3frac (problem 3.3.3)

Percentage Accurate: 69.1% → 99.7%

Time: 12.9s

Alternatives: 9

Speedup: 1.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% 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

\[\begin{array}{l} \\ \mathsf{fma}\left(2, \frac{\frac{1}{x}}{x}, \mathsf{fma}\left(2, {x}^{-4}, 2\right)\right) \cdot {x}^{-3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma 2.0 (/ (/ 1.0 x) x) (fma 2.0 (pow x -4.0) 2.0)) (pow x -3.0)))

C

double code(double x) {
	return fma(2.0, ((1.0 / x) / x), fma(2.0, pow(x, -4.0), 2.0)) * pow(x, -3.0);
}

Julia

function code(x)
	return Float64(fma(2.0, Float64(Float64(1.0 / x) / x), fma(2.0, (x ^ -4.0), 2.0)) * (x ^ -3.0))
end

Wolfram

code[x_] := N[(N[(2.0 * N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[Power[x, -4.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-+r- 71.1%

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

   3. sub-neg 71.1%

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

   4. remove-double-neg 71.1%

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

   5. neg-sub0 71.1%

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

   6. associate-+l- 71.1%

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

   7. neg-sub0 71.1%

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

   8. distribute-neg-frac2 71.1%

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

   9. distribute-frac-neg2 71.1%

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

   10. associate-+r+ 71.1%

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

   11. +-commutative 71.1%

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

   12. remove-double-neg 71.1%

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

   13. distribute-neg-frac2 71.1%

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

   14. sub0-neg 71.1%

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

   15. associate-+l- 71.1%

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

   16. neg-sub0 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in x around inf 99.2%

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

   1. associate-+r+ 99.2%

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

   2. +-commutative 99.2%

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

   3. associate-+l+ 99.2%

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

   4. associate-*r/ 99.2%

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

   5. metadata-eval 99.2%

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

7. Simplified 99.2%

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

   1. div-inv 99.2%

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

   2. div-inv 99.2%

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

   3. fma-define 99.2%

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

   4. pow-flip 99.2%

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

   5. metadata-eval 99.2%

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

   6. +-commutative 99.2%

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

   7. div-inv 99.2%

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

   8. fma-define 99.2%

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

   9. pow-flip 99.2%

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

   10. metadata-eval 99.2%

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

   11. pow-flip 100.0%

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

   12. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

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

    1. metadata-eval 100.0%

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

    2. pow-prod-up 100.0%

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

    3. inv-pow 100.0%

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

    4. inv-pow 100.0%

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

11. Applied egg-rr 100.0%

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

    1. un-div-inv 100.0%

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

13. Applied egg-rr 100.0%

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

Alternative 2: 98.9% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (/ 2.0 (pow x 2.0)) (+ 2.0 (/ 2.0 (pow x 4.0)))) (pow x 3.0)))

C

double code(double x) {
	return ((2.0 / pow(x, 2.0)) + (2.0 + (2.0 / pow(x, 4.0)))) / pow(x, 3.0);
}

Fortran

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

Java

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

Python

def code(x):
	return ((2.0 / math.pow(x, 2.0)) + (2.0 + (2.0 / math.pow(x, 4.0)))) / math.pow(x, 3.0)

Julia

function code(x)
	return Float64(Float64(Float64(2.0 / (x ^ 2.0)) + Float64(2.0 + Float64(2.0 / (x ^ 4.0)))) / (x ^ 3.0))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 / (x ^ 2.0)) + (2.0 + (2.0 / (x ^ 4.0)))) / (x ^ 3.0);
end

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-+r- 71.1%

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

   3. sub-neg 71.1%

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

   4. remove-double-neg 71.1%

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

   5. neg-sub0 71.1%

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

   6. associate-+l- 71.1%

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

   7. neg-sub0 71.1%

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

   8. distribute-neg-frac2 71.1%

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

   9. distribute-frac-neg2 71.1%

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

   10. associate-+r+ 71.1%

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

   11. +-commutative 71.1%

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

   12. remove-double-neg 71.1%

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

   13. distribute-neg-frac2 71.1%

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

   14. sub0-neg 71.1%

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

   15. associate-+l- 71.1%

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

   16. neg-sub0 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in x around inf 99.2%

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

   1. associate-+r+ 99.2%

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

   2. +-commutative 99.2%

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

   3. associate-+l+ 99.2%

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

   4. associate-*r/ 99.2%

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

   5. metadata-eval 99.2%

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

7. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{{x}^{2}} + \left(2 + \frac{2}{{x}^{4}}\right)}{{x}^{3}}} \]
8. Add Preprocessing

Alternative 3: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \frac{2}{{x}^{2}}}{{x}^{3}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-+r- 71.1%

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

   3. sub-neg 71.1%

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

   4. remove-double-neg 71.1%

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

   5. neg-sub0 71.1%

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

   6. associate-+l- 71.1%

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

   7. neg-sub0 71.1%

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

   8. distribute-neg-frac2 71.1%

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

   9. distribute-frac-neg2 71.1%

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

   10. associate-+r+ 71.1%

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

   11. +-commutative 71.1%

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

   12. remove-double-neg 71.1%

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

   13. distribute-neg-frac2 71.1%

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

   14. sub0-neg 71.1%

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

   15. associate-+l- 71.1%

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

   16. neg-sub0 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in x around inf 99.1%

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

   1. associate-*r/ 99.1%

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

   2. metadata-eval 99.1%

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

7. Simplified 99.1%

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

Alternative 4: 98.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-+r- 71.1%

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

   3. sub-neg 71.1%

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

   4. remove-double-neg 71.1%

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

   5. neg-sub0 71.1%

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

   6. associate-+l- 71.1%

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

   7. neg-sub0 71.1%

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

   8. distribute-neg-frac2 71.1%

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

   9. distribute-frac-neg2 71.1%

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

   10. associate-+r+ 71.1%

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

   11. +-commutative 71.1%

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

   12. remove-double-neg 71.1%

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

   13. distribute-neg-frac2 71.1%

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

   14. sub0-neg 71.1%

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

   15. associate-+l- 71.1%

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

   16. neg-sub0 71.1%

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

3. Simplified 71.1%

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

   1. associate-+r- 71.1%

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

   2. flip-- 71.1%

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

   3. inv-pow 71.1%

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

   4. inv-pow 71.1%

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

   5. pow-prod-up 62.9%

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

   6. metadata-eval 62.9%

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

6. Applied egg-rr 62.9%

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

7. Taylor expanded in x around inf 91.2%

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

   1. sub-neg 91.2%

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

   2. associate-*r/ 91.2%

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

   3. metadata-eval 91.2%

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

   4. metadata-eval 91.2%

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

9. Simplified 91.2%

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

    1. *-un-lft-identity 91.2%

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

    2. associate-/l/ 91.2%

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

    3. +-commutative 91.2%

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

11. Applied egg-rr 91.2%

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

12. Taylor expanded in x around inf 98.4%

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

    1. sub-neg 98.4%

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

    2. associate-*r/ 98.4%

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

    3. metadata-eval 98.4%

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

    4. metadata-eval 98.4%

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

14. Simplified 98.4%

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

15. Final simplification 98.4%

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

Alternative 5: 98.2% accurate, 0.1× speedup

Math

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

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 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-+r- 71.1%

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

   3. sub-neg 71.1%

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

   4. remove-double-neg 71.1%

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

   5. neg-sub0 71.1%

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

   6. associate-+l- 71.1%

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

   7. neg-sub0 71.1%

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

   8. distribute-neg-frac2 71.1%

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

   9. distribute-frac-neg2 71.1%

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

   10. associate-+r+ 71.1%

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

   11. +-commutative 71.1%

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

   12. remove-double-neg 71.1%

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

   13. distribute-neg-frac2 71.1%

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

   14. sub0-neg 71.1%

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

   15. associate-+l- 71.1%

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

   16. neg-sub0 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in x around inf 98.4%

   \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
6. Add Preprocessing

Alternative 6: 72.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-+r- 71.1%

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

   3. sub-neg 71.1%

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

   4. remove-double-neg 71.1%

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

   5. neg-sub0 71.1%

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

   6. associate-+l- 71.1%

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

   7. neg-sub0 71.1%

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

   8. distribute-neg-frac2 71.1%

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

   9. distribute-frac-neg2 71.1%

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

   10. associate-+r+ 71.1%

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

   11. +-commutative 71.1%

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

   12. remove-double-neg 71.1%

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

   13. distribute-neg-frac2 71.1%

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

   14. sub0-neg 71.1%

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

   15. associate-+l- 71.1%

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

   16. neg-sub0 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in x around inf 70.2%

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

   1. associate-*r/ 70.2%

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

   2. neg-mul-1 70.2%

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

   3. distribute-neg-in 70.2%

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

   4. metadata-eval 70.2%

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

   5. distribute-neg-frac 70.2%

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

   6. metadata-eval 70.2%

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

7. Simplified 70.2%

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

   1. frac-add 70.3%

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

   2. *-un-lft-identity 70.3%

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

9. Applied egg-rr 70.3%

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

10. Taylor expanded in x around 0 74.0%

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

11. Final simplification 74.0%

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

Alternative 7: 67.8% 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 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-+r- 71.1%

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

   3. sub-neg 71.1%

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

   4. remove-double-neg 71.1%

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

   5. neg-sub0 71.1%

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

   6. associate-+l- 71.1%

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

   7. neg-sub0 71.1%

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

   8. distribute-neg-frac2 71.1%

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

   9. distribute-frac-neg2 71.1%

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

   10. associate-+r+ 71.1%

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

   11. +-commutative 71.1%

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

   12. remove-double-neg 71.1%

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

   13. distribute-neg-frac2 71.1%

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

   14. sub0-neg 71.1%

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

   15. associate-+l- 71.1%

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

   16. neg-sub0 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in x around inf 70.1%

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

Alternative 8: 5.0% 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 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-+r- 71.1%

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

   3. sub-neg 71.1%

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

   4. remove-double-neg 71.1%

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

   5. neg-sub0 71.1%

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

   6. associate-+l- 71.1%

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

   7. neg-sub0 71.1%

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

   8. distribute-neg-frac2 71.1%

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

   9. distribute-frac-neg2 71.1%

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

   10. associate-+r+ 71.1%

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

   11. +-commutative 71.1%

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

   12. remove-double-neg 71.1%

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

   13. distribute-neg-frac2 71.1%

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

   14. sub0-neg 71.1%

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

   15. associate-+l- 71.1%

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

   16. neg-sub0 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in x around inf 70.1%

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

6. Taylor expanded in x around 0 5.3%

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

Alternative 9: 5.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

   2. associate-+r- 71.1%

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

   3. sub-neg 71.1%

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

   4. remove-double-neg 71.1%

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

   5. neg-sub0 71.1%

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

   6. associate-+l- 71.1%

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

   7. neg-sub0 71.1%

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

   8. distribute-neg-frac2 71.1%

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

   9. distribute-frac-neg2 71.1%

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

   10. associate-+r+ 71.1%

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

   11. +-commutative 71.1%

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

   12. remove-double-neg 71.1%

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

   13. distribute-neg-frac2 71.1%

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

   14. sub0-neg 71.1%

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

   15. associate-+l- 71.1%

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

   16. neg-sub0 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in x around 0 5.3%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
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 2024089 
(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))))