3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 99.7%

Time: 11.3s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

7. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. div-inv 99.2%

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

   3. +-commutative 99.2%

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

   4. div-inv 99.2%

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

   5. fma-define 99.2%

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

   6. pow-flip 99.2%

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

   7. metadata-eval 99.2%

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

   8. pow-flip 99.9%

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

   9. metadata-eval 99.9%

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

9. Applied egg-rr 99.9%

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

    1. *-lft-identity 99.9%

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

    2. *-commutative 99.9%

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

11. Simplified 99.9%

    \[\leadsto \color{blue}{{x}^{-3} \cdot \left(2 + \mathsf{fma}\left(2, {x}^{-4}, \frac{2}{{x}^{2}}\right)\right)} \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

7. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. div-inv 99.2%

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

   3. +-commutative 99.2%

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

   4. div-inv 99.2%

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

   5. fma-define 99.2%

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

   6. pow-flip 99.2%

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

   7. metadata-eval 99.2%

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

   8. pow-flip 99.9%

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

   9. metadata-eval 99.9%

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

9. Applied egg-rr 99.9%

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

    1. *-lft-identity 99.9%

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

    2. *-commutative 99.9%

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

11. Simplified 99.9%

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

12. Taylor expanded in x around inf 99.8%

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

Alternative 3: 99.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return ((2.0 + (2.0 / Math.pow(x, 2.0))) / x) * ((1.0 / x) * (1.0 / x));
}

Python

def code(x):
	return ((2.0 + (2.0 / math.pow(x, 2.0))) / x) * ((1.0 / x) * (1.0 / x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

7. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. unpow3 99.1%

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

   3. unpow2 99.1%

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

   4. times-frac 99.7%

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

   5. pow-flip 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

   8. div-inv 99.7%

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

   9. fma-define 99.7%

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

   10. pow-flip 99.7%

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

   11. metadata-eval 99.7%

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in x around inf 99.6%

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

    1. metadata-eval 99.6%

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

    2. pow-prod-up 99.5%

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

    3. inv-pow 99.5%

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

    4. inv-pow 99.5%

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

12. Applied egg-rr 99.5%

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

13. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot x} \cdot \frac{2 + \frac{2}{{x}^{2}}}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (1.0 / (x * x)) * ((2.0 + (2.0 / Math.pow(x, 2.0))) / x);
}

Python

def code(x):
	return (1.0 / (x * x)) * ((2.0 + (2.0 / math.pow(x, 2.0))) / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

7. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. unpow3 99.1%

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

   3. unpow2 99.1%

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

   4. times-frac 99.7%

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

   5. pow-flip 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

   8. div-inv 99.7%

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

   9. fma-define 99.7%

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

   10. pow-flip 99.7%

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

   11. metadata-eval 99.7%

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in x around inf 99.6%

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

    1. metadata-eval 99.6%

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

    2. pow-prod-up 99.5%

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

    3. inv-pow 99.5%

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

    4. inv-pow 99.5%

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

    5. frac-2neg 99.5%

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

    6. metadata-eval 99.5%

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

    7. pow1 99.5%

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

    8. metadata-eval 99.5%

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

    9. sqrt-pow1 84.3%

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

    10. pow2 84.3%

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

    11. sqr-neg 84.3%

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

    12. sqrt-prod 32.1%

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

    13. add-sqr-sqrt 70.5%

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

    14. clear-num 70.5%

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

    15. frac-times 70.5%

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

    16. metadata-eval 70.5%

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

    17. frac-2neg 70.5%

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

    18. pow1 70.5%

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

    19. metadata-eval 70.5%

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

    20. sqrt-pow1 85.8%

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

    21. pow2 85.8%

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

    22. sqr-neg 85.8%

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

    23. sqrt-prod 47.3%

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

    24. add-sqr-sqrt 99.6%

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

    25. metadata-eval 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

    \[\leadsto \frac{1}{x \cdot x} \cdot \frac{2 + \frac{2}{{x}^{2}}}{x} \]
14. Add Preprocessing

Alternative 5: 99.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around inf 98.6%

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

   1. div-inv 98.6%

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

   2. pow-flip 99.2%

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

   3. metadata-eval 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 6: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

3. Final simplification 72.2%

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

Alternative 7: 68.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around inf 71.2%

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

6. Taylor expanded in x around inf 71.2%

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

Alternative 8: 68.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around inf 71.2%

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

6. Final simplification 71.2%

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

Alternative 9: 67.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around inf 71.2%

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

6. Taylor expanded in x around inf 71.0%

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

Alternative 10: 6.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around inf 71.2%

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

   1. frac-2neg 71.2%

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

   2. metadata-eval 71.2%

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

   3. clear-num 71.2%

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

   4. frac-add 71.2%

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

   5. frac-2neg 71.2%

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

   6. pow1 71.2%

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

   7. metadata-eval 71.2%

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

   8. sqrt-pow1 10.9%

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

   9. pow2 10.9%

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

   10. sqr-neg 10.9%

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

   11. sqrt-prod 22.7%

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

   12. add-sqr-sqrt 51.6%

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

   13. metadata-eval 51.6%

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

   14. frac-2neg 51.6%

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

   15. pow1 51.6%

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

   16. metadata-eval 51.6%

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

   17. sqrt-pow1 52.2%

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

   18. pow2 52.2%

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

   19. sqr-neg 52.2%

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

   20. sqrt-prod 23.3%

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

   21. add-sqr-sqrt 52.7%

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

   22. metadata-eval 52.7%

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

7. Applied egg-rr 52.7%

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

   1. *-rgt-identity 52.7%

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

   2. +-commutative 52.7%

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

   3. /-rgt-identity 52.7%

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

   4. *-commutative 52.7%

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

   5. *-commutative 52.7%

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

   6. mul-1-neg 52.7%

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

   7. unsub-neg 52.7%

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

   8. +-commutative 52.7%

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

   9. distribute-neg-in 52.7%

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

   10. metadata-eval 52.7%

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

   11. unsub-neg 52.7%

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

   12. /-rgt-identity 52.7%

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

   13. *-commutative 52.7%

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

   14. distribute-neg-in 52.7%

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

   15. mul-1-neg 52.7%

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

   16. metadata-eval 52.7%

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

   17. +-commutative 52.7%

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

   18. mul-1-neg 52.7%

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

   19. unsub-neg 52.7%

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

9. Simplified 52.7%

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

10. Taylor expanded in x around 0 6.2%

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

Alternative 11: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around inf 71.2%

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

6. Taylor expanded in x around 0 5.2%

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

Alternative 12: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-+r- 72.1%

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

   3. sub-neg 72.1%

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

   4. remove-double-neg 72.1%

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

   5. neg-sub0 72.1%

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

   6. associate-+l- 72.1%

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

   7. neg-sub0 72.1%

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

   8. distribute-neg-frac2 72.1%

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

   9. distribute-frac-neg2 72.1%

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

   10. associate-+r+ 72.2%

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

   11. +-commutative 72.2%

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

   12. remove-double-neg 72.2%

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

   13. distribute-neg-frac2 72.2%

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

   14. sub0-neg 72.2%

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

   15. associate-+l- 72.2%

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

   16. neg-sub0 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in x around 0 5.1%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
6. 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 2024110 
(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))))