3frac (problem 3.3.3)

Percentage Accurate: 69.8% → 99.8%

Time: 11.6s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.8% 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: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+r- 70.8%

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

   3. sub-neg 70.8%

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

   4. remove-double-neg 70.8%

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

   5. neg-sub0 70.8%

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

   6. associate-+l- 70.8%

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

   7. neg-sub0 70.8%

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

   8. distribute-neg-frac2 70.8%

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

   9. distribute-frac-neg2 70.8%

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

   10. associate-+r+ 70.7%

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

   11. +-commutative 70.7%

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

   12. remove-double-neg 70.7%

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

   13. distribute-neg-frac2 70.7%

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

   14. sub0-neg 70.7%

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

   15. associate-+l- 70.7%

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

   16. neg-sub0 70.7%

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

3. Simplified 70.7%

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

   1. frac-sub 18.6%

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

   2. frac-add 19.2%

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

   3. *-un-lft-identity 19.2%

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

   4. fma-define 18.5%

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

   5. *-rgt-identity 18.5%

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

   6. fma-neg 18.5%

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

6. Applied egg-rr 18.5%

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

8. Simplified 19.2%

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

9. Taylor expanded in x around 0 99.4%

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

    1. *-un-lft-identity 99.4%

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

    2. associate-/r* 99.8%

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

    3. metadata-eval 99.8%

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

    4. frac-2neg 99.8%

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

11. Applied egg-rr 99.8%

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

    1. *-lft-identity 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

    \[\leadsto \frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, -1\right)} \]
15. Add Preprocessing

Alternative 2: 99.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+r- 70.8%

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

   3. sub-neg 70.8%

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

   4. remove-double-neg 70.8%

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

   5. neg-sub0 70.8%

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

   6. associate-+l- 70.8%

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

   7. neg-sub0 70.8%

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

   8. distribute-neg-frac2 70.8%

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

   9. distribute-frac-neg2 70.8%

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

   10. associate-+r+ 70.7%

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

   11. +-commutative 70.7%

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

   12. remove-double-neg 70.7%

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

   13. distribute-neg-frac2 70.7%

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

   14. sub0-neg 70.7%

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

   15. associate-+l- 70.7%

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

   16. neg-sub0 70.7%

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

3. Simplified 70.7%

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

   1. frac-sub 18.6%

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

   2. frac-add 19.2%

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

   3. *-un-lft-identity 19.2%

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

   4. fma-define 18.5%

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

   5. *-rgt-identity 18.5%

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

   6. fma-neg 18.5%

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

6. Applied egg-rr 18.5%

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

8. Simplified 19.2%

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

9. Taylor expanded in x around 0 99.4%

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

    1. *-un-lft-identity 99.4%

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

    2. associate-/r* 99.8%

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

    3. metadata-eval 99.8%

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

    4. frac-2neg 99.8%

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

11. Applied egg-rr 99.8%

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

    1. *-lft-identity 99.8%

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

    2. associate-/l/ 99.4%

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

    3. *-commutative 99.4%

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

    4. fma-undefine 99.4%

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

    5. unpow2 99.4%

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

    6. distribute-rgt-in 99.4%

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

    7. unpow2 99.4%

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

    8. unpow3 99.5%

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

    9. neg-mul-1 99.5%

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

    10. unsub-neg 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 3: 98.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+r- 70.8%

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

   3. sub-neg 70.8%

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

   4. remove-double-neg 70.8%

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

   5. neg-sub0 70.8%

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

   6. associate-+l- 70.8%

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

   7. neg-sub0 70.8%

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

   8. distribute-neg-frac2 70.8%

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

   9. distribute-frac-neg2 70.8%

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

   10. associate-+r+ 70.7%

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

   11. +-commutative 70.7%

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

   12. remove-double-neg 70.7%

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

   13. distribute-neg-frac2 70.7%

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

   14. sub0-neg 70.7%

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

   15. associate-+l- 70.7%

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

   16. neg-sub0 70.7%

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

3. Simplified 70.7%

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

5. Taylor expanded in x around inf 98.4%

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

   1. clear-num 98.4%

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

   2. associate-/r/ 98.4%

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

   3. pow-flip 98.9%

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

   4. metadata-eval 98.9%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 4: 69.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+r- 70.8%

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

   3. sub-neg 70.8%

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

   4. remove-double-neg 70.8%

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

   5. neg-sub0 70.8%

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

   6. associate-+l- 70.8%

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

   7. neg-sub0 70.8%

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

   8. distribute-neg-frac2 70.8%

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

   9. distribute-frac-neg2 70.8%

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

   10. associate-+r+ 70.7%

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

   11. +-commutative 70.7%

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

   12. remove-double-neg 70.7%

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

   13. distribute-neg-frac2 70.7%

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

   14. sub0-neg 70.7%

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

   15. associate-+l- 70.7%

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

   16. neg-sub0 70.7%

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

3. Simplified 70.7%

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

   1. clear-num 70.7%

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

   2. frac-sub 18.6%

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

   3. *-un-lft-identity 18.6%

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

   4. div-inv 18.6%

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

   5. metadata-eval 18.6%

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

   6. div-inv 18.6%

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

   7. metadata-eval 18.6%

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

6. Applied egg-rr 18.6%

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

   1. associate-/r* 70.8%

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

   2. *-rgt-identity 70.8%

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

   3. associate--r+ 70.8%

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

8. Simplified 70.8%

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

9. Final simplification 70.8%

   \[\leadsto \frac{1}{x + -1} + \frac{\frac{-1 - \left(x + x \cdot -0.5\right)}{x \cdot -0.5}}{-1 - x} \]
10. Add Preprocessing

Alternative 5: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Final simplification 70.7%

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

Alternative 6: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+r- 70.8%

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

   3. sub-neg 70.8%

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

   4. remove-double-neg 70.8%

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

   5. neg-sub0 70.8%

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

   6. associate-+l- 70.8%

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

   7. neg-sub0 70.8%

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

   8. distribute-neg-frac2 70.8%

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

   9. distribute-frac-neg2 70.8%

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

   10. associate-+r+ 70.7%

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

   11. +-commutative 70.7%

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

   12. remove-double-neg 70.7%

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

   13. distribute-neg-frac2 70.7%

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

   14. sub0-neg 70.7%

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

   15. associate-+l- 70.7%

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

   16. neg-sub0 70.7%

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

3. Simplified 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+l- 70.8%

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

6. Applied egg-rr 70.8%

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

7. Final simplification 70.8%

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

Alternative 7: 68.4% 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 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+r- 70.8%

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

   3. sub-neg 70.8%

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

   4. remove-double-neg 70.8%

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

   5. neg-sub0 70.8%

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

   6. associate-+l- 70.8%

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

   7. neg-sub0 70.8%

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

   8. distribute-neg-frac2 70.8%

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

   9. distribute-frac-neg2 70.8%

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

   10. associate-+r+ 70.7%

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

   11. +-commutative 70.7%

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

   12. remove-double-neg 70.7%

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

   13. distribute-neg-frac2 70.7%

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

   14. sub0-neg 70.7%

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

   15. associate-+l- 70.7%

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

   16. neg-sub0 70.7%

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

3. Simplified 70.7%

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

5. Taylor expanded in x around inf 69.6%

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

6. Final simplification 69.6%

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

Alternative 8: 68.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+r- 70.8%

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

   3. sub-neg 70.8%

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

   4. remove-double-neg 70.8%

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

   5. neg-sub0 70.8%

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

   6. associate-+l- 70.8%

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

   7. neg-sub0 70.8%

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

   8. distribute-neg-frac2 70.8%

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

   9. distribute-frac-neg2 70.8%

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

   10. associate-+r+ 70.7%

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

   11. +-commutative 70.7%

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

   12. remove-double-neg 70.7%

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

   13. distribute-neg-frac2 70.7%

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

   14. sub0-neg 70.7%

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

   15. associate-+l- 70.7%

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

   16. neg-sub0 70.7%

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

3. Simplified 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+l- 70.8%

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

6. Applied egg-rr 70.8%

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

8. Applied egg-rr 6.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x + -1} + \left(\frac{2}{x} + \frac{-1}{x + -1}\right)\right)\right)} \]
9. Step-by-step derivation

   1. expm1-define 69.5%

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

   2. sub-neg 69.5%

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

   3. metadata-eval 69.5%

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

   4. +-commutative 69.5%

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

   5. log1p-undefine 69.5%

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

   6. rem-exp-log 69.6%

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

   7. +-commutative 69.6%

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

   8. associate-+l+ 69.6%

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

   9. associate-+r+ 69.6%

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

   10. metadata-eval 69.6%

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

   11. distribute-neg-frac 69.6%

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

   12. sub-neg 69.6%

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

   13. +-inverses 69.6%

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

   14. metadata-eval 69.6%

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

10. Simplified 69.6%

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

11. Final simplification 69.6%

    \[\leadsto -1 + \left(\frac{2}{x} + 1\right) \]
12. 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 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+r- 70.8%

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

   3. sub-neg 70.8%

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

   4. remove-double-neg 70.8%

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

   5. neg-sub0 70.8%

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

   6. associate-+l- 70.8%

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

   7. neg-sub0 70.8%

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

   8. distribute-neg-frac2 70.8%

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

   9. distribute-frac-neg2 70.8%

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

   10. associate-+r+ 70.7%

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

   11. +-commutative 70.7%

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

   12. remove-double-neg 70.7%

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

   13. distribute-neg-frac2 70.7%

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

   14. sub0-neg 70.7%

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

   15. associate-+l- 70.7%

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

   16. neg-sub0 70.7%

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

3. Simplified 70.7%

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

5. Taylor expanded in x around 0 5.2%

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

6. Final simplification 5.2%

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

Alternative 10: 6.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+r- 70.8%

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

   3. sub-neg 70.8%

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

   4. remove-double-neg 70.8%

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

   5. neg-sub0 70.8%

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

   6. associate-+l- 70.8%

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

   7. neg-sub0 70.8%

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

   8. distribute-neg-frac2 70.8%

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

   9. distribute-frac-neg2 70.8%

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

   10. associate-+r+ 70.7%

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

   11. +-commutative 70.7%

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

   12. remove-double-neg 70.7%

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

   13. distribute-neg-frac2 70.7%

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

   14. sub0-neg 70.7%

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

   15. associate-+l- 70.7%

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

   16. neg-sub0 70.7%

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

3. Simplified 70.7%

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

   1. +-commutative 70.7%

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

   2. associate-+l- 70.8%

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

6. Applied egg-rr 70.8%

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

8. Applied egg-rr 6.3%

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

   1. associate-+r+ 6.3%

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

   2. metadata-eval 6.3%

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

   3. distribute-neg-frac 6.3%

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

   4. sub-neg 6.3%

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

   5. associate--l+ 6.3%

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

   6. +-inverses 6.3%

      \[\leadsto \frac{2}{x} + \color{blue}{0} \]

   7. +-rgt-identity 6.3%

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

10. Simplified 6.3%

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

11. Final simplification 6.3%

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

Developer target: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024074 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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