3frac (problem 3.3.3)

Percentage Accurate: 70.0% → 99.7%

Time: 14.3s

Alternatives: 10

Speedup: 1.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.7% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 + (2.0d0 * ((x ** (-2.0d0)) + ((x ** (-6.0d0)) + (x ** (-4.0d0)))))) * (x ** (-3.0d0))
end function

Java

public static double code(double x) {
	return (2.0 + (2.0 * (Math.pow(x, -2.0) + (Math.pow(x, -6.0) + Math.pow(x, -4.0))))) * Math.pow(x, -3.0);
}

Python

def code(x):
	return (2.0 + (2.0 * (math.pow(x, -2.0) + (math.pow(x, -6.0) + math.pow(x, -4.0))))) * math.pow(x, -3.0)

Julia

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

MATLAB

function tmp = code(x)
	tmp = (2.0 + (2.0 * ((x ^ -2.0) + ((x ^ -6.0) + (x ^ -4.0))))) * (x ^ -3.0);
end

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 66.8%

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

   3. sub-neg 66.8%

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

   4. remove-double-neg 66.8%

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

   5. neg-sub0 66.8%

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

   6. associate-+l- 66.8%

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

   7. neg-sub0 66.8%

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

   8. distribute-neg-frac2 66.8%

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

   9. distribute-frac-neg2 66.8%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 98.9%

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

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

   3. +-commutative 98.9%

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

   4. associate-*r/ 98.9%

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

   5. metadata-eval 98.9%

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

7. Simplified 98.9%

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

   1. div-inv 98.9%

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

   2. div-inv 98.9%

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

   3. fma-define 98.9%

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

   4. pow-flip 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. div-inv 98.9%

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

   8. fma-define 98.9%

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

   9. pow-flip 98.9%

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

   10. metadata-eval 98.9%

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

   11. div-inv 98.9%

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

   12. pow-flip 98.9%

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

   13. metadata-eval 98.9%

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

   14. pow-flip 99.5%

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

   15. metadata-eval 99.5%

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

9. Applied egg-rr 99.5%

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

    1. fma-undefine 99.5%

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

    2. distribute-lft-out 99.5%

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

11. Simplified 99.5%

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

    1. fma-undefine 99.5%

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

13. Applied egg-rr 99.5%

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

    1. distribute-lft-out 99.5%

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

15. Simplified 99.5%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 66.8%

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

   3. sub-neg 66.8%

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

   4. remove-double-neg 66.8%

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

   5. neg-sub0 66.8%

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

   6. associate-+l- 66.8%

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

   7. neg-sub0 66.8%

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

   8. distribute-neg-frac2 66.8%

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

   9. distribute-frac-neg2 66.8%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 98.9%

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

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

   3. +-commutative 98.9%

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

   4. associate-*r/ 98.9%

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

   5. metadata-eval 98.9%

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

7. Simplified 98.9%

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

   1. div-inv 98.9%

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

   2. div-inv 98.9%

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

   3. fma-define 98.9%

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

   4. pow-flip 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. div-inv 98.9%

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

   8. fma-define 98.9%

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

   9. pow-flip 98.9%

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

   10. metadata-eval 98.9%

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

   11. div-inv 98.9%

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

   12. pow-flip 98.9%

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

   13. metadata-eval 98.9%

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

   14. pow-flip 99.5%

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

   15. metadata-eval 99.5%

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

9. Applied egg-rr 99.5%

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

    1. fma-undefine 99.5%

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

    2. distribute-lft-out 99.5%

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

11. Simplified 99.5%

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

12. Taylor expanded in x around inf 99.0%

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

13. Final simplification 99.0%

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

Alternative 3: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{8 + \frac{\frac{12}{x} - 10}{x}}{x} - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{-1}{x - -1}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ (- (/ (+ 8.0 (/ (- (/ 12.0 x) 10.0) x)) x) 6.0) (pow x 3.0))
  (* (+ x -1.0) (+ (/ -2.0 x) (/ -1.0 (- x -1.0))))))

C

double code(double x) {
	return ((((8.0 + (((12.0 / x) - 10.0) / x)) / x) - 6.0) / pow(x, 3.0)) / ((x + -1.0) * ((-2.0 / x) + (-1.0 / (x - -1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((8.0d0 + (((12.0d0 / x) - 10.0d0) / x)) / x) - 6.0d0) / (x ** 3.0d0)) / ((x + (-1.0d0)) * (((-2.0d0) / x) + ((-1.0d0) / (x - (-1.0d0)))))
end function

Java

public static double code(double x) {
	return ((((8.0 + (((12.0 / x) - 10.0) / x)) / x) - 6.0) / Math.pow(x, 3.0)) / ((x + -1.0) * ((-2.0 / x) + (-1.0 / (x - -1.0))));
}

Python

def code(x):
	return ((((8.0 + (((12.0 / x) - 10.0) / x)) / x) - 6.0) / math.pow(x, 3.0)) / ((x + -1.0) * ((-2.0 / x) + (-1.0 / (x - -1.0))))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(8.0 + Float64(Float64(Float64(12.0 / x) - 10.0) / x)) / x) - 6.0) / (x ^ 3.0)) / Float64(Float64(x + -1.0) * Float64(Float64(-2.0 / x) + Float64(-1.0 / Float64(x - -1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = ((((8.0 + (((12.0 / x) - 10.0) / x)) / x) - 6.0) / (x ^ 3.0)) / ((x + -1.0) * ((-2.0 / x) + (-1.0 / (x - -1.0))));
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(8.0 + N[(N[(N[(12.0 / x), $MachinePrecision] - 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 6.0), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(N[(-2.0 / x), $MachinePrecision] + N[(-1.0 / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 66.8%

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

   3. sub-neg 66.8%

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

   4. remove-double-neg 66.8%

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

   5. neg-sub0 66.8%

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

   6. associate-+l- 66.8%

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

   7. neg-sub0 66.8%

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

   8. distribute-neg-frac2 66.8%

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

   9. distribute-frac-neg2 66.8%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

   1. flip-- 19.5%

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

   2. frac-add 18.8%

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

   3. *-un-lft-identity 18.8%

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

   4. frac-times 13.7%

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

   5. metadata-eval 13.7%

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

   6. pow2 13.7%

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

   7. inv-pow 13.7%

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

   8. inv-pow 13.7%

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

   9. pow-prod-up 14.9%

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

   10. metadata-eval 14.9%

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

6. Applied egg-rr 14.9%

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

7. Taylor expanded in x around -inf 98.1%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{6 + -1 \cdot \frac{8 + -1 \cdot \frac{10 - 12 \cdot \frac{1}{x}}{x}}{x}}{{x}^{3}}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]
8. Step-by-step derivation

   1. mul-1-neg 98.1%

      \[\leadsto \frac{\color{blue}{-\frac{6 + -1 \cdot \frac{8 + -1 \cdot \frac{10 - 12 \cdot \frac{1}{x}}{x}}{x}}{{x}^{3}}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   2. distribute-neg-frac2 98.1%

      \[\leadsto \frac{\color{blue}{\frac{6 + -1 \cdot \frac{8 + -1 \cdot \frac{10 - 12 \cdot \frac{1}{x}}{x}}{x}}{-{x}^{3}}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   3. mul-1-neg 98.1%

      \[\leadsto \frac{\frac{6 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{10 - 12 \cdot \frac{1}{x}}{x}}{x}\right)}}{-{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   4. unsub-neg 98.1%

      \[\leadsto \frac{\frac{\color{blue}{6 - \frac{8 + -1 \cdot \frac{10 - 12 \cdot \frac{1}{x}}{x}}{x}}}{-{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   5. mul-1-neg 98.1%

      \[\leadsto \frac{\frac{6 - \frac{8 + \color{blue}{\left(-\frac{10 - 12 \cdot \frac{1}{x}}{x}\right)}}{x}}{-{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   6. unsub-neg 98.1%

      \[\leadsto \frac{\frac{6 - \frac{\color{blue}{8 - \frac{10 - 12 \cdot \frac{1}{x}}{x}}}{x}}{-{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   7. associate-*r/ 98.1%

      \[\leadsto \frac{\frac{6 - \frac{8 - \frac{10 - \color{blue}{\frac{12 \cdot 1}{x}}}{x}}{x}}{-{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   8. metadata-eval 98.1%

      \[\leadsto \frac{\frac{6 - \frac{8 - \frac{10 - \frac{\color{blue}{12}}{x}}{x}}{x}}{-{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

9. Simplified 98.1%

   \[\leadsto \frac{\color{blue}{\frac{6 - \frac{8 - \frac{10 - \frac{12}{x}}{x}}{x}}{-{x}^{3}}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

10. Final simplification 98.1%

    \[\leadsto \frac{\frac{\frac{8 + \frac{\frac{12}{x} - 10}{x}}{x} - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{-1}{x - -1}\right)} \]
11. Add Preprocessing

Alternative 4: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{8 + \frac{-10}{x}}{x} - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{-1}{x - -1}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ (- (/ (+ 8.0 (/ -10.0 x)) x) 6.0) (pow x 3.0))
  (* (+ x -1.0) (+ (/ -2.0 x) (/ -1.0 (- x -1.0))))))

C

double code(double x) {
	return ((((8.0 + (-10.0 / x)) / x) - 6.0) / pow(x, 3.0)) / ((x + -1.0) * ((-2.0 / x) + (-1.0 / (x - -1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((((8.0d0 + ((-10.0d0) / x)) / x) - 6.0d0) / (x ** 3.0d0)) / ((x + (-1.0d0)) * (((-2.0d0) / x) + ((-1.0d0) / (x - (-1.0d0)))))
end function

Java

public static double code(double x) {
	return ((((8.0 + (-10.0 / x)) / x) - 6.0) / Math.pow(x, 3.0)) / ((x + -1.0) * ((-2.0 / x) + (-1.0 / (x - -1.0))));
}

Python

def code(x):
	return ((((8.0 + (-10.0 / x)) / x) - 6.0) / math.pow(x, 3.0)) / ((x + -1.0) * ((-2.0 / x) + (-1.0 / (x - -1.0))))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(8.0 + Float64(-10.0 / x)) / x) - 6.0) / (x ^ 3.0)) / Float64(Float64(x + -1.0) * Float64(Float64(-2.0 / x) + Float64(-1.0 / Float64(x - -1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = ((((8.0 + (-10.0 / x)) / x) - 6.0) / (x ^ 3.0)) / ((x + -1.0) * ((-2.0 / x) + (-1.0 / (x - -1.0))));
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(8.0 + N[(-10.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 6.0), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(N[(-2.0 / x), $MachinePrecision] + N[(-1.0 / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 66.8%

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

   3. sub-neg 66.8%

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

   4. remove-double-neg 66.8%

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

   5. neg-sub0 66.8%

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

   6. associate-+l- 66.8%

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

   7. neg-sub0 66.8%

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

   8. distribute-neg-frac2 66.8%

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

   9. distribute-frac-neg2 66.8%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

   1. flip-- 19.5%

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

   2. frac-add 18.8%

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

   3. *-un-lft-identity 18.8%

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

   4. frac-times 13.7%

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

   5. metadata-eval 13.7%

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

   6. pow2 13.7%

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

   7. inv-pow 13.7%

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

   8. inv-pow 13.7%

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

   9. pow-prod-up 14.9%

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

   10. metadata-eval 14.9%

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

6. Applied egg-rr 14.9%

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

7. Taylor expanded in x around inf 97.8%

   \[\leadsto \frac{\color{blue}{\frac{8 \cdot \frac{1}{x} - \left(6 + \frac{10}{{x}^{2}}\right)}{{x}^{3}}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]
8. Step-by-step derivation

   1. +-commutative 97.8%

      \[\leadsto \frac{\frac{8 \cdot \frac{1}{x} - \color{blue}{\left(\frac{10}{{x}^{2}} + 6\right)}}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   2. associate--r+ 97.8%

      \[\leadsto \frac{\frac{\color{blue}{\left(8 \cdot \frac{1}{x} - \frac{10}{{x}^{2}}\right) - 6}}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   3. associate-*r/ 97.8%

      \[\leadsto \frac{\frac{\left(\color{blue}{\frac{8 \cdot 1}{x}} - \frac{10}{{x}^{2}}\right) - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   4. metadata-eval 97.8%

      \[\leadsto \frac{\frac{\left(\frac{\color{blue}{8}}{x} - \frac{10}{{x}^{2}}\right) - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   5. unpow2 97.8%

      \[\leadsto \frac{\frac{\left(\frac{8}{x} - \frac{10}{\color{blue}{x \cdot x}}\right) - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   6. associate-/r* 97.8%

      \[\leadsto \frac{\frac{\left(\frac{8}{x} - \color{blue}{\frac{\frac{10}{x}}{x}}\right) - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   7. metadata-eval 97.8%

      \[\leadsto \frac{\frac{\left(\frac{8}{x} - \frac{\frac{\color{blue}{10 \cdot 1}}{x}}{x}\right) - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   8. associate-*r/ 97.8%

      \[\leadsto \frac{\frac{\left(\frac{8}{x} - \frac{\color{blue}{10 \cdot \frac{1}{x}}}{x}\right) - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   9. div-sub 97.8%

      \[\leadsto \frac{\frac{\color{blue}{\frac{8 - 10 \cdot \frac{1}{x}}{x}} - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   10. sub-neg 97.8%

       \[\leadsto \frac{\frac{\frac{\color{blue}{8 + \left(-10 \cdot \frac{1}{x}\right)}}{x} - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   11. associate-*r/ 97.8%

       \[\leadsto \frac{\frac{\frac{8 + \left(-\color{blue}{\frac{10 \cdot 1}{x}}\right)}{x} - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   12. metadata-eval 97.8%

       \[\leadsto \frac{\frac{\frac{8 + \left(-\frac{\color{blue}{10}}{x}\right)}{x} - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   13. distribute-neg-frac 97.8%

       \[\leadsto \frac{\frac{\frac{8 + \color{blue}{\frac{-10}{x}}}{x} - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

   14. metadata-eval 97.8%

       \[\leadsto \frac{\frac{\frac{8 + \frac{\color{blue}{-10}}{x}}{x} - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

9. Simplified 97.8%

   \[\leadsto \frac{\color{blue}{\frac{\frac{8 + \frac{-10}{x}}{x} - 6}{{x}^{3}}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{1}{-1 - x}\right)} \]

10. Final simplification 97.8%

    \[\leadsto \frac{\frac{\frac{8 + \frac{-10}{x}}{x} - 6}{{x}^{3}}}{\left(x + -1\right) \cdot \left(\frac{-2}{x} + \frac{-1}{x - -1}\right)} \]
11. Add Preprocessing

Alternative 5: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot {x}^{-3} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 66.8%

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

   3. sub-neg 66.8%

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

   4. remove-double-neg 66.8%

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

   5. neg-sub0 66.8%

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

   6. associate-+l- 66.8%

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

   7. neg-sub0 66.8%

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

   8. distribute-neg-frac2 66.8%

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

   9. distribute-frac-neg2 66.8%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 97.2%

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

   1. div-inv 97.2%

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

   2. pow-flip 97.8%

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

   3. metadata-eval 97.8%

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

7. Applied egg-rr 97.8%

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

Alternative 6: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 66.8%

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

   3. sub-neg 66.8%

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

   4. remove-double-neg 66.8%

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

   5. neg-sub0 66.8%

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

   6. associate-+l- 66.8%

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

   7. neg-sub0 66.8%

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

   8. distribute-neg-frac2 66.8%

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

   9. distribute-frac-neg2 66.8%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 65.6%

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

   1. +-commutative 65.6%

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

   2. associate--r+ 65.6%

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

   3. unpow2 65.6%

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

   4. associate-/r* 65.6%

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

   5. div-sub 65.6%

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

   6. sub-neg 65.6%

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

   7. metadata-eval 65.6%

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

   8. +-commutative 65.6%

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

7. Simplified 65.6%

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

   1. frac-add 65.6%

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

   2. *-un-lft-identity 65.6%

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

   3. sub-neg 65.6%

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

   4. metadata-eval 65.6%

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

9. Applied egg-rr 65.6%

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

10. Taylor expanded in x around inf 95.9%

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

11. Final simplification 95.9%

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

Alternative 7: 97.8% accurate, 1.7× speedup

Math

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

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(Float64(2.0 / x) / Float64(x * Float64(x + -1.0)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 66.8%

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

   3. sub-neg 66.8%

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

   4. remove-double-neg 66.8%

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

   5. neg-sub0 66.8%

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

   6. associate-+l- 66.8%

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

   7. neg-sub0 66.8%

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

   8. distribute-neg-frac2 66.8%

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

   9. distribute-frac-neg2 66.8%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 65.6%

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

   1. +-commutative 65.6%

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

   2. associate--r+ 65.6%

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

   3. unpow2 65.6%

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

   4. associate-/r* 65.6%

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

   5. div-sub 65.6%

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

   6. sub-neg 65.6%

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

   7. metadata-eval 65.6%

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

   8. +-commutative 65.6%

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

7. Simplified 65.6%

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

   1. frac-add 65.6%

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

   2. *-un-lft-identity 65.6%

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

   3. sub-neg 65.6%

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

   4. metadata-eval 65.6%

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

9. Applied egg-rr 65.6%

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

10. Taylor expanded in x around inf 95.7%

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

11. Final simplification 95.7%

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

Alternative 8: 68.8% 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 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 66.8%

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

   3. sub-neg 66.8%

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

   4. remove-double-neg 66.8%

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

   5. neg-sub0 66.8%

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

   6. associate-+l- 66.8%

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

   7. neg-sub0 66.8%

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

   8. distribute-neg-frac2 66.8%

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

   9. distribute-frac-neg2 66.8%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 64.1%

   \[\leadsto \frac{1}{x + -1} + \color{blue}{\frac{-1}{x}} \]
6. 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 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 66.8%

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

   3. sub-neg 66.8%

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

   4. remove-double-neg 66.8%

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

   5. neg-sub0 66.8%

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

   6. associate-+l- 66.8%

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

   7. neg-sub0 66.8%

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

   8. distribute-neg-frac2 66.8%

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

   9. distribute-frac-neg2 66.8%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around inf 64.1%

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

6. Taylor expanded in x around 0 4.9%

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

Alternative 10: 5.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

   1. +-commutative 67.0%

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

   2. associate-+r- 66.8%

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

   3. sub-neg 66.8%

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

   4. remove-double-neg 66.8%

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

   5. neg-sub0 66.8%

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

   6. associate-+l- 66.8%

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

   7. neg-sub0 66.8%

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

   8. distribute-neg-frac2 66.8%

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

   9. distribute-frac-neg2 66.8%

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

   10. associate-+r+ 67.0%

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

   11. +-commutative 67.0%

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

   12. remove-double-neg 67.0%

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

   13. distribute-neg-frac2 67.0%

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

   14. sub0-neg 67.0%

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

   15. associate-+l- 67.0%

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

   16. neg-sub0 67.0%

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

3. Simplified 67.0%

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

5. Taylor expanded in x around 0 4.9%

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

Developer target: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024108 
(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))))