} + \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(\frac{1}{{x}^{3}} + \frac{1}{{x}^{7}}\right)}\right)} - 1\right) \]

   6. pow-flip 66.3%

      \[\leadsto \frac{2}{{x}^{5}} + \left(e^{\mathsf{log1p}\left(2 \cdot \left(\color{blue}{{x}^{\left(-3\right)}} + \frac{1}{{x}^{7}}\right)\right)} - 1\right) \]

   7. metadata-eval 66.3%

      \[\leadsto \frac{2}{{x}^{5}} + \left(e^{\mathsf{log1p}\left(2 \cdot \left({x}^{\color{blue}{-3}} + \frac{1}{{x}^{7}}\right)\right)} - 1\right) \]

   8. pow-flip 66.3%

      \[\leadsto \frac{2}{{x}^{5}} + \left(e^{\mathsf{log1p}\left(2 \cdot \left({x}^{-3} + \color{blue}{{x}^{\left(-7\right)}}\right)\right)} - 1\right) \]

   9. metadata-eval 66.3%

      \[\leadsto \frac{2}{{x}^{5}} + \left(e^{\mathsf{log1p}\left(2 \cdot \left({x}^{-3} + {x}^{\color{blue}{-7}}\right)\right)} - 1\right) \]

9. Applied egg-rr 66.3%

   \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \left({x}^{-3} + {x}^{-7}\right)\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 99.7%

       \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left({x}^{-3} + {x}^{-7}\right)\right)\right)} \]

    2. expm1-log1p 99.7%

       \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{2 \cdot \left({x}^{-3} + {x}^{-7}\right)} \]

11. Simplified 99.7%

    \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{2 \cdot \left({x}^{-3} + {x}^{-7}\right)} \]
12. Step-by-step derivation

    1. div-inv 99.7%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}}} + 2 \cdot \left({x}^{-3} + {x}^{-7}\right) \]

    2. fma-def 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{5}}, 2 \cdot \left({x}^{-3} + {x}^{-7}\right)\right)} \]

    3. pow-flip 99.7%

       \[\leadsto \mathsf{fma}\left(2, \color{blue}{{x}^{\left(-5\right)}}, 2 \cdot \left({x}^{-3} + {x}^{-7}\right)\right) \]

    4. metadata-eval 99.7%

       \[\leadsto \mathsf{fma}\left(2, {x}^{\color{blue}{-5}}, 2 \cdot \left({x}^{-3} + {x}^{-7}\right)\right) \]

13. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(2, {x}^{-5}, 2 \cdot \left({x}^{-3} + {x}^{-7}\right)\right)} \]
14. Step-by-step derivation

    1. fma-udef 99.7%

       \[\leadsto \color{blue}{2 \cdot {x}^{-5} + 2 \cdot \left({x}^{-3} + {x}^{-7}\right)} \]

    2. distribute-lft-out 99.7%

       \[\leadsto \color{blue}{2 \cdot \left({x}^{-5} + \left({x}^{-3} + {x}^{-7}\right)\right)} \]

15. Simplified 99.7%

    \[\leadsto \color{blue}{2 \cdot \left({x}^{-5} + \left({x}^{-3} + {x}^{-7}\right)\right)} \]

16. Final simplification 99.7%

    \[\leadsto 2 \cdot \left({x}^{-5} + \left({x}^{-3} + {x}^{-7}\right)\right) \]
17. Add Preprocessing

Alternative 2: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left({x}^{-5} + {x}^{-3}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (+ (pow x -5.0) (pow x -3.0))))

C

double code(double x) {
	return 2.0 * (pow(x, -5.0) + pow(x, -3.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * ((x ** (-5.0d0)) + (x ** (-3.0d0)))
end function

Java

public static double code(double x) {
	return 2.0 * (Math.pow(x, -5.0) + Math.pow(x, -3.0));
}

Python

def code(x):
	return 2.0 * (math.pow(x, -5.0) + math.pow(x, -3.0))

Julia

function code(x)
	return Float64(2.0 * Float64((x ^ -5.0) + (x ^ -3.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * ((x ^ -5.0) + (x ^ -3.0));
end

Wolfram

code[x_] := N[(2.0 * N[(N[Power[x, -5.0], $MachinePrecision] + N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

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

   1. associate-+l- 66.8%

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

   2. sub-neg 66.8%

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

   3. +-commutative 66.8%

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

   4. neg-sub0 66.8%

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

   5. associate-+l- 66.8%

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

   6. neg-sub0 66.8%

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

   7. distribute-neg-frac 66.8%

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

   8. metadata-eval 66.8%

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

   9. sub-neg 66.8%

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

   10. metadata-eval 66.8%

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

3. Simplified 66.8%

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

5. Taylor expanded in x around inf 98.5%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}} \]
6. Step-by-step derivation

   1. associate-*r/ 98.5%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} + 2 \cdot \frac{1}{{x}^{5}} \]

   2. metadata-eval 98.5%

      \[\leadsto \frac{\color{blue}{2}}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}} \]

   3. associate-*r/ 98.5%

      \[\leadsto \frac{2}{{x}^{3}} + \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} \]

   4. metadata-eval 98.5%

      \[\leadsto \frac{2}{{x}^{3}} + \frac{\color{blue}{2}}{{x}^{5}} \]

7. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)\right)} \]

   2. expm1-udef 65.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)} - 1} \]

   3. div-inv 65.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}} + \frac{2}{{x}^{5}}\right)} - 1 \]

   4. div-inv 65.7%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \frac{1}{{x}^{3}} + \color{blue}{2 \cdot \frac{1}{{x}^{5}}}\right)} - 1 \]

   5. distribute-lft-out 65.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(\frac{1}{{x}^{3}} + \frac{1}{{x}^{5}}\right)}\right)} - 1 \]

   6. pow-flip 65.7%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(\color{blue}{{x}^{\left(-3\right)}} + \frac{1}{{x}^{5}}\right)\right)} - 1 \]

   7. metadata-eval 65.7%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left({x}^{\color{blue}{-3}} + \frac{1}{{x}^{5}}\right)\right)} - 1 \]

   8. pow-flip 65.7%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left({x}^{-3} + \color{blue}{{x}^{\left(-5\right)}}\right)\right)} - 1 \]

   9. metadata-eval 65.7%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left({x}^{-3} + {x}^{\color{blue}{-5}}\right)\right)} - 1 \]

9. Applied egg-rr 65.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left({x}^{-3} + {x}^{-5}\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 99.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left({x}^{-3} + {x}^{-5}\right)\right)\right)} \]

    2. expm1-log1p 99.5%

       \[\leadsto \color{blue}{2 \cdot \left({x}^{-3} + {x}^{-5}\right)} \]

11. Simplified 99.5%

    \[\leadsto \color{blue}{2 \cdot \left({x}^{-3} + {x}^{-5}\right)} \]

12. Final simplification 99.5%

    \[\leadsto 2 \cdot \left({x}^{-5} + {x}^{-3}\right) \]
13. Add Preprocessing

Alternative 3: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

   1. associate-+l- 66.8%

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

   2. sub-neg 66.8%

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

   3. +-commutative 66.8%

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

   4. neg-sub0 66.8%

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

   5. associate-+l- 66.8%

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

   6. neg-sub0 66.8%

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

   7. distribute-neg-frac 66.8%

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

   8. metadata-eval 66.8%

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

   9. sub-neg 66.8%

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

   10. metadata-eval 66.8%

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

3. Simplified 66.8%

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

   1. clear-num 66.8%

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

   2. frac-2neg 66.8%

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

   3. metadata-eval 66.8%

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

   4. frac-add 19.0%

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

   5. *-un-lft-identity 19.0%

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

   6. +-commutative 19.0%

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

   7. distribute-neg-in 19.0%

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

   8. metadata-eval 19.0%

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

   9. sub-neg 19.0%

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

   10. div-inv 19.0%

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

   11. metadata-eval 19.0%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot \color{blue}{-0.5}\right) \cdot -1}{\frac{x}{-2} \cdot \left(-\left(x + -1\right)\right)} \]

   12. div-inv 19.0%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-\left(x + -1\right)\right)} \]

   13. metadata-eval 19.0%

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

   14. +-commutative 19.0%

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

   15. distribute-neg-in 19.0%

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

   16. metadata-eval 19.0%

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

   17. sub-neg 19.0%

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

6. Applied egg-rr 19.0%

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

   1. associate-*l* 19.0%

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

   2. metadata-eval 19.0%

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

   3. *-commutative 19.0%

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

8. Simplified 19.0%

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

10. Applied egg-rr 17.1%

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

    1. sub-neg 17.1%

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

    2. associate-/l* 66.8%

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

    3. div-sub 66.8%

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

    4. sub-neg 66.8%

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

    5. *-commutative 66.8%

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

    6. associate-/r* 66.8%

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

    7. metadata-eval 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{\color{blue}{2}}{x} + \left(-\frac{x \cdot 0.5}{x \cdot 0.5}\right)} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    8. *-inverses 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + \left(-\color{blue}{1}\right)} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    9. metadata-eval 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + \color{blue}{-1}} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    10. +-commutative 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(-\color{blue}{\left(1 + x\right)}\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    11. distribute-neg-in 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \color{blue}{\left(\left(-1\right) + \left(-x\right)\right)}}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    12. metadata-eval 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(\color{blue}{-1} + \left(-x\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    13. unsub-neg 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \color{blue}{\left(-1 - x\right)}}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    14. *-commutative 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(-1 - x\right)}{\color{blue}{\frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5} \cdot \left(x + 1\right)}} \]

12. Simplified 66.8%

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

13. Taylor expanded in x around inf 66.1%

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

    1. +-commutative 66.1%

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

    2. unpow2 66.1%

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

    3. distribute-rgt-out 66.1%

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

15. Simplified 66.1%

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

16. Taylor expanded in x around inf 98.4%

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

    1. associate-*r/ 98.4%

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

    2. metadata-eval 98.4%

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

    3. associate-*r/ 98.4%

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

    4. metadata-eval 98.4%

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

18. Simplified 98.4%

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

19. Final simplification 98.4%

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

Alternative 4: 98.2% accurate, 0.1× speedup

Math

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

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 66.8%

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

   1. associate-+l- 66.8%

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

   2. sub-neg 66.8%

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

   3. +-commutative 66.8%

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

   4. neg-sub0 66.8%

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

   5. associate-+l- 66.8%

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

   6. neg-sub0 66.8%

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

   7. distribute-neg-frac 66.8%

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

   8. metadata-eval 66.8%

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

   9. sub-neg 66.8%

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

   10. metadata-eval 66.8%

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

3. Simplified 66.8%

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

5. Taylor expanded in x around inf 97.7%

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

6. Final simplification 97.7%

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

Alternative 5: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{2}{x}}{\frac{\left(x + 1\right) \cdot \left(1 - x\right)}{\frac{2}{x} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right) + \left(x + 1\right) \cdot \left(-1 + x \cdot 0.5\right)}{\left(1 - x\right) \cdot \left(\left(x + 1\right) \cdot \left(x \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))) 5e-29)
   (/ (/ 2.0 x) (/ (* (+ x 1.0) (- 1.0 x)) (+ (/ 2.0 x) -1.0)))
   (/
    (+ (* (- 1.0 x) (* x 0.5)) (* (+ x 1.0) (+ -1.0 (* x 0.5))))
    (* (- 1.0 x) (* (+ x 1.0) (* x 0.5))))))

C

double code(double x) {
	double tmp;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 5e-29) {
		tmp = (2.0 / x) / (((x + 1.0) * (1.0 - x)) / ((2.0 / x) + -1.0));
	} else {
		tmp = (((1.0 - x) * (x * 0.5)) + ((x + 1.0) * (-1.0 + (x * 0.5)))) / ((1.0 - x) * ((x + 1.0) * (x * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))) <= 5d-29) then
        tmp = (2.0d0 / x) / (((x + 1.0d0) * (1.0d0 - x)) / ((2.0d0 / x) + (-1.0d0)))
    else
        tmp = (((1.0d0 - x) * (x * 0.5d0)) + ((x + 1.0d0) * ((-1.0d0) + (x * 0.5d0)))) / ((1.0d0 - x) * ((x + 1.0d0) * (x * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 5e-29) {
		tmp = (2.0 / x) / (((x + 1.0) * (1.0 - x)) / ((2.0 / x) + -1.0));
	} else {
		tmp = (((1.0 - x) * (x * 0.5)) + ((x + 1.0) * (-1.0 + (x * 0.5)))) / ((1.0 - x) * ((x + 1.0) * (x * 0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 5e-29:
		tmp = (2.0 / x) / (((x + 1.0) * (1.0 - x)) / ((2.0 / x) + -1.0))
	else:
		tmp = (((1.0 - x) * (x * 0.5)) + ((x + 1.0) * (-1.0 + (x * 0.5)))) / ((1.0 - x) * ((x + 1.0) * (x * 0.5)))
	return tmp

Julia

function code(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))) <= 5e-29)
		tmp = Float64(Float64(2.0 / x) / Float64(Float64(Float64(x + 1.0) * Float64(1.0 - x)) / Float64(Float64(2.0 / x) + -1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - x) * Float64(x * 0.5)) + Float64(Float64(x + 1.0) * Float64(-1.0 + Float64(x * 0.5)))) / Float64(Float64(1.0 - x) * Float64(Float64(x + 1.0) * Float64(x * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x + -1.0))) <= 5e-29)
		tmp = (2.0 / x) / (((x + 1.0) * (1.0 - x)) / ((2.0 / x) + -1.0));
	else
		tmp = (((1.0 - x) * (x * 0.5)) + ((x + 1.0) * (-1.0 + (x * 0.5)))) / ((1.0 - x) * ((x + 1.0) * (x * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := 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], 5e-29], N[(N[(2.0 / x), $MachinePrecision] / N[(N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 - x), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - x), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 4.99999999999999986e-29

   1. Initial program 67.3%

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

      1. associate-+l- 67.3%

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

      2. sub-neg 67.3%

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

      3. +-commutative 67.3%

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

      4. neg-sub0 67.3%

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

      5. associate-+l- 67.3%

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

      6. neg-sub0 67.3%

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

      7. distribute-neg-frac 67.3%

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

      8. metadata-eval 67.3%

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

      9. sub-neg 67.3%

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

      10. metadata-eval 67.3%

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

   3. Simplified 67.3%

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

      1. clear-num 67.3%

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

      2. frac-2neg 67.3%

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

      3. metadata-eval 67.3%

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

      4. frac-add 18.2%

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

      5. *-un-lft-identity 18.2%

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

      6. +-commutative 18.2%

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

      7. distribute-neg-in 18.2%

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

      8. metadata-eval 18.2%

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

      9. sub-neg 18.2%

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

      10. div-inv 18.2%

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

      11. metadata-eval 18.2%

          \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot \color{blue}{-0.5}\right) \cdot -1}{\frac{x}{-2} \cdot \left(-\left(x + -1\right)\right)} \]

      12. div-inv 18.2%

          \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-\left(x + -1\right)\right)} \]

      13. metadata-eval 18.2%

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

      14. +-commutative 18.2%

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

      15. distribute-neg-in 18.2%

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

      16. metadata-eval 18.2%

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

      17. sub-neg 18.2%

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

   6. Applied egg-rr 18.2%

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

      1. associate-*l* 18.2%

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

      2. metadata-eval 18.2%

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

      3. *-commutative 18.2%

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

   8. Simplified 18.2%

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

   10. Applied egg-rr 16.3%

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

       1. sub-neg 16.3%

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

       2. associate-/l* 67.3%

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

       3. div-sub 67.3%

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

       4. sub-neg 67.3%

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

       5. *-commutative 67.3%

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

       6. associate-/r* 67.3%

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

       7. metadata-eval 67.3%

          \[\leadsto \frac{\frac{1 - x}{\frac{\color{blue}{2}}{x} + \left(-\frac{x \cdot 0.5}{x \cdot 0.5}\right)} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

       8. *-inverses 67.3%

          \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + \left(-\color{blue}{1}\right)} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

       9. metadata-eval 67.3%

          \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + \color{blue}{-1}} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

       10. +-commutative 67.3%

           \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(-\color{blue}{\left(1 + x\right)}\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

       11. distribute-neg-in 67.3%

           \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \color{blue}{\left(\left(-1\right) + \left(-x\right)\right)}}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

       12. metadata-eval 67.3%

           \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(\color{blue}{-1} + \left(-x\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

       13. unsub-neg 67.3%

           \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \color{blue}{\left(-1 - x\right)}}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

       14. *-commutative 67.3%

           \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(-1 - x\right)}{\color{blue}{\frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5} \cdot \left(x + 1\right)}} \]

   12. Simplified 67.3%

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

   13. Taylor expanded in x around inf 98.4%

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

   if 4.99999999999999986e-29 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 46.3%

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

      1. associate-+l- 46.3%

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

      2. sub-neg 46.3%

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

      3. +-commutative 46.3%

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

      4. neg-sub0 46.3%

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

      5. associate-+l- 46.3%

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

      6. neg-sub0 46.3%

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

      7. distribute-neg-frac 46.3%

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

      8. metadata-eval 46.3%

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

      9. sub-neg 46.3%

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

      10. metadata-eval 46.3%

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

   3. Simplified 46.3%

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

      1. clear-num 46.3%

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

      2. frac-2neg 46.3%

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

      3. metadata-eval 46.3%

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

      4. frac-add 47.9%

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

      5. *-un-lft-identity 47.9%

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

      6. +-commutative 47.9%

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

      7. distribute-neg-in 47.9%

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

      8. metadata-eval 47.9%

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

      9. sub-neg 47.9%

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

      10. div-inv 47.9%

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

      11. metadata-eval 47.9%

          \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot \color{blue}{-0.5}\right) \cdot -1}{\frac{x}{-2} \cdot \left(-\left(x + -1\right)\right)} \]

      12. div-inv 47.9%

          \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-\left(x + -1\right)\right)} \]

      13. metadata-eval 47.9%

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

      14. +-commutative 47.9%

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

      15. distribute-neg-in 47.9%

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

      16. metadata-eval 47.9%

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

      17. sub-neg 47.9%

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

   6. Applied egg-rr 47.9%

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

      1. associate-*l* 47.9%

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

      2. metadata-eval 47.9%

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

      3. *-commutative 47.9%

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

   8. Simplified 47.9%

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

   10. Applied egg-rr 85.5%

       \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right) - \left(x + 1\right) \cdot \left(1 - x \cdot 0.5\right)}{\left(\left(x + 1\right) \cdot \left(x \cdot 0.5\right)\right) \cdot \left(1 - x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{2}{x}}{\frac{\left(x + 1\right) \cdot \left(1 - x\right)}{\frac{2}{x} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right) + \left(x + 1\right) \cdot \left(-1 + x \cdot 0.5\right)}{\left(1 - x\right) \cdot \left(\left(x + 1\right) \cdot \left(x \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% 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 66.8%

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

   1. associate-+l- 66.8%

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

   2. sub-neg 66.8%

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

   3. +-commutative 66.8%

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

   4. neg-sub0 66.8%

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

   5. associate-+l- 66.8%

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

   6. neg-sub0 66.8%

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

   7. distribute-neg-frac 66.8%

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

   8. metadata-eval 66.8%

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

   9. sub-neg 66.8%

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

   10. metadata-eval 66.8%

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

3. Simplified 66.8%

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

5. Taylor expanded in x around inf 65.0%

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

6. Final simplification 65.0%

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

Alternative 7: 97.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

   1. associate-+l- 66.8%

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

   2. sub-neg 66.8%

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

   3. +-commutative 66.8%

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

   4. neg-sub0 66.8%

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

   5. associate-+l- 66.8%

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

   6. neg-sub0 66.8%

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

   7. distribute-neg-frac 66.8%

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

   8. metadata-eval 66.8%

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

   9. sub-neg 66.8%

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

   10. metadata-eval 66.8%

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

3. Simplified 66.8%

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

   1. clear-num 66.8%

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

   2. frac-2neg 66.8%

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

   3. metadata-eval 66.8%

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

   4. frac-add 19.0%

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

   5. *-un-lft-identity 19.0%

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

   6. +-commutative 19.0%

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

   7. distribute-neg-in 19.0%

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

   8. metadata-eval 19.0%

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

   9. sub-neg 19.0%

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

   10. div-inv 19.0%

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

   11. metadata-eval 19.0%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot \color{blue}{-0.5}\right) \cdot -1}{\frac{x}{-2} \cdot \left(-\left(x + -1\right)\right)} \]

   12. div-inv 19.0%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-\left(x + -1\right)\right)} \]

   13. metadata-eval 19.0%

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

   14. +-commutative 19.0%

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

   15. distribute-neg-in 19.0%

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

   16. metadata-eval 19.0%

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

   17. sub-neg 19.0%

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

6. Applied egg-rr 19.0%

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

   1. associate-*l* 19.0%

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

   2. metadata-eval 19.0%

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

   3. *-commutative 19.0%

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

8. Simplified 19.0%

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

10. Applied egg-rr 17.1%

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

    1. sub-neg 17.1%

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

    2. associate-/l* 66.8%

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

    3. div-sub 66.8%

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

    4. sub-neg 66.8%

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

    5. *-commutative 66.8%

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

    6. associate-/r* 66.8%

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

    7. metadata-eval 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{\color{blue}{2}}{x} + \left(-\frac{x \cdot 0.5}{x \cdot 0.5}\right)} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    8. *-inverses 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + \left(-\color{blue}{1}\right)} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    9. metadata-eval 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + \color{blue}{-1}} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    10. +-commutative 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(-\color{blue}{\left(1 + x\right)}\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    11. distribute-neg-in 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \color{blue}{\left(\left(-1\right) + \left(-x\right)\right)}}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    12. metadata-eval 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(\color{blue}{-1} + \left(-x\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    13. unsub-neg 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \color{blue}{\left(-1 - x\right)}}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    14. *-commutative 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(-1 - x\right)}{\color{blue}{\frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5} \cdot \left(x + 1\right)}} \]

12. Simplified 66.8%

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

13. Taylor expanded in x around inf 66.1%

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

    1. +-commutative 66.1%

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

    2. unpow2 66.1%

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

    3. distribute-rgt-out 66.1%

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

15. Simplified 66.1%

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

16. Taylor expanded in x around inf 96.9%

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

17. Final simplification 96.9%

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

Alternative 8: 52.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

   1. associate-+l- 66.8%

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

   2. sub-neg 66.8%

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

   3. +-commutative 66.8%

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

   4. neg-sub0 66.8%

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

   5. associate-+l- 66.8%

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

   6. neg-sub0 66.8%

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

   7. distribute-neg-frac 66.8%

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

   8. metadata-eval 66.8%

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

   9. sub-neg 66.8%

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

   10. metadata-eval 66.8%

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

3. Simplified 66.8%

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

   1. clear-num 66.8%

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

   2. frac-2neg 66.8%

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

   3. metadata-eval 66.8%

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

   4. frac-add 19.0%

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

   5. *-un-lft-identity 19.0%

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

   6. +-commutative 19.0%

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

   7. distribute-neg-in 19.0%

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

   8. metadata-eval 19.0%

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

   9. sub-neg 19.0%

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

   10. div-inv 19.0%

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

   11. metadata-eval 19.0%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot \color{blue}{-0.5}\right) \cdot -1}{\frac{x}{-2} \cdot \left(-\left(x + -1\right)\right)} \]

   12. div-inv 19.0%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-\left(x + -1\right)\right)} \]

   13. metadata-eval 19.0%

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

   14. +-commutative 19.0%

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

   15. distribute-neg-in 19.0%

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

   16. metadata-eval 19.0%

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

   17. sub-neg 19.0%

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

6. Applied egg-rr 19.0%

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

   1. associate-*l* 19.0%

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

   2. metadata-eval 19.0%

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

   3. *-commutative 19.0%

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

8. Simplified 19.0%

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

10. Applied egg-rr 17.1%

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

    1. sub-neg 17.1%

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

    2. associate-/l* 66.8%

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

    3. div-sub 66.8%

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

    4. sub-neg 66.8%

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

    5. *-commutative 66.8%

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

    6. associate-/r* 66.8%

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

    7. metadata-eval 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{\color{blue}{2}}{x} + \left(-\frac{x \cdot 0.5}{x \cdot 0.5}\right)} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    8. *-inverses 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + \left(-\color{blue}{1}\right)} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    9. metadata-eval 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + \color{blue}{-1}} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    10. +-commutative 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(-\color{blue}{\left(1 + x\right)}\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    11. distribute-neg-in 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \color{blue}{\left(\left(-1\right) + \left(-x\right)\right)}}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    12. metadata-eval 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(\color{blue}{-1} + \left(-x\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    13. unsub-neg 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \color{blue}{\left(-1 - x\right)}}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    14. *-commutative 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(-1 - x\right)}{\color{blue}{\frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5} \cdot \left(x + 1\right)}} \]

12. Simplified 66.8%

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

13. Taylor expanded in x around inf 66.1%

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

    1. +-commutative 66.1%

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

    2. unpow2 66.1%

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

    3. distribute-rgt-out 66.1%

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

15. Simplified 66.1%

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

16. Taylor expanded in x around 0 51.6%

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

17. Final simplification 51.6%

    \[\leadsto \frac{-1}{x \cdot \left(2 + x\right)} \]
18. 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 66.8%

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

   1. associate-+l- 66.8%

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

   2. sub-neg 66.8%

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

   3. +-commutative 66.8%

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

   4. neg-sub0 66.8%

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

   5. associate-+l- 66.8%

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

   6. neg-sub0 66.8%

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

   7. distribute-neg-frac 66.8%

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

   8. metadata-eval 66.8%

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

   9. sub-neg 66.8%

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

   10. metadata-eval 66.8%

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

3. Simplified 66.8%

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

5. Taylor expanded in x around 0 4.9%

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

6. Final simplification 4.9%

   \[\leadsto \frac{-2}{x} \]
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 66.8%

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

   1. associate-+l- 66.8%

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

   2. sub-neg 66.8%

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

   3. +-commutative 66.8%

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

   4. neg-sub0 66.8%

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

   5. associate-+l- 66.8%

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

   6. neg-sub0 66.8%

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

   7. distribute-neg-frac 66.8%

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

   8. metadata-eval 66.8%

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

   9. sub-neg 66.8%

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

   10. metadata-eval 66.8%

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

3. Simplified 66.8%

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

5. Taylor expanded in x around inf 65.0%

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

6. Taylor expanded in x around 0 4.9%

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

7. Final simplification 4.9%

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

Alternative 11: 5.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -0.5 x))

C

double code(double x) {
	return -0.5 / x;
}

Fortran

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

Java

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

Python

def code(x):
	return -0.5 / x

Julia

function code(x)
	return Float64(-0.5 / x)
end

MATLAB

function tmp = code(x)
	tmp = -0.5 / x;
end

Wolfram

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

Derivation

1. Initial program 66.8%

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

   1. associate-+l- 66.8%

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

   2. sub-neg 66.8%

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

   3. +-commutative 66.8%

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

   4. neg-sub0 66.8%

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

   5. associate-+l- 66.8%

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

   6. neg-sub0 66.8%

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

   7. distribute-neg-frac 66.8%

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

   8. metadata-eval 66.8%

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

   9. sub-neg 66.8%

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

   10. metadata-eval 66.8%

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

3. Simplified 66.8%

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

   1. clear-num 66.8%

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

   2. frac-2neg 66.8%

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

   3. metadata-eval 66.8%

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

   4. frac-add 19.0%

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

   5. *-un-lft-identity 19.0%

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

   6. +-commutative 19.0%

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

   7. distribute-neg-in 19.0%

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

   8. metadata-eval 19.0%

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

   9. sub-neg 19.0%

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

   10. div-inv 19.0%

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

   11. metadata-eval 19.0%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot \color{blue}{-0.5}\right) \cdot -1}{\frac{x}{-2} \cdot \left(-\left(x + -1\right)\right)} \]

   12. div-inv 19.0%

       \[\leadsto \frac{1}{1 + x} + \frac{\left(1 - x\right) + \left(x \cdot -0.5\right) \cdot -1}{\color{blue}{\left(x \cdot \frac{1}{-2}\right)} \cdot \left(-\left(x + -1\right)\right)} \]

   13. metadata-eval 19.0%

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

   14. +-commutative 19.0%

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

   15. distribute-neg-in 19.0%

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

   16. metadata-eval 19.0%

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

   17. sub-neg 19.0%

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

6. Applied egg-rr 19.0%

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

   1. associate-*l* 19.0%

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

   2. metadata-eval 19.0%

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

   3. *-commutative 19.0%

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

8. Simplified 19.0%

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

10. Applied egg-rr 17.1%

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

    1. sub-neg 17.1%

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

    2. associate-/l* 66.8%

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

    3. div-sub 66.8%

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

    4. sub-neg 66.8%

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

    5. *-commutative 66.8%

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

    6. associate-/r* 66.8%

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

    7. metadata-eval 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{\color{blue}{2}}{x} + \left(-\frac{x \cdot 0.5}{x \cdot 0.5}\right)} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    8. *-inverses 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + \left(-\color{blue}{1}\right)} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    9. metadata-eval 66.8%

       \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + \color{blue}{-1}} + \left(-\left(x + 1\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    10. +-commutative 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(-\color{blue}{\left(1 + x\right)}\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    11. distribute-neg-in 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \color{blue}{\left(\left(-1\right) + \left(-x\right)\right)}}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    12. metadata-eval 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(\color{blue}{-1} + \left(-x\right)\right)}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    13. unsub-neg 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \color{blue}{\left(-1 - x\right)}}{\left(x + 1\right) \cdot \frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5}} \]

    14. *-commutative 66.8%

        \[\leadsto \frac{\frac{1 - x}{\frac{2}{x} + -1} + \left(-1 - x\right)}{\color{blue}{\frac{\left(1 - x\right) \cdot \left(x \cdot 0.5\right)}{1 - x \cdot 0.5} \cdot \left(x + 1\right)}} \]

12. Simplified 66.8%

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

13. Taylor expanded in x around inf 66.1%

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

    1. +-commutative 66.1%

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

    2. unpow2 66.1%

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

    3. distribute-rgt-out 66.1%

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

15. Simplified 66.1%

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

16. Taylor expanded in x around 0 5.0%

    \[\leadsto \color{blue}{\frac{-0.5}{x}} \]

17. Final simplification 5.0%

    \[\leadsto \frac{-0.5}{x} \]
18. 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 2024019 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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