3frac (problem 3.3.3)

Percentage Accurate: 70.1% → 99.7%

Time: 10.2s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(2, {x}^{-2}, 2\right) + \mathsf{fma}\left(2, {x}^{-6}, 2 \cdot \left({x}^{-4} + {x}^{-8}\right)\right)\right) \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+
   (fma 2.0 (pow x -2.0) 2.0)
   (fma 2.0 (pow x -6.0) (* 2.0 (+ (pow x -4.0) (pow x -8.0)))))
  (pow x -3.0)))

C

double code(double x) {
	return (fma(2.0, pow(x, -2.0), 2.0) + fma(2.0, pow(x, -6.0), (2.0 * (pow(x, -4.0) + pow(x, -8.0))))) * pow(x, -3.0);
}

Julia

function code(x)
	return Float64(Float64(fma(2.0, (x ^ -2.0), 2.0) + fma(2.0, (x ^ -6.0), Float64(2.0 * Float64((x ^ -4.0) + (x ^ -8.0))))) * (x ^ -3.0))
end

Wolfram

code[x_] := N[(N[(N[(2.0 * N[Power[x, -2.0], $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * N[Power[x, -6.0], $MachinePrecision] + N[(2.0 * N[(N[Power[x, -4.0], $MachinePrecision] + N[Power[x, -8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+r- 68.7%

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

   3. sub-neg 68.7%

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

   4. remove-double-neg 68.7%

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

   5. neg-sub0 68.7%

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

   6. associate-+l- 68.7%

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

   7. neg-sub0 68.7%

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

   8. distribute-neg-frac2 68.7%

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

   9. distribute-frac-neg2 68.7%

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

   10. associate-+r+ 68.7%

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

   11. +-commutative 68.7%

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

   12. remove-double-neg 68.7%

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

   13. distribute-neg-frac2 68.7%

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

   14. sub0-neg 68.7%

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

   15. associate-+l- 68.7%

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

   16. neg-sub0 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{2 + \left(2 \cdot \frac{1}{{x}^{2}} + \left(2 \cdot \frac{1}{{x}^{6}} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{8}}\right)\right)\right)}{{x}^{3}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 98.8%

      \[\leadsto \color{blue}{1 \cdot \frac{2 + \left(2 \cdot \frac{1}{{x}^{2}} + \left(2 \cdot \frac{1}{{x}^{6}} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{8}}\right)\right)\right)}{{x}^{3}}} \]

   2. div-inv 98.8%

      \[\leadsto 1 \cdot \color{blue}{\left(\left(2 + \left(2 \cdot \frac{1}{{x}^{2}} + \left(2 \cdot \frac{1}{{x}^{6}} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{8}}\right)\right)\right)\right) \cdot \frac{1}{{x}^{3}}\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{1 \cdot \left(\left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-6}, \mathsf{fma}\left(2, {x}^{-4}, 2 \cdot {x}^{-8}\right)\right)\right)\right) \cdot {x}^{-3}\right)} \]
8. Step-by-step derivation

   1. associate-*r* 99.6%

      \[\leadsto \color{blue}{\left(1 \cdot \left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-6}, \mathsf{fma}\left(2, {x}^{-4}, 2 \cdot {x}^{-8}\right)\right)\right)\right)\right) \cdot {x}^{-3}} \]

9. Simplified 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right) + \mathsf{fma}\left(2, {x}^{-6}, 2 \cdot \left({x}^{-4} + {x}^{-8}\right)\right)\right) \cdot {x}^{-3}} \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+r- 68.7%

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

   3. sub-neg 68.7%

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

   4. remove-double-neg 68.7%

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

   5. neg-sub0 68.7%

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

   6. associate-+l- 68.7%

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

   7. neg-sub0 68.7%

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

   8. distribute-neg-frac2 68.7%

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

   9. distribute-frac-neg2 68.7%

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

   10. associate-+r+ 68.7%

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

   11. +-commutative 68.7%

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

   12. remove-double-neg 68.7%

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

   13. distribute-neg-frac2 68.7%

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

   14. sub0-neg 68.7%

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

   15. associate-+l- 68.7%

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

   16. neg-sub0 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{\frac{2 + \left(2 \cdot \frac{1}{{x}^{2}} + \left(2 \cdot \frac{1}{{x}^{6}} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{8}}\right)\right)\right)}{{x}^{3}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 98.8%

      \[\leadsto \color{blue}{1 \cdot \frac{2 + \left(2 \cdot \frac{1}{{x}^{2}} + \left(2 \cdot \frac{1}{{x}^{6}} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{8}}\right)\right)\right)}{{x}^{3}}} \]

   2. div-inv 98.8%

      \[\leadsto 1 \cdot \color{blue}{\left(\left(2 + \left(2 \cdot \frac{1}{{x}^{2}} + \left(2 \cdot \frac{1}{{x}^{6}} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{8}}\right)\right)\right)\right) \cdot \frac{1}{{x}^{3}}\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{1 \cdot \left(\left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-6}, \mathsf{fma}\left(2, {x}^{-4}, 2 \cdot {x}^{-8}\right)\right)\right)\right) \cdot {x}^{-3}\right)} \]
8. Step-by-step derivation

   1. associate-*r* 99.6%

      \[\leadsto \color{blue}{\left(1 \cdot \left(2 + \mathsf{fma}\left(2, {x}^{-2}, \mathsf{fma}\left(2, {x}^{-6}, \mathsf{fma}\left(2, {x}^{-4}, 2 \cdot {x}^{-8}\right)\right)\right)\right)\right) \cdot {x}^{-3}} \]

9. Simplified 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2, {x}^{-2}, 2\right) + \mathsf{fma}\left(2, {x}^{-6}, 2 \cdot \left({x}^{-4} + {x}^{-8}\right)\right)\right) \cdot {x}^{-3}} \]

10. Taylor expanded in x around inf 98.9%

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

    1. associate-*r/ 98.9%

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

    2. metadata-eval 98.9%

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

12. Simplified 98.9%

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

    1. unpow2 98.9%

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

14. Applied egg-rr 98.9%

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

15. Final simplification 98.9%

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

Alternative 3: 70.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+r- 68.7%

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

   3. sub-neg 68.7%

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

   4. remove-double-neg 68.7%

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

   5. neg-sub0 68.7%

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

   6. associate-+l- 68.7%

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

   7. neg-sub0 68.7%

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

   8. distribute-neg-frac2 68.7%

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

   9. distribute-frac-neg2 68.7%

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

   10. associate-+r+ 68.7%

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

   11. +-commutative 68.7%

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

   12. remove-double-neg 68.7%

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

   13. distribute-neg-frac2 68.7%

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

   14. sub0-neg 68.7%

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

   15. associate-+l- 68.7%

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

   16. neg-sub0 68.7%

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

3. Simplified 68.7%

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

   1. frac-sub 17.3%

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

   2. div-inv 16.6%

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

   3. *-rgt-identity 16.6%

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

   4. fmm-def 16.6%

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

6. Applied egg-rr 16.6%

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

   1. associate-*r/ 17.3%

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

   2. *-rgt-identity 17.3%

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

   3. associate-/r* 68.8%

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

   4. fmm-undef 68.8%

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

   5. div-sub 68.7%

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

   6. *-inverses 68.7%

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

8. Simplified 68.7%

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

9. Final simplification 68.7%

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

Alternative 4: 70.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+r- 68.7%

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

   3. sub-neg 68.7%

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

   4. remove-double-neg 68.7%

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

   5. neg-sub0 68.7%

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

   6. associate-+l- 68.7%

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

   7. neg-sub0 68.7%

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

   8. distribute-neg-frac2 68.7%

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

   9. distribute-frac-neg2 68.7%

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

   10. associate-+r+ 68.7%

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

   11. +-commutative 68.7%

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

   12. remove-double-neg 68.7%

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

   13. distribute-neg-frac2 68.7%

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

   14. sub0-neg 68.7%

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

   15. associate-+l- 68.7%

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

   16. neg-sub0 68.7%

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

3. Simplified 68.7%

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

   1. frac-2neg 68.7%

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

   2. frac-2neg 68.7%

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

   3. metadata-eval 68.7%

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

   4. frac-sub 17.3%

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

   5. metadata-eval 17.3%

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

   6. sub-neg 17.3%

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

   7. distribute-neg-in 17.3%

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

   8. metadata-eval 17.3%

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

   9. neg-mul-1 17.3%

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

   10. *-commutative 17.3%

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

   11. sub-neg 17.3%

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

   12. *-commutative 17.3%

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

   13. neg-mul-1 17.3%

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

   14. sub-neg 17.3%

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

   15. distribute-neg-in 17.3%

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

   16. metadata-eval 17.3%

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

   17. neg-mul-1 17.3%

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

   18. *-commutative 17.3%

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

   19. sub-neg 17.3%

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

   20. *-commutative 17.3%

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

6. Applied egg-rr 17.3%

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

   1. *-commutative 17.3%

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

   2. sub-neg 17.3%

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

   3. remove-double-neg 17.3%

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

   4. distribute-rgt-in 17.3%

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

   5. metadata-eval 17.3%

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

   6. neg-mul-1 17.3%

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

   7. remove-double-neg 17.3%

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

   8. sub-neg 17.3%

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

   9. remove-double-neg 17.3%

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

   10. distribute-lft-in 17.3%

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

   11. *-rgt-identity 17.3%

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

   12. neg-mul-1 17.3%

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

   13. distribute-rgt-in 17.3%

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

   14. sub-neg 17.3%

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

8. Simplified 17.3%

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

   1. clear-num 20.2%

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

   2. inv-pow 20.2%

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

   3. +-commutative 20.2%

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

   4. *-commutative 20.2%

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

   5. fma-define 20.2%

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

10. Applied egg-rr 20.2%

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

    1. unpow-1 20.2%

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

    2. associate-/l* 68.7%

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

    3. fma-undefine 68.7%

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

    4. *-commutative 68.7%

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

    5. distribute-lft1-in 68.7%

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

    6. distribute-rgt1-in 68.7%

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

    7. *-commutative 68.7%

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

    8. associate-+r- 68.8%

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

    9. *-lft-identity 68.8%

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

    10. distribute-rgt-out-- 68.8%

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

    11. metadata-eval 68.8%

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

    12. *-rgt-identity 68.8%

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

12. Simplified 68.8%

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

Alternative 5: 70.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+r- 68.7%

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

   3. sub-neg 68.7%

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

   4. remove-double-neg 68.7%

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

   5. neg-sub0 68.7%

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

   6. associate-+l- 68.7%

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

   7. neg-sub0 68.7%

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

   8. distribute-neg-frac2 68.7%

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

   9. distribute-frac-neg2 68.7%

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

   10. associate-+r+ 68.7%

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

   11. +-commutative 68.7%

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

   12. remove-double-neg 68.7%

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

   13. distribute-neg-frac2 68.7%

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

   14. sub0-neg 68.7%

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

   15. associate-+l- 68.7%

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

   16. neg-sub0 68.7%

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

3. Simplified 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+l- 68.7%

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

6. Applied egg-rr 68.7%

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

   1. frac-sub 17.3%

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

   2. *-un-lft-identity 17.3%

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

8. Applied egg-rr 17.3%

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

   1. associate-/r* 68.7%

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

   2. metadata-eval 68.7%

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

   3. sub-neg 68.7%

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

   4. *-rgt-identity 68.7%

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

   5. associate--l- 68.8%

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

   6. associate-+r- 68.7%

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

   7. metadata-eval 68.7%

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

   8. neg-sub0 68.7%

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

10. Simplified 68.7%

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

11. Final simplification 68.7%

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

Alternative 6: 70.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (-2.0 / x) + ((1.0 / (x + -1.0)) + (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))) + (1.0d0 / (x - (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+r- 68.7%

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

   3. sub-neg 68.7%

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

   4. remove-double-neg 68.7%

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

   5. neg-sub0 68.7%

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

   6. associate-+l- 68.7%

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

   7. neg-sub0 68.7%

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

   8. distribute-neg-frac2 68.7%

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

   9. distribute-frac-neg2 68.7%

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

   10. associate-+r+ 68.7%

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

   11. +-commutative 68.7%

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

   12. remove-double-neg 68.7%

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

   13. distribute-neg-frac2 68.7%

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

   14. sub0-neg 68.7%

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

   15. associate-+l- 68.7%

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

   16. neg-sub0 68.7%

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

3. Simplified 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+l- 68.7%

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

6. Applied egg-rr 68.7%

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

7. Final simplification 68.7%

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

Alternative 7: 70.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

3. Final simplification 68.7%

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

Alternative 8: 68.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+r- 68.7%

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

   3. sub-neg 68.7%

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

   4. remove-double-neg 68.7%

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

   5. neg-sub0 68.7%

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

   6. associate-+l- 68.7%

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

   7. neg-sub0 68.7%

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

   8. distribute-neg-frac2 68.7%

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

   9. distribute-frac-neg2 68.7%

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

   10. associate-+r+ 68.7%

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

   11. +-commutative 68.7%

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

   12. remove-double-neg 68.7%

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

   13. distribute-neg-frac2 68.7%

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

   14. sub0-neg 68.7%

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

   15. associate-+l- 68.7%

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

   16. neg-sub0 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around inf 66.9%

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

   1. associate-*r/ 66.9%

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

   2. neg-mul-1 66.9%

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

   3. distribute-neg-in 66.9%

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

   4. metadata-eval 66.9%

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

   5. distribute-neg-frac 66.9%

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

   6. metadata-eval 66.9%

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

7. Simplified 66.9%

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

8. Final simplification 66.9%

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

Alternative 9: 51.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+r- 68.7%

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

   3. sub-neg 68.7%

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

   4. remove-double-neg 68.7%

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

   5. neg-sub0 68.7%

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

   6. associate-+l- 68.7%

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

   7. neg-sub0 68.7%

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

   8. distribute-neg-frac2 68.7%

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

   9. distribute-frac-neg2 68.7%

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

   10. associate-+r+ 68.7%

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

   11. +-commutative 68.7%

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

   12. remove-double-neg 68.7%

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

   13. distribute-neg-frac2 68.7%

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

   14. sub0-neg 68.7%

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

   15. associate-+l- 68.7%

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

   16. neg-sub0 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around 0 3.5%

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

   1. frac-add 1.9%

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

   2. *-un-lft-identity 1.9%

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

   3. sub-neg 1.9%

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

   4. metadata-eval 1.9%

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

7. Applied egg-rr 1.9%

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

8. Taylor expanded in x around 0 53.0%

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

   1. *-commutative 53.0%

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

10. Simplified 53.0%

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

11. Final simplification 53.0%

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

Alternative 10: 3.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+r- 68.7%

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

   3. sub-neg 68.7%

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

   4. remove-double-neg 68.7%

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

   5. neg-sub0 68.7%

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

   6. associate-+l- 68.7%

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

   7. neg-sub0 68.7%

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

   8. distribute-neg-frac2 68.7%

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

   9. distribute-frac-neg2 68.7%

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

   10. associate-+r+ 68.7%

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

   11. +-commutative 68.7%

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

   12. remove-double-neg 68.7%

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

   13. distribute-neg-frac2 68.7%

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

   14. sub0-neg 68.7%

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

   15. associate-+l- 68.7%

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

   16. neg-sub0 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around 0 3.5%

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

6. Taylor expanded in x around -inf 3.5%

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

   1. mul-1-neg 3.5%

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

   2. unsub-neg 3.5%

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

   3. sub-neg 3.5%

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

   4. distribute-neg-frac 3.5%

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

   5. metadata-eval 3.5%

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

8. Simplified 3.5%

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

9. Final simplification 3.5%

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

Alternative 11: 3.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. associate-+r- 68.7%

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

   3. sub-neg 68.7%

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

   4. remove-double-neg 68.7%

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

   5. neg-sub0 68.7%

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

   6. associate-+l- 68.7%

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

   7. neg-sub0 68.7%

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

   8. distribute-neg-frac2 68.7%

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

   9. distribute-frac-neg2 68.7%

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

   10. associate-+r+ 68.7%

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

   11. +-commutative 68.7%

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

   12. remove-double-neg 68.7%

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

   13. distribute-neg-frac2 68.7%

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

   14. sub0-neg 68.7%

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

   15. associate-+l- 68.7%

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

   16. neg-sub0 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around 0 3.5%

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

6. Taylor expanded in x around inf 3.5%

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
7. 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 2024179 
(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))))