3frac (problem 3.3.3)

Percentage Accurate: 69.6% → 99.8%

Time: 10.2s

Alternatives: 8

Speedup: 1.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.6% 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, 1.4× speedup

Math

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

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(Float64(Float64(-2.0 / x) / 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[(N[(N[(-2.0 / x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.2%

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

   3. sub-neg 71.2%

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

   4. remove-double-neg 71.2%

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

   5. neg-sub0 71.2%

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

   6. associate-+l- 71.2%

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

   7. neg-sub0 71.2%

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

   8. distribute-neg-frac2 71.2%

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

   9. distribute-frac-neg2 71.2%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

   1. frac-sub 19.2%

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

   2. frac-add 21.1%

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

   3. *-un-lft-identity 21.1%

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

   4. fma-define 20.1%

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

   5. *-rgt-identity 20.1%

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

   6. fma-neg 20.1%

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

6. Applied egg-rr 20.1%

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

   1. fma-undefine 21.1%

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

   2. +-commutative 21.1%

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

   3. fma-define 20.2%

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

   4. fma-neg 20.2%

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

8. Simplified 20.2%

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

9. Taylor expanded in x around 0 99.4%

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

    1. div-inv 99.4%

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

11. Applied egg-rr 99.4%

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

    1. associate-*r/ 99.4%

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

    2. metadata-eval 99.4%

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

    3. associate-*r* 99.4%

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

    4. *-commutative 99.4%

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

    5. metadata-eval 99.4%

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

    6. sub-neg 99.4%

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

    7. associate-/r* 99.8%

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

    8. sub-neg 99.8%

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

    9. metadata-eval 99.8%

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

13. Simplified 99.8%

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

    1. *-commutative 99.8%

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

    2. associate-/r* 99.8%

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

    3. add-sqr-sqrt 49.5%

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

    4. sqrt-unprod 82.0%

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

    5. sqr-neg 82.0%

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

    6. sqrt-unprod 32.4%

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

    7. add-sqr-sqrt 67.9%

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

    8. div-inv 67.9%

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

    9. add-sqr-sqrt 32.4%

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

    10. sqrt-unprod 81.9%

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

    11. sqr-neg 81.9%

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

    12. sqrt-unprod 49.4%

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

    13. add-sqr-sqrt 99.7%

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

15. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.8%

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

    2. associate-*r/ 99.8%

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

    3. metadata-eval 99.8%

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

17. Simplified 99.8%

    \[\leadsto \frac{\color{blue}{\frac{\frac{-2}{x}}{x + -1}}}{-1 - x} \]
18. Add Preprocessing

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

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(Float64(-2.0 / Float64(x * 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[(N[(-2.0 / N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.2%

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

   3. sub-neg 71.2%

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

   4. remove-double-neg 71.2%

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

   5. neg-sub0 71.2%

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

   6. associate-+l- 71.2%

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

   7. neg-sub0 71.2%

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

   8. distribute-neg-frac2 71.2%

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

   9. distribute-frac-neg2 71.2%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

   1. frac-sub 19.2%

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

   2. frac-add 21.1%

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

   3. *-un-lft-identity 21.1%

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

   4. fma-define 20.1%

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

   5. *-rgt-identity 20.1%

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

   6. fma-neg 20.1%

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

6. Applied egg-rr 20.1%

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

   1. fma-undefine 21.1%

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

   2. +-commutative 21.1%

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

   3. fma-define 20.2%

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

   4. fma-neg 20.2%

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

8. Simplified 20.2%

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

9. Taylor expanded in x around 0 99.4%

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

    1. div-inv 99.4%

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

11. Applied egg-rr 99.4%

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

    1. associate-*r/ 99.4%

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

    2. metadata-eval 99.4%

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

    3. associate-*r* 99.4%

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

    4. *-commutative 99.4%

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

    5. metadata-eval 99.4%

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

    6. sub-neg 99.4%

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

    7. associate-/r* 99.8%

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

    8. sub-neg 99.8%

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

    9. metadata-eval 99.8%

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

13. Simplified 99.8%

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

Alternative 3: 99.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.2%

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

   3. sub-neg 71.2%

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

   4. remove-double-neg 71.2%

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

   5. neg-sub0 71.2%

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

   6. associate-+l- 71.2%

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

   7. neg-sub0 71.2%

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

   8. distribute-neg-frac2 71.2%

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

   9. distribute-frac-neg2 71.2%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

   1. frac-sub 19.2%

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

   2. frac-add 21.1%

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

   3. *-un-lft-identity 21.1%

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

   4. fma-define 20.1%

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

   5. *-rgt-identity 20.1%

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

   6. fma-neg 20.1%

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

6. Applied egg-rr 20.1%

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

   1. fma-undefine 21.1%

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

   2. +-commutative 21.1%

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

   3. fma-define 20.2%

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

   4. fma-neg 20.2%

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

8. Simplified 20.2%

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

9. Taylor expanded in x around 0 99.4%

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

Alternative 4: 97.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.2%

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

   3. sub-neg 71.2%

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

   4. remove-double-neg 71.2%

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

   5. neg-sub0 71.2%

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

   6. associate-+l- 71.2%

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

   7. neg-sub0 71.2%

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

   8. distribute-neg-frac2 71.2%

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

   9. distribute-frac-neg2 71.2%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

   1. frac-sub 19.2%

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

   2. frac-add 21.1%

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

   3. *-un-lft-identity 21.1%

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

   4. fma-define 20.1%

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

   5. *-rgt-identity 20.1%

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

   6. fma-neg 20.1%

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

6. Applied egg-rr 20.1%

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

   1. fma-undefine 21.1%

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

   2. +-commutative 21.1%

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

   3. fma-define 20.2%

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

   4. fma-neg 20.2%

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

8. Simplified 20.2%

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

9. Taylor expanded in x around 0 99.4%

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

    1. div-inv 99.4%

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

11. Applied egg-rr 99.4%

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

    1. associate-*r/ 99.4%

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

    2. metadata-eval 99.4%

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

    3. associate-*r* 99.4%

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

    4. *-commutative 99.4%

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

    5. metadata-eval 99.4%

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

    6. sub-neg 99.4%

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

    7. associate-/r* 99.8%

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

    8. sub-neg 99.8%

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

    9. metadata-eval 99.8%

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

13. Simplified 99.8%

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

14. Taylor expanded in x around inf 96.9%

    \[\leadsto \frac{\frac{-2}{x \cdot \color{blue}{x}}}{-1 - x} \]
15. Add Preprocessing

Alternative 5: 97.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.2%

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

   3. sub-neg 71.2%

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

   4. remove-double-neg 71.2%

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

   5. neg-sub0 71.2%

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

   6. associate-+l- 71.2%

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

   7. neg-sub0 71.2%

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

   8. distribute-neg-frac2 71.2%

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

   9. distribute-frac-neg2 71.2%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

   1. frac-sub 19.2%

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

   2. frac-add 21.1%

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

   3. *-un-lft-identity 21.1%

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

   4. fma-define 20.1%

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

   5. *-rgt-identity 20.1%

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

   6. fma-neg 20.1%

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

6. Applied egg-rr 20.1%

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

   1. fma-undefine 21.1%

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

   2. +-commutative 21.1%

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

   3. fma-define 20.2%

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

   4. fma-neg 20.2%

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

8. Simplified 20.2%

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

9. Taylor expanded in x around 0 99.4%

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

10. Taylor expanded in x around inf 96.5%

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

Alternative 6: 68.3% 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 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.2%

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

   3. sub-neg 71.2%

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

   4. remove-double-neg 71.2%

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

   5. neg-sub0 71.2%

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

   6. associate-+l- 71.2%

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

   7. neg-sub0 71.2%

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

   8. distribute-neg-frac2 71.2%

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

   9. distribute-frac-neg2 71.2%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in x around inf 68.6%

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

Alternative 7: 53.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.2%

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

   3. sub-neg 71.2%

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

   4. remove-double-neg 71.2%

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

   5. neg-sub0 71.2%

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

   6. associate-+l- 71.2%

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

   7. neg-sub0 71.2%

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

   8. distribute-neg-frac2 71.2%

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

   9. distribute-frac-neg2 71.2%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

   1. frac-sub 19.2%

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

   2. frac-add 21.1%

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

   3. *-un-lft-identity 21.1%

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

   4. fma-define 20.1%

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

   5. *-rgt-identity 20.1%

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

   6. fma-neg 20.1%

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

6. Applied egg-rr 20.1%

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

   1. fma-undefine 21.1%

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

   2. +-commutative 21.1%

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

   3. fma-define 20.2%

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

   4. fma-neg 20.2%

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

8. Simplified 20.2%

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

9. Taylor expanded in x around 0 99.4%

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

10. Taylor expanded in x around 0 53.8%

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

    1. neg-mul-1 53.8%

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

12. Simplified 53.8%

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

13. Taylor expanded in x around inf 53.8%

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

Alternative 8: 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 71.3%

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

   1. +-commutative 71.3%

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

   2. associate-+r- 71.2%

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

   3. sub-neg 71.2%

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

   4. remove-double-neg 71.2%

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

   5. neg-sub0 71.2%

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

   6. associate-+l- 71.2%

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

   7. neg-sub0 71.2%

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

   8. distribute-neg-frac2 71.2%

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

   9. distribute-frac-neg2 71.2%

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

   10. associate-+r+ 71.3%

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

   11. +-commutative 71.3%

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

   12. remove-double-neg 71.3%

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

   13. distribute-neg-frac2 71.3%

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

   14. sub0-neg 71.3%

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

   15. associate-+l- 71.3%

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

   16. neg-sub0 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in x around 0 5.1%

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

Developer Target 1: 99.2% 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 2024118 
(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))))