3frac (problem 3.3.3)

Percentage Accurate: 70.2% → 99.8%

Time: 12.9s

Alternatives: 7

Speedup: 1.4×

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: 70.2% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. +-commutative 70.1%

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

   2. associate-+r- 70.1%

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

   3. sub-neg 70.1%

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

   4. remove-double-neg 70.1%

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

   5. neg-sub0 70.1%

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

   6. associate-+l- 70.1%

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

   7. neg-sub0 70.1%

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

   8. distribute-neg-frac2 70.1%

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

   9. distribute-frac-neg2 70.1%

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

   10. associate-+r+ 70.1%

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

   11. +-commutative 70.1%

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

   12. remove-double-neg 70.1%

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

   13. distribute-neg-frac2 70.1%

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

   14. sub0-neg 70.1%

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

   15. associate-+l- 70.1%

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

   16. neg-sub0 70.1%

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

3. Simplified 70.1%

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

   1. frac-sub 16.7%

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

   2. frac-add 19.9%

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

   3. *-un-lft-identity 19.9%

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

   4. fma-define 19.0%

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

   5. *-rgt-identity 19.0%

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

   6. fma-neg 19.0%

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

6. Applied egg-rr 19.0%

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

8. Simplified 19.9%

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

   1. *-commutative 19.9%

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

   2. distribute-lft-in 19.9%

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

10. Applied egg-rr 19.9%

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

    1. distribute-lft-out 19.9%

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

    2. *-commutative 19.9%

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

    3. associate-*r* 19.9%

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

    4. *-commutative 19.9%

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

    5. associate-*l* 19.9%

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

12. Simplified 19.9%

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

13. Taylor expanded in x around 0 99.0%

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

    1. *-commutative 99.0%

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

    2. associate-*r* 99.0%

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

    3. *-commutative 99.0%

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

    4. associate-/r* 99.8%

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

    5. div-inv 99.8%

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

15. Applied egg-rr 99.8%

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

16. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. +-commutative 70.1%

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

   2. associate-+r- 70.1%

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

   3. sub-neg 70.1%

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

   4. remove-double-neg 70.1%

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

   5. neg-sub0 70.1%

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

   6. associate-+l- 70.1%

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

   7. neg-sub0 70.1%

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

   8. distribute-neg-frac2 70.1%

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

   9. distribute-frac-neg2 70.1%

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

   10. associate-+r+ 70.1%

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

   11. +-commutative 70.1%

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

   12. remove-double-neg 70.1%

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

   13. distribute-neg-frac2 70.1%

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

   14. sub0-neg 70.1%

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

   15. associate-+l- 70.1%

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

   16. neg-sub0 70.1%

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

3. Simplified 70.1%

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

   1. frac-sub 16.7%

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

   2. frac-add 19.9%

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

   3. *-un-lft-identity 19.9%

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

   4. fma-define 19.0%

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

   5. *-rgt-identity 19.0%

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

   6. fma-neg 19.0%

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

6. Applied egg-rr 19.0%

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

8. Simplified 19.9%

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

   1. *-commutative 19.9%

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

   2. distribute-lft-in 19.9%

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

10. Applied egg-rr 19.9%

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

    1. distribute-lft-out 19.9%

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

    2. *-commutative 19.9%

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

    3. associate-*r* 19.9%

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

    4. *-commutative 19.9%

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

    5. associate-*l* 19.9%

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

12. Simplified 19.9%

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

13. Taylor expanded in x around 0 99.0%

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

14. Final simplification 99.0%

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

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. +-commutative 70.1%

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

   2. associate-+r- 70.1%

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

   3. sub-neg 70.1%

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

   4. remove-double-neg 70.1%

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

   5. neg-sub0 70.1%

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

   6. associate-+l- 70.1%

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

   7. neg-sub0 70.1%

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

   8. distribute-neg-frac2 70.1%

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

   9. distribute-frac-neg2 70.1%

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

   10. associate-+r+ 70.1%

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

   11. +-commutative 70.1%

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

   12. remove-double-neg 70.1%

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

   13. distribute-neg-frac2 70.1%

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

   14. sub0-neg 70.1%

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

   15. associate-+l- 70.1%

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

   16. neg-sub0 70.1%

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

3. Simplified 70.1%

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

   1. frac-sub 16.7%

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

   2. frac-add 19.9%

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

   3. *-un-lft-identity 19.9%

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

   4. fma-define 19.0%

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

   5. *-rgt-identity 19.0%

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

   6. fma-neg 19.0%

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

6. Applied egg-rr 19.0%

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

8. Simplified 19.9%

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

   1. *-commutative 19.9%

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

   2. distribute-lft-in 19.9%

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

10. Applied egg-rr 19.9%

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

    1. distribute-lft-out 19.9%

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

    2. *-commutative 19.9%

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

    3. associate-*r* 19.9%

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

    4. *-commutative 19.9%

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

    5. associate-*l* 19.9%

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

12. Simplified 19.9%

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

13. Taylor expanded in x around 0 99.0%

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

    1. *-commutative 99.0%

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

    2. associate-*r* 99.0%

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

    3. *-commutative 99.0%

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

    4. div-inv 99.0%

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

    5. *-commutative 99.0%

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

    6. *-commutative 99.0%

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

    7. associate-*l* 99.0%

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

15. Applied egg-rr 99.0%

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

    1. associate-*r/ 99.0%

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

    2. metadata-eval 99.0%

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

    3. associate-/r* 99.8%

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

17. Simplified 99.8%

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

18. Final simplification 99.8%

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

Alternative 4: 68.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. +-commutative 70.1%

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

   2. associate-+r- 70.1%

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

   3. sub-neg 70.1%

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

   4. remove-double-neg 70.1%

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

   5. neg-sub0 70.1%

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

   6. associate-+l- 70.1%

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

   7. neg-sub0 70.1%

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

   8. distribute-neg-frac2 70.1%

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

   9. distribute-frac-neg2 70.1%

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

   10. associate-+r+ 70.1%

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

   11. +-commutative 70.1%

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

   12. remove-double-neg 70.1%

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

   13. distribute-neg-frac2 70.1%

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

   14. sub0-neg 70.1%

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

   15. associate-+l- 70.1%

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

   16. neg-sub0 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in x around inf 69.0%

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

6. Final simplification 69.0%

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

Alternative 5: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. +-commutative 70.1%

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

   2. associate-+r- 70.1%

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

   3. sub-neg 70.1%

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

   4. remove-double-neg 70.1%

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

   5. neg-sub0 70.1%

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

   6. associate-+l- 70.1%

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

   7. neg-sub0 70.1%

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

   8. distribute-neg-frac2 70.1%

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

   9. distribute-frac-neg2 70.1%

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

   10. associate-+r+ 70.1%

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

   11. +-commutative 70.1%

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

   12. remove-double-neg 70.1%

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

   13. distribute-neg-frac2 70.1%

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

   14. sub0-neg 70.1%

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

   15. associate-+l- 70.1%

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

   16. neg-sub0 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in x around 0 5.1%

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

6. Final simplification 5.1%

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

Alternative 6: 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 70.1%

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

   1. +-commutative 70.1%

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

   2. associate-+r- 70.1%

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

   3. sub-neg 70.1%

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

   4. remove-double-neg 70.1%

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

   5. neg-sub0 70.1%

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

   6. associate-+l- 70.1%

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

   7. neg-sub0 70.1%

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

   8. distribute-neg-frac2 70.1%

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

   9. distribute-frac-neg2 70.1%

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

   10. associate-+r+ 70.1%

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

   11. +-commutative 70.1%

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

   12. remove-double-neg 70.1%

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

   13. distribute-neg-frac2 70.1%

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

   14. sub0-neg 70.1%

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

   15. associate-+l- 70.1%

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

   16. neg-sub0 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in x around inf 69.0%

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

6. Taylor expanded in x around 0 5.1%

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

7. Final simplification 5.1%

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

Alternative 7: 3.3% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 70.1%

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

   1. +-commutative 70.1%

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

   2. associate-+r- 70.1%

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

   3. sub-neg 70.1%

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

   4. remove-double-neg 70.1%

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

   5. neg-sub0 70.1%

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

   6. associate-+l- 70.1%

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

   7. neg-sub0 70.1%

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

   8. distribute-neg-frac2 70.1%

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

   9. distribute-frac-neg2 70.1%

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

   10. associate-+r+ 70.1%

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

   11. +-commutative 70.1%

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

   12. remove-double-neg 70.1%

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

   13. distribute-neg-frac2 70.1%

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

   14. sub0-neg 70.1%

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

   15. associate-+l- 70.1%

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

   16. neg-sub0 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in x around 0 3.4%

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

   1. associate-*r/ 3.4%

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

   2. metadata-eval 3.4%

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

7. Simplified 3.4%

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

8. Taylor expanded in x around inf 3.4%

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

9. Final simplification 3.4%

   \[\leadsto 1 \]
10. 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 2024052 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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