3frac (problem 3.3.3)

Percentage Accurate: 70.0% → 99.8%

Time: 10.1s

Alternatives: 10

Speedup: 1.4×

• could not determine a ground truth

  1. x = 4.1702726105485936e+95

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: 70.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. +-commutative 66.0%

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

   2. associate-+r- 65.9%

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

   3. sub-neg 65.9%

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

   4. remove-double-neg 65.9%

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

   5. neg-sub0 65.9%

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

   6. associate-+l- 65.9%

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

   7. neg-sub0 65.9%

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

   8. distribute-neg-frac2 65.9%

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

   9. distribute-frac-neg2 65.9%

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

   10. associate-+r+ 66.0%

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

   11. +-commutative 66.0%

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

   12. remove-double-neg 66.0%

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

   13. distribute-neg-frac2 66.0%

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

   14. sub0-neg 66.0%

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

   15. associate-+l- 66.0%

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

   16. neg-sub0 66.0%

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

3. Simplified 66.0%

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

   1. +-commutative 66.0%

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

   2. frac-sub 17.2%

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

   3. frac-add 19.5%

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

   4. fma-define 18.7%

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

   5. *-rgt-identity 18.7%

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

   6. fma-neg 18.7%

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

6. Applied egg-rr 18.7%

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

7. Taylor expanded in x around 0 99.4%

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

   1. clear-num 99.4%

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

   2. associate-/r/ 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.9%

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

   5. *-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. +-commutative 66.0%

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

   2. associate-+r- 65.9%

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

   3. sub-neg 65.9%

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

   4. remove-double-neg 65.9%

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

   5. neg-sub0 65.9%

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

   6. associate-+l- 65.9%

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

   7. neg-sub0 65.9%

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

   8. distribute-neg-frac2 65.9%

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

   9. distribute-frac-neg2 65.9%

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

   10. associate-+r+ 66.0%

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

   11. +-commutative 66.0%

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

   12. remove-double-neg 66.0%

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

   13. distribute-neg-frac2 66.0%

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

   14. sub0-neg 66.0%

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

   15. associate-+l- 66.0%

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

   16. neg-sub0 66.0%

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

3. Simplified 66.0%

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

   1. +-commutative 66.0%

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

   2. frac-sub 17.2%

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

   3. frac-add 19.5%

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

   4. fma-define 18.7%

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

   5. *-rgt-identity 18.7%

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

   6. fma-neg 18.7%

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

6. Applied egg-rr 18.7%

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

7. Taylor expanded in x around 0 99.4%

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

   1. associate-/r* 99.8%

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

   2. div-inv 99.8%

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

   3. associate-/r* 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

    \[\leadsto \frac{\frac{-2}{x}}{-1 - x} \cdot \frac{1}{x + -1} \]
11. Add Preprocessing

Alternative 3: 99.3% 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 66.0%

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

   1. +-commutative 66.0%

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

   2. associate-+r- 65.9%

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

   3. sub-neg 65.9%

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

   4. remove-double-neg 65.9%

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

   5. neg-sub0 65.9%

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

   6. associate-+l- 65.9%

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

   7. neg-sub0 65.9%

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

   8. distribute-neg-frac2 65.9%

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

   9. distribute-frac-neg2 65.9%

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

   10. associate-+r+ 66.0%

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

   11. +-commutative 66.0%

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

   12. remove-double-neg 66.0%

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

   13. distribute-neg-frac2 66.0%

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

   14. sub0-neg 66.0%

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

   15. associate-+l- 66.0%

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

   16. neg-sub0 66.0%

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

3. Simplified 66.0%

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

   1. +-commutative 66.0%

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

   2. frac-sub 17.2%

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

   3. frac-add 19.5%

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

   4. fma-define 18.7%

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

   5. *-rgt-identity 18.7%

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

   6. fma-neg 18.7%

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

6. Applied egg-rr 18.7%

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

7. Taylor expanded in x around 0 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. +-commutative 66.0%

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

   2. associate-+r- 65.9%

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

   3. sub-neg 65.9%

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

   4. remove-double-neg 65.9%

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

   5. neg-sub0 65.9%

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

   6. associate-+l- 65.9%

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

   7. neg-sub0 65.9%

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

   8. distribute-neg-frac2 65.9%

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

   9. distribute-frac-neg2 65.9%

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

   10. associate-+r+ 66.0%

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

   11. +-commutative 66.0%

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

   12. remove-double-neg 66.0%

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

   13. distribute-neg-frac2 66.0%

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

   14. sub0-neg 66.0%

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

   15. associate-+l- 66.0%

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

   16. neg-sub0 66.0%

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

3. Simplified 66.0%

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

   1. +-commutative 66.0%

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

   2. frac-sub 17.2%

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

   3. frac-add 19.5%

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

   4. fma-define 18.7%

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

   5. *-rgt-identity 18.7%

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

   6. fma-neg 18.7%

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

6. Applied egg-rr 18.7%

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

7. Taylor expanded in x around 0 99.4%

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

   1. frac-2neg 99.4%

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

   2. metadata-eval 99.4%

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

   3. div-inv 99.4%

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

   4. *-commutative 99.4%

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

   5. distribute-lft-neg-in 99.4%

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

   6. +-commutative 99.4%

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

   7. distribute-neg-in 99.4%

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

   8. metadata-eval 99.4%

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

   9. sub-neg 99.4%

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

9. Applied egg-rr 99.4%

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

    1. associate-*r/ 99.4%

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

    2. metadata-eval 99.4%

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

    3. associate-*r* 99.4%

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

    4. *-commutative 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 5: 68.6% 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 66.0%

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

   1. +-commutative 66.0%

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

   2. associate-+r- 65.9%

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

   3. sub-neg 65.9%

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

   4. remove-double-neg 65.9%

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

   5. neg-sub0 65.9%

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

   6. associate-+l- 65.9%

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

   7. neg-sub0 65.9%

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

   8. distribute-neg-frac2 65.9%

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

   9. distribute-frac-neg2 65.9%

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

   10. associate-+r+ 66.0%

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

   11. +-commutative 66.0%

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

   12. remove-double-neg 66.0%

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

   13. distribute-neg-frac2 66.0%

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

   14. sub0-neg 66.0%

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

   15. associate-+l- 66.0%

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

   16. neg-sub0 66.0%

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

3. Simplified 66.0%

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

5. Taylor expanded in x around inf 64.6%

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

6. Final simplification 64.6%

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

Alternative 6: 54.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. +-commutative 66.0%

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

   2. associate-+r- 65.9%

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

   3. sub-neg 65.9%

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

   4. remove-double-neg 65.9%

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

   5. neg-sub0 65.9%

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

   6. associate-+l- 65.9%

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

   7. neg-sub0 65.9%

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

   8. distribute-neg-frac2 65.9%

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

   9. distribute-frac-neg2 65.9%

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

   10. associate-+r+ 66.0%

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

   11. +-commutative 66.0%

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

   12. remove-double-neg 66.0%

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

   13. distribute-neg-frac2 66.0%

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

   14. sub0-neg 66.0%

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

   15. associate-+l- 66.0%

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

   16. neg-sub0 66.0%

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

3. Simplified 66.0%

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

   1. +-commutative 66.0%

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

   2. frac-sub 17.2%

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

   3. frac-add 19.5%

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

   4. fma-define 18.7%

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

   5. *-rgt-identity 18.7%

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

   6. fma-neg 18.7%

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

6. Applied egg-rr 18.7%

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

7. Taylor expanded in x around 0 99.4%

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

8. Taylor expanded in x around 0 50.1%

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

   1. mul-1-neg 50.1%

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

10. Simplified 50.1%

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

11. Final simplification 50.1%

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

Alternative 7: 54.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. +-commutative 66.0%

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

   2. associate-+r- 65.9%

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

   3. sub-neg 65.9%

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

   4. remove-double-neg 65.9%

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

   5. neg-sub0 65.9%

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

   6. associate-+l- 65.9%

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

   7. neg-sub0 65.9%

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

   8. distribute-neg-frac2 65.9%

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

   9. distribute-frac-neg2 65.9%

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

   10. associate-+r+ 66.0%

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

   11. +-commutative 66.0%

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

   12. remove-double-neg 66.0%

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

   13. distribute-neg-frac2 66.0%

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

   14. sub0-neg 66.0%

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

   15. associate-+l- 66.0%

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

   16. neg-sub0 66.0%

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

3. Simplified 66.0%

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

5. Taylor expanded in x around inf 64.6%

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

   1. frac-2neg 64.6%

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

   2. metadata-eval 64.6%

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

   3. frac-2neg 64.6%

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

   4. metadata-eval 64.6%

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

   5. frac-add 64.6%

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

   6. add-sqr-sqrt 26.2%

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

   7. sqrt-unprod 10.9%

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

   8. sqr-neg 10.9%

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

   9. sqrt-prod 27.0%

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

   10. add-sqr-sqrt 48.1%

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

   11. neg-mul-1 48.1%

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

   12. add-sqr-sqrt 21.1%

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

   13. sqrt-unprod 8.5%

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

   14. sqr-neg 8.5%

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

   15. sqrt-prod 30.0%

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

   16. add-sqr-sqrt 64.6%

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

   17. add-sqr-sqrt 30.4%

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

   18. sqrt-unprod 64.4%

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

   19. sqr-neg 64.4%

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

   20. sqrt-prod 34.0%

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

   21. add-sqr-sqrt 64.8%

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

7. Applied egg-rr 64.8%

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

   1. *-rgt-identity 64.8%

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

   2. sub-neg 64.8%

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

   3. associate--r+ 50.1%

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

   4. +-inverses 50.1%

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

   5. metadata-eval 50.1%

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

   6. *-commutative 50.1%

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

   7. distribute-rgt-neg-in 50.1%

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

   8. neg-mul-1 50.1%

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

   9. *-commutative 50.1%

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

   10. associate-*l* 50.1%

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

   11. *-commutative 50.1%

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

   12. neg-mul-1 50.1%

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

   13. neg-sub0 50.1%

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

   14. +-commutative 50.1%

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

   15. associate--r+ 50.1%

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

   16. metadata-eval 50.1%

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

9. Simplified 50.1%

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

10. Final simplification 50.1%

    \[\leadsto \frac{1}{x \cdot \left(1 - x\right)} \]
11. 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 66.0%

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

   1. +-commutative 66.0%

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

   2. associate-+r- 65.9%

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

   3. sub-neg 65.9%

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

   4. remove-double-neg 65.9%

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

   5. neg-sub0 65.9%

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

   6. associate-+l- 65.9%

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

   7. neg-sub0 65.9%

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

   8. distribute-neg-frac2 65.9%

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

   9. distribute-frac-neg2 65.9%

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

   10. associate-+r+ 66.0%

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

   11. +-commutative 66.0%

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

   12. remove-double-neg 66.0%

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

   13. distribute-neg-frac2 66.0%

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

   14. sub0-neg 66.0%

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

   15. associate-+l- 66.0%

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

   16. neg-sub0 66.0%

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

3. Simplified 66.0%

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

5. Taylor expanded in x around 0 4.9%

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

6. Final simplification 4.9%

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

Alternative 9: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. +-commutative 66.0%

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

   2. associate-+r- 65.9%

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

   3. sub-neg 65.9%

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

   4. remove-double-neg 65.9%

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

   5. neg-sub0 65.9%

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

   6. associate-+l- 65.9%

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

   7. neg-sub0 65.9%

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

   8. distribute-neg-frac2 65.9%

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

   9. distribute-frac-neg2 65.9%

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

   10. associate-+r+ 66.0%

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

   11. +-commutative 66.0%

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

   12. remove-double-neg 66.0%

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

   13. distribute-neg-frac2 66.0%

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

   14. sub0-neg 66.0%

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

   15. associate-+l- 66.0%

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

   16. neg-sub0 66.0%

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

3. Simplified 66.0%

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

5. Taylor expanded in x around inf 64.6%

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

6. Taylor expanded in x around 0 4.9%

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

7. Final simplification 4.9%

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

Alternative 10: 6.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. +-commutative 66.0%

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

   2. associate-+r- 65.9%

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

   3. sub-neg 65.9%

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

   4. remove-double-neg 65.9%

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

   5. neg-sub0 65.9%

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

   6. associate-+l- 65.9%

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

   7. neg-sub0 65.9%

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

   8. distribute-neg-frac2 65.9%

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

   9. distribute-frac-neg2 65.9%

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

   10. associate-+r+ 66.0%

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

   11. +-commutative 66.0%

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

   12. remove-double-neg 66.0%

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

   13. distribute-neg-frac2 66.0%

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

   14. sub0-neg 66.0%

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

   15. associate-+l- 66.0%

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

   16. neg-sub0 66.0%

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

3. Simplified 66.0%

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

5. Taylor expanded in x around inf 64.6%

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

   1. frac-2neg 64.6%

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

   2. metadata-eval 64.6%

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

   3. frac-2neg 64.6%

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

   4. metadata-eval 64.6%

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

   5. frac-add 64.6%

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

   6. add-sqr-sqrt 26.2%

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

   7. sqrt-unprod 10.9%

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

   8. sqr-neg 10.9%

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

   9. sqrt-prod 27.0%

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

   10. add-sqr-sqrt 48.1%

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

   11. neg-mul-1 48.1%

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

   12. add-sqr-sqrt 21.1%

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

   13. sqrt-unprod 8.5%

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

   14. sqr-neg 8.5%

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

   15. sqrt-prod 30.0%

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

   16. add-sqr-sqrt 64.6%

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

   17. add-sqr-sqrt 30.4%

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

   18. sqrt-unprod 64.4%

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

   19. sqr-neg 64.4%

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

   20. sqrt-prod 34.0%

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

   21. add-sqr-sqrt 64.8%

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

7. Applied egg-rr 64.8%

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

   1. *-rgt-identity 64.8%

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

   2. sub-neg 64.8%

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

   3. associate--r+ 50.1%

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

   4. +-inverses 50.1%

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

   5. metadata-eval 50.1%

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

   6. *-commutative 50.1%

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

   7. distribute-rgt-neg-in 50.1%

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

   8. neg-mul-1 50.1%

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

   9. *-commutative 50.1%

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

   10. associate-*l* 50.1%

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

   11. *-commutative 50.1%

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

   12. neg-mul-1 50.1%

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

   13. neg-sub0 50.1%

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

   14. +-commutative 50.1%

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

   15. associate--r+ 50.1%

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

   16. metadata-eval 50.1%

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

9. Simplified 50.1%

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

10. Taylor expanded in x around 0 6.3%

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

11. Final simplification 6.3%

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

Developer target: 99.3% 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 2024073 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

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

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