3frac (problem 3.3.3)

Percentage Accurate: 69.9% → 99.8%

Time: 9.5s

Alternatives: 10

Speedup: 2.1×

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: 69.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

   1. clear-num 65.0%

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

   2. frac-sub 18.5%

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

   3. *-un-lft-identity 18.5%

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

   4. div-inv 18.5%

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

   5. metadata-eval 18.5%

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

   6. div-inv 18.5%

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

   7. metadata-eval 18.5%

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

6. Applied egg-rr 18.5%

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

   1. sub-neg 18.5%

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

   2. *-rgt-identity 18.5%

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

   3. sub-neg 18.5%

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

   4. neg-mul-1 18.5%

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

   5. *-commutative 18.5%

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

   6. associate-+l+ 18.5%

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

   7. neg-mul-1 18.5%

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

   8. *-commutative 18.5%

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

   9. associate-*l* 18.5%

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

   10. distribute-lft-out 18.5%

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

   11. metadata-eval 18.5%

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

   12. metadata-eval 18.5%

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

   13. *-commutative 18.5%

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

   14. associate-*r* 18.5%

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

   15. *-commutative 18.5%

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

8. Simplified 18.5%

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

   1. +-commutative 18.5%

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

   2. frac-add 19.3%

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

   3. +-commutative 19.3%

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

   4. fma-define 19.3%

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

   5. associate-*l* 19.3%

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

   6. associate-*l* 19.3%

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

10. Applied egg-rr 19.3%

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

11. Taylor expanded in x around 0 99.1%

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

    1. inv-pow 99.1%

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

    2. unpow-prod-down 99.8%

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

13. Applied egg-rr 99.8%

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

    1. unpow-1 99.8%

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

    2. associate-*r/ 99.8%

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

    3. *-rgt-identity 99.8%

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

    4. *-commutative 99.8%

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

    5. associate-/r* 99.8%

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

    6. metadata-eval 99.8%

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

15. Simplified 99.8%

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

Alternative 2: 99.0% 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 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 98.7%

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

   1. div-inv 98.7%

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

   2. pow-flip 99.5%

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

   3. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

Alternative 3: 99.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

   1. clear-num 65.0%

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

   2. frac-sub 18.5%

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

   3. *-un-lft-identity 18.5%

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

   4. div-inv 18.5%

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

   5. metadata-eval 18.5%

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

   6. div-inv 18.5%

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

   7. metadata-eval 18.5%

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

6. Applied egg-rr 18.5%

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

   1. sub-neg 18.5%

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

   2. *-rgt-identity 18.5%

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

   3. sub-neg 18.5%

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

   4. neg-mul-1 18.5%

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

   5. *-commutative 18.5%

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

   6. associate-+l+ 18.5%

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

   7. neg-mul-1 18.5%

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

   8. *-commutative 18.5%

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

   9. associate-*l* 18.5%

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

   10. distribute-lft-out 18.5%

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

   11. metadata-eval 18.5%

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

   12. metadata-eval 18.5%

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

   13. *-commutative 18.5%

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

   14. associate-*r* 18.5%

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

   15. *-commutative 18.5%

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

8. Simplified 18.5%

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

   1. +-commutative 18.5%

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

   2. frac-add 19.3%

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

   3. +-commutative 19.3%

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

   4. fma-define 19.3%

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

   5. associate-*l* 19.3%

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

   6. associate-*l* 19.3%

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

10. Applied egg-rr 19.3%

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

11. Taylor expanded in x around 0 99.1%

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

    1. *-rgt-identity 99.1%

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

    2. distribute-lft-in 75.7%

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

    3. *-rgt-identity 75.7%

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

    4. *-rgt-identity 75.7%

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

13. Applied egg-rr 75.7%

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

    1. distribute-lft-out 99.1%

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

    2. associate-*r* 99.1%

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

    3. associate-*l* 99.1%

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

15. Simplified 99.1%

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

16. Final simplification 99.1%

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

Alternative 4: 97.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

   1. clear-num 65.0%

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

   2. frac-sub 18.5%

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

   3. *-un-lft-identity 18.5%

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

   4. div-inv 18.5%

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

   5. metadata-eval 18.5%

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

   6. div-inv 18.5%

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

   7. metadata-eval 18.5%

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

6. Applied egg-rr 18.5%

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

   1. sub-neg 18.5%

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

   2. *-rgt-identity 18.5%

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

   3. sub-neg 18.5%

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

   4. neg-mul-1 18.5%

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

   5. *-commutative 18.5%

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

   6. associate-+l+ 18.5%

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

   7. neg-mul-1 18.5%

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

   8. *-commutative 18.5%

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

   9. associate-*l* 18.5%

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

   10. distribute-lft-out 18.5%

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

   11. metadata-eval 18.5%

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

   12. metadata-eval 18.5%

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

   13. *-commutative 18.5%

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

   14. associate-*r* 18.5%

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

   15. *-commutative 18.5%

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

8. Simplified 18.5%

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

   1. +-commutative 18.5%

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

   2. frac-add 19.3%

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

   3. +-commutative 19.3%

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

   4. fma-define 19.3%

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

   5. associate-*l* 19.3%

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

   6. associate-*l* 19.3%

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

10. Applied egg-rr 19.3%

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

11. Taylor expanded in x around 0 99.1%

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

12. Taylor expanded in x around inf 97.0%

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

    1. *-commutative 97.0%

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

14. Simplified 97.0%

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

15. Final simplification 97.0%

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

Alternative 5: 72.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. neg-mul-1 64.4%

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

   3. distribute-neg-in 64.4%

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

   4. metadata-eval 64.4%

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

   5. distribute-neg-frac 64.4%

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

   6. metadata-eval 64.4%

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

7. Simplified 64.4%

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

   1. frac-add 64.4%

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

   2. *-un-lft-identity 64.4%

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

9. Applied egg-rr 64.4%

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

10. Taylor expanded in x around 0 68.9%

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

11. Final simplification 68.9%

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

Alternative 6: 72.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. neg-mul-1 64.4%

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

   3. distribute-neg-in 64.4%

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

   4. metadata-eval 64.4%

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

   5. distribute-neg-frac 64.4%

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

   6. metadata-eval 64.4%

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

7. Simplified 64.4%

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

   1. frac-add 64.4%

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

   2. *-un-lft-identity 64.4%

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

9. Applied egg-rr 64.4%

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

10. Taylor expanded in x around 0 68.9%

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

11. Taylor expanded in x around inf 68.9%

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

Alternative 7: 52.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. neg-mul-1 64.4%

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

   3. distribute-neg-in 64.4%

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

   4. metadata-eval 64.4%

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

   5. distribute-neg-frac 64.4%

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

   6. metadata-eval 64.4%

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

7. Simplified 64.4%

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

8. Taylor expanded in x around 0 48.9%

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

   1. sub-neg 48.9%

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

   2. *-commutative 48.9%

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

   3. metadata-eval 48.9%

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

   4. distribute-lft1-in 48.9%

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

   5. distribute-rgt1-in 48.9%

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

   6. *-commutative 48.9%

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

   7. neg-mul-1 48.9%

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

   8. sub-neg 48.9%

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

   9. unpow2 48.9%

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

   10. associate-/r* 4.9%

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

   11. sub-neg 4.9%

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

   12. neg-mul-1 4.9%

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

   13. *-commutative 4.9%

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

   14. distribute-rgt1-in 4.9%

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

   15. distribute-lft1-in 4.9%

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

   16. *-commutative 4.9%

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

   17. metadata-eval 4.9%

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

   18. sub-neg 4.9%

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

   19. div-sub 4.9%

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

   20. neg-mul-1 4.9%

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

   21. distribute-frac-neg 4.9%

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

   22. *-inverses 4.9%

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

   23. metadata-eval 4.9%

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

10. Simplified 4.9%

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

    1. sub-neg 4.9%

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

    2. add-sqr-sqrt 2.6%

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

    3. sqrt-unprod 4.9%

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

    4. frac-times 4.9%

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

    5. metadata-eval 4.9%

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

    6. metadata-eval 4.9%

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

    7. frac-times 4.9%

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

    8. sqrt-unprod 2.3%

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

    9. add-sqr-sqrt 4.9%

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

12. Applied egg-rr 4.9%

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

    1. distribute-neg-frac 4.9%

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

    2. metadata-eval 4.9%

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

14. Simplified 4.9%

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

15. Taylor expanded in x around 0 49.8%

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

Alternative 8: 6.3% accurate, 3.8× 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 / Float64(-x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. neg-mul-1 64.4%

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

   3. distribute-neg-in 64.4%

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

   4. metadata-eval 64.4%

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

   5. distribute-neg-frac 64.4%

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

   6. metadata-eval 64.4%

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

7. Simplified 64.4%

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

8. Taylor expanded in x around 0 48.9%

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

   1. sub-neg 48.9%

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

   2. *-commutative 48.9%

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

   3. metadata-eval 48.9%

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

   4. distribute-lft1-in 48.9%

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

   5. distribute-rgt1-in 48.9%

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

   6. *-commutative 48.9%

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

   7. neg-mul-1 48.9%

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

   8. sub-neg 48.9%

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

   9. unpow2 48.9%

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

   10. associate-/r* 4.9%

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

   11. sub-neg 4.9%

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

   12. neg-mul-1 4.9%

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

   13. *-commutative 4.9%

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

   14. distribute-rgt1-in 4.9%

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

   15. distribute-lft1-in 4.9%

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

   16. *-commutative 4.9%

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

   17. metadata-eval 4.9%

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

   18. sub-neg 4.9%

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

   19. div-sub 4.9%

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

   20. neg-mul-1 4.9%

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

   21. distribute-frac-neg 4.9%

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

   22. *-inverses 4.9%

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

   23. metadata-eval 4.9%

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

10. Simplified 4.9%

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

11. Taylor expanded in x around inf 4.9%

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

    1. add-sqr-sqrt 2.3%

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

    2. sqrt-unprod 47.5%

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

    3. frac-times 49.7%

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

    4. metadata-eval 49.7%

       \[\leadsto \sqrt{\frac{\color{blue}{1}}{x \cdot x}} \]

    5. metadata-eval 49.7%

       \[\leadsto \sqrt{\frac{\color{blue}{1 \cdot 1}}{x \cdot x}} \]

    6. frac-times 47.5%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{x} \cdot \frac{1}{x}}} \]

    7. sqrt-prod 3.5%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}} \]

    8. add-sqr-sqrt 6.2%

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

    9. frac-2neg 6.2%

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

    10. metadata-eval 6.2%

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

    11. distribute-frac-neg2 6.2%

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

13. Applied egg-rr 6.2%

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

14. Final simplification 6.2%

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

Alternative 9: 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 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. neg-mul-1 64.4%

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

   3. distribute-neg-in 64.4%

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

   4. metadata-eval 64.4%

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

   5. distribute-neg-frac 64.4%

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

   6. metadata-eval 64.4%

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

7. Simplified 64.4%

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

8. Taylor expanded in x around 0 48.9%

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

   1. sub-neg 48.9%

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

   2. *-commutative 48.9%

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

   3. metadata-eval 48.9%

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

   4. distribute-lft1-in 48.9%

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

   5. distribute-rgt1-in 48.9%

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

   6. *-commutative 48.9%

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

   7. neg-mul-1 48.9%

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

   8. sub-neg 48.9%

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

   9. unpow2 48.9%

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

   10. associate-/r* 4.9%

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

   11. sub-neg 4.9%

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

   12. neg-mul-1 4.9%

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

   13. *-commutative 4.9%

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

   14. distribute-rgt1-in 4.9%

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

   15. distribute-lft1-in 4.9%

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

   16. *-commutative 4.9%

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

   17. metadata-eval 4.9%

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

   18. sub-neg 4.9%

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

   19. div-sub 4.9%

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

   20. neg-mul-1 4.9%

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

   21. distribute-frac-neg 4.9%

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

   22. *-inverses 4.9%

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

   23. metadata-eval 4.9%

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

10. Simplified 4.9%

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

11. Taylor expanded in x around inf 4.9%

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

Alternative 10: 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 65.0%

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

   1. +-commutative 65.0%

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

   2. associate-+r- 65.0%

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

   3. sub-neg 65.0%

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

   4. remove-double-neg 65.0%

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

   5. neg-sub0 65.0%

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

   6. associate-+l- 65.0%

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

   7. neg-sub0 65.0%

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

   8. distribute-neg-frac2 65.0%

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

   9. distribute-frac-neg2 65.0%

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

   10. associate-+r+ 65.0%

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

   11. +-commutative 65.0%

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

   12. remove-double-neg 65.0%

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

   13. distribute-neg-frac2 65.0%

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

   14. sub0-neg 65.0%

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

   15. associate-+l- 65.0%

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

   16. neg-sub0 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in x around 0 4.9%

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

Developer Target 1: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024123 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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