3frac (problem 3.3.3)

Percentage Accurate: 68.5% → 99.7%

Time: 12.5s

Alternatives: 8

Speedup: 0.1×

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: 68.5% 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

\[\begin{array}{l} \\ \frac{2}{{x}^{5}} + \left(2 \cdot {x}^{-3} + \frac{2}{{x}^{7}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ 2.0 (pow x 5.0)) (+ (* 2.0 (pow x -3.0)) (/ 2.0 (pow x 7.0)))))

C

double code(double x) {
	return (2.0 / pow(x, 5.0)) + ((2.0 * pow(x, -3.0)) + (2.0 / pow(x, 7.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 / (x ** 5.0d0)) + ((2.0d0 * (x ** (-3.0d0))) + (2.0d0 / (x ** 7.0d0)))
end function

Java

public static double code(double x) {
	return (2.0 / Math.pow(x, 5.0)) + ((2.0 * Math.pow(x, -3.0)) + (2.0 / Math.pow(x, 7.0)));
}

Python

def code(x):
	return (2.0 / math.pow(x, 5.0)) + ((2.0 * math.pow(x, -3.0)) + (2.0 / math.pow(x, 7.0)))

Julia

function code(x)
	return Float64(Float64(2.0 / (x ^ 5.0)) + Float64(Float64(2.0 * (x ^ -3.0)) + Float64(2.0 / (x ^ 7.0))))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 / (x ^ 5.0)) + ((2.0 * (x ^ -3.0)) + (2.0 / (x ^ 7.0)));
end

Wolfram

code[x_] := N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

   1. sub-neg 68.0%

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

   2. distribute-neg-frac 68.0%

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

   3. metadata-eval 68.0%

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

   4. metadata-eval 68.0%

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

   5. metadata-eval 68.0%

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

   6. associate-/r* 68.0%

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

   7. metadata-eval 68.0%

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

   8. neg-mul-1 68.0%

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

   9. +-commutative 68.0%

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

   10. associate-+l+ 68.0%

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

   11. +-commutative 68.0%

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

   12. neg-mul-1 68.0%

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

   13. metadata-eval 68.0%

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

   14. associate-/r* 68.0%

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

   15. metadata-eval 68.0%

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

   16. metadata-eval 68.0%

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

   17. +-commutative 68.0%

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

   18. +-commutative 68.0%

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

   19. sub-neg 68.0%

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

   20. metadata-eval 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 99.1%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.1%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right) \]

   2. metadata-eval 99.1%

      \[\leadsto \frac{\color{blue}{2}}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right) \]

   3. +-commutative 99.1%

      \[\leadsto \frac{2}{{x}^{5}} + \color{blue}{\left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{7}}\right)} \]

   4. associate-*r/ 99.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{\frac{2 \cdot 1}{{x}^{3}}} + 2 \cdot \frac{1}{{x}^{7}}\right) \]

   5. metadata-eval 99.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{\color{blue}{2}}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{7}}\right) \]

   6. associate-*r/ 99.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \color{blue}{\frac{2 \cdot 1}{{x}^{7}}}\right) \]

   7. metadata-eval 99.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \frac{\color{blue}{2}}{{x}^{7}}\right) \]

7. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 99.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)\right)} + \frac{2}{{x}^{7}}\right) \]

   2. expm1-udef 67.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)} - 1\right)} + \frac{2}{{x}^{7}}\right) \]

   3. div-inv 67.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)} - 1\right) + \frac{2}{{x}^{7}}\right) \]

   4. pow-flip 67.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)} - 1\right) + \frac{2}{{x}^{7}}\right) \]

   5. metadata-eval 67.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\left(e^{\mathsf{log1p}\left(2 \cdot {x}^{\color{blue}{-3}}\right)} - 1\right) + \frac{2}{{x}^{7}}\right) \]

9. Applied egg-rr 67.9%

   \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)} - 1\right)} + \frac{2}{{x}^{7}}\right) \]
10. Step-by-step derivation

    1. expm1-def 99.5%

       \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)\right)} + \frac{2}{{x}^{7}}\right) \]

    2. expm1-log1p 99.5%

       \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{2 \cdot {x}^{-3}} + \frac{2}{{x}^{7}}\right) \]

11. Simplified 99.5%

    \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{2 \cdot {x}^{-3}} + \frac{2}{{x}^{7}}\right) \]

12. Final simplification 99.5%

    \[\leadsto \frac{2}{{x}^{5}} + \left(2 \cdot {x}^{-3} + \frac{2}{{x}^{7}}\right) \]
13. Add Preprocessing

Alternative 2: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{{x}^{5}} + \frac{2}{{x}^{3}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ 2.0 (pow x 5.0)) (/ 2.0 (pow x 3.0))))

C

double code(double x) {
	return (2.0 / pow(x, 5.0)) + (2.0 / pow(x, 3.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 / (x ** 5.0d0)) + (2.0d0 / (x ** 3.0d0))
end function

Java

public static double code(double x) {
	return (2.0 / Math.pow(x, 5.0)) + (2.0 / Math.pow(x, 3.0));
}

Python

def code(x):
	return (2.0 / math.pow(x, 5.0)) + (2.0 / math.pow(x, 3.0))

Julia

function code(x)
	return Float64(Float64(2.0 / (x ^ 5.0)) + Float64(2.0 / (x ^ 3.0)))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 / (x ^ 5.0)) + (2.0 / (x ^ 3.0));
end

Wolfram

code[x_] := N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

   1. sub-neg 68.0%

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

   2. distribute-neg-frac 68.0%

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

   3. metadata-eval 68.0%

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

   4. metadata-eval 68.0%

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

   5. metadata-eval 68.0%

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

   6. associate-/r* 68.0%

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

   7. metadata-eval 68.0%

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

   8. neg-mul-1 68.0%

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

   9. +-commutative 68.0%

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

   10. associate-+l+ 68.0%

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

   11. +-commutative 68.0%

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

   12. neg-mul-1 68.0%

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

   13. metadata-eval 68.0%

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

   14. associate-/r* 68.0%

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

   15. metadata-eval 68.0%

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

   16. metadata-eval 68.0%

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

   17. +-commutative 68.0%

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

   18. +-commutative 68.0%

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

   19. sub-neg 68.0%

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

   20. metadata-eval 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 98.9%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}} \]
6. Step-by-step derivation

   1. associate-*r/ 98.9%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} + 2 \cdot \frac{1}{{x}^{5}} \]

   2. metadata-eval 98.9%

      \[\leadsto \frac{\color{blue}{2}}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}} \]

   3. associate-*r/ 98.9%

      \[\leadsto \frac{2}{{x}^{3}} + \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} \]

   4. metadata-eval 98.9%

      \[\leadsto \frac{2}{{x}^{3}} + \frac{\color{blue}{2}}{{x}^{5}} \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}} \]

8. Final simplification 98.9%

   \[\leadsto \frac{2}{{x}^{5}} + \frac{2}{{x}^{3}} \]
9. Add Preprocessing

Alternative 3: 99.0% 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 68.0%

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

   1. sub-neg 68.0%

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

   2. distribute-neg-frac 68.0%

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

   3. metadata-eval 68.0%

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

   4. metadata-eval 68.0%

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

   5. metadata-eval 68.0%

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

   6. associate-/r* 68.0%

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

   7. metadata-eval 68.0%

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

   8. neg-mul-1 68.0%

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

   9. +-commutative 68.0%

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

   10. associate-+l+ 68.0%

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

   11. +-commutative 68.0%

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

   12. neg-mul-1 68.0%

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

   13. metadata-eval 68.0%

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

   14. associate-/r* 68.0%

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

   15. metadata-eval 68.0%

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

   16. metadata-eval 68.0%

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

   17. +-commutative 68.0%

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

   18. +-commutative 68.0%

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

   19. sub-neg 68.0%

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

   20. metadata-eval 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 98.4%

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

   1. expm1-log1p-u 99.1%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)\right)} + \frac{2}{{x}^{7}}\right) \]

   2. expm1-udef 67.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{{x}^{3}}\right)} - 1\right)} + \frac{2}{{x}^{7}}\right) \]

   3. div-inv 67.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{{x}^{3}}}\right)} - 1\right) + \frac{2}{{x}^{7}}\right) \]

   4. pow-flip 67.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{x}^{\left(-3\right)}}\right)} - 1\right) + \frac{2}{{x}^{7}}\right) \]

   5. metadata-eval 67.9%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\left(e^{\mathsf{log1p}\left(2 \cdot {x}^{\color{blue}{-3}}\right)} - 1\right) + \frac{2}{{x}^{7}}\right) \]

7. Applied egg-rr 67.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 99.5%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-3}\right)\right)} + \frac{2}{{x}^{7}}\right) \]

   2. expm1-log1p 99.5%

      \[\leadsto \frac{2}{{x}^{5}} + \left(\color{blue}{2 \cdot {x}^{-3}} + \frac{2}{{x}^{7}}\right) \]

9. Simplified 98.8%

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

10. Final simplification 98.8%

    \[\leadsto 2 \cdot {x}^{-3} \]
11. Add Preprocessing

Alternative 4: 68.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 0.5 - \frac{0.5}{x}\\ \frac{x + t\_0 \cdot -2}{x \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* x 0.5) (/ 0.5 x)))) (/ (+ x (* t_0 -2.0)) (* x t_0))))

C

double code(double x) {
	double t_0 = (x * 0.5) - (0.5 / x);
	return (x + (t_0 * -2.0)) / (x * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (x * 0.5d0) - (0.5d0 / x)
    code = (x + (t_0 * (-2.0d0))) / (x * t_0)
end function

Java

public static double code(double x) {
	double t_0 = (x * 0.5) - (0.5 / x);
	return (x + (t_0 * -2.0)) / (x * t_0);
}

Python

def code(x):
	t_0 = (x * 0.5) - (0.5 / x)
	return (x + (t_0 * -2.0)) / (x * t_0)

Julia

function code(x)
	t_0 = Float64(Float64(x * 0.5) - Float64(0.5 / x))
	return Float64(Float64(x + Float64(t_0 * -2.0)) / Float64(x * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = (x * 0.5) - (0.5 / x);
	tmp = (x + (t_0 * -2.0)) / (x * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * 0.5), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]}, N[(N[(x + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 68.0%

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

   1. sub-neg 68.0%

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

   2. distribute-neg-frac 68.0%

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

   3. metadata-eval 68.0%

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

   4. metadata-eval 68.0%

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

   5. metadata-eval 68.0%

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

   6. associate-/r* 68.0%

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

   7. metadata-eval 68.0%

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

   8. neg-mul-1 68.0%

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

   9. +-commutative 68.0%

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

   10. associate-+l+ 68.0%

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

   11. +-commutative 68.0%

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

   12. neg-mul-1 68.0%

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

   13. metadata-eval 68.0%

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

   14. associate-/r* 68.0%

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

   15. metadata-eval 68.0%

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

   16. metadata-eval 68.0%

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

   17. +-commutative 68.0%

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

   18. +-commutative 68.0%

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

   19. sub-neg 68.0%

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

   20. metadata-eval 68.0%

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

3. Simplified 68.0%

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

   1. frac-add 18.0%

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

   2. clear-num 20.7%

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

   3. +-commutative 20.7%

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

   4. *-un-lft-identity 20.7%

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

   5. *-rgt-identity 20.7%

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

   6. +-commutative 20.7%

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

   7. +-commutative 20.7%

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

6. Applied egg-rr 20.7%

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

7. Taylor expanded in x around 0 68.0%

   \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{0.5 \cdot x - 0.5 \cdot \frac{1}{x}}} \]
8. Step-by-step derivation

   1. *-commutative 68.0%

      \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{x \cdot 0.5} - 0.5 \cdot \frac{1}{x}} \]

   2. associate-*r/ 68.0%

      \[\leadsto \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \color{blue}{\frac{0.5 \cdot 1}{x}}} \]

   3. metadata-eval 68.0%

      \[\leadsto \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{\color{blue}{0.5}}{x}} \]

9. Simplified 68.0%

   \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{x \cdot 0.5 - \frac{0.5}{x}}} \]
10. Step-by-step derivation

    1. +-commutative 68.0%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 0.5 - \frac{0.5}{x}} + \frac{-2}{x}} \]

    2. frac-add 68.1%

       \[\leadsto \color{blue}{\frac{1 \cdot x + \left(x \cdot 0.5 - \frac{0.5}{x}\right) \cdot -2}{\left(x \cdot 0.5 - \frac{0.5}{x}\right) \cdot x}} \]

    3. *-un-lft-identity 68.1%

       \[\leadsto \frac{\color{blue}{x} + \left(x \cdot 0.5 - \frac{0.5}{x}\right) \cdot -2}{\left(x \cdot 0.5 - \frac{0.5}{x}\right) \cdot x} \]

11. Applied egg-rr 68.1%

    \[\leadsto \color{blue}{\frac{x + \left(x \cdot 0.5 - \frac{0.5}{x}\right) \cdot -2}{\left(x \cdot 0.5 - \frac{0.5}{x}\right) \cdot x}} \]

12. Final simplification 68.1%

    \[\leadsto \frac{x + \left(x \cdot 0.5 - \frac{0.5}{x}\right) \cdot -2}{x \cdot \left(x \cdot 0.5 - \frac{0.5}{x}\right)} \]
13. Add Preprocessing

Alternative 5: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{0.5}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ -2.0 x) (/ 1.0 (- (* x 0.5) (/ 0.5 x)))))

C

double code(double x) {
	return (-2.0 / x) + (1.0 / ((x * 0.5) - (0.5 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-2.0d0) / x) + (1.0d0 / ((x * 0.5d0) - (0.5d0 / x)))
end function

Java

public static double code(double x) {
	return (-2.0 / x) + (1.0 / ((x * 0.5) - (0.5 / x)));
}

Python

def code(x):
	return (-2.0 / x) + (1.0 / ((x * 0.5) - (0.5 / x)))

Julia

function code(x)
	return Float64(Float64(-2.0 / x) + Float64(1.0 / Float64(Float64(x * 0.5) - Float64(0.5 / x))))
end

MATLAB

function tmp = code(x)
	tmp = (-2.0 / x) + (1.0 / ((x * 0.5) - (0.5 / x)));
end

Wolfram

code[x_] := N[(N[(-2.0 / x), $MachinePrecision] + N[(1.0 / N[(N[(x * 0.5), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

   1. sub-neg 68.0%

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

   2. distribute-neg-frac 68.0%

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

   3. metadata-eval 68.0%

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

   4. metadata-eval 68.0%

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

   5. metadata-eval 68.0%

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

   6. associate-/r* 68.0%

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

   7. metadata-eval 68.0%

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

   8. neg-mul-1 68.0%

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

   9. +-commutative 68.0%

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

   10. associate-+l+ 68.0%

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

   11. +-commutative 68.0%

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

   12. neg-mul-1 68.0%

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

   13. metadata-eval 68.0%

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

   14. associate-/r* 68.0%

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

   15. metadata-eval 68.0%

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

   16. metadata-eval 68.0%

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

   17. +-commutative 68.0%

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

   18. +-commutative 68.0%

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

   19. sub-neg 68.0%

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

   20. metadata-eval 68.0%

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

3. Simplified 68.0%

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

   1. frac-add 18.0%

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

   2. clear-num 20.7%

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

   3. +-commutative 20.7%

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

   4. *-un-lft-identity 20.7%

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

   5. *-rgt-identity 20.7%

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

   6. +-commutative 20.7%

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

   7. +-commutative 20.7%

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

6. Applied egg-rr 20.7%

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

7. Taylor expanded in x around 0 68.0%

   \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{0.5 \cdot x - 0.5 \cdot \frac{1}{x}}} \]
8. Step-by-step derivation

   1. *-commutative 68.0%

      \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{x \cdot 0.5} - 0.5 \cdot \frac{1}{x}} \]

   2. associate-*r/ 68.0%

      \[\leadsto \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \color{blue}{\frac{0.5 \cdot 1}{x}}} \]

   3. metadata-eval 68.0%

      \[\leadsto \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{\color{blue}{0.5}}{x}} \]

9. Simplified 68.0%

   \[\leadsto \frac{-2}{x} + \frac{1}{\color{blue}{x \cdot 0.5 - \frac{0.5}{x}}} \]

10. Final simplification 68.0%

    \[\leadsto \frac{-2}{x} + \frac{1}{x \cdot 0.5 - \frac{0.5}{x}} \]
11. Add Preprocessing

Alternative 6: 67.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. sub-neg 68.0%

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

   2. distribute-neg-frac 68.0%

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

   3. metadata-eval 68.0%

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

   4. metadata-eval 68.0%

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

   5. metadata-eval 68.0%

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

   6. associate-/r* 68.0%

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

   7. metadata-eval 68.0%

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

   8. neg-mul-1 68.0%

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

   9. +-commutative 68.0%

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

   10. associate-+l+ 68.0%

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

   11. +-commutative 68.0%

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

   12. neg-mul-1 68.0%

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

   13. metadata-eval 68.0%

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

   14. associate-/r* 68.0%

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

   15. metadata-eval 68.0%

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

   16. metadata-eval 68.0%

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

   17. +-commutative 68.0%

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

   18. +-commutative 68.0%

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

   19. sub-neg 68.0%

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

   20. metadata-eval 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around inf 66.5%

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

6. Final simplification 66.5%

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

Alternative 7: 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 68.0%

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

   1. sub-neg 68.0%

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

   2. distribute-neg-frac 68.0%

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

   3. metadata-eval 68.0%

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

   4. metadata-eval 68.0%

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

   5. metadata-eval 68.0%

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

   6. associate-/r* 68.0%

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

   7. metadata-eval 68.0%

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

   8. neg-mul-1 68.0%

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

   9. +-commutative 68.0%

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

   10. associate-+l+ 68.0%

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

   11. +-commutative 68.0%

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

   12. neg-mul-1 68.0%

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

   13. metadata-eval 68.0%

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

   14. associate-/r* 68.0%

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

   15. metadata-eval 68.0%

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

   16. metadata-eval 68.0%

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

   17. +-commutative 68.0%

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

   18. +-commutative 68.0%

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

   19. sub-neg 68.0%

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

   20. metadata-eval 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in x around 0 5.1%

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

6. Final simplification 5.1%

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

Alternative 8: 3.4% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 68.0%

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

   1. associate-+l- 68.0%

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

   2. sub-neg 68.0%

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

   3. +-commutative 68.0%

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

   4. neg-sub0 68.0%

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

   5. associate-+l- 68.0%

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

   6. neg-sub0 68.0%

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

   7. distribute-neg-frac 68.0%

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

   8. metadata-eval 68.0%

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

   9. sub-neg 68.0%

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

   10. metadata-eval 68.0%

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

3. Simplified 68.0%

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

   1. frac-2neg 68.0%

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

   2. metadata-eval 68.0%

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

   3. frac-add 18.1%

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

   4. +-commutative 18.1%

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

   5. distribute-neg-in 18.1%

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

   6. metadata-eval 18.1%

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

   7. sub-neg 18.1%

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

   8. *-commutative 18.1%

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

   9. neg-mul-1 18.1%

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

   10. +-commutative 18.1%

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

   11. distribute-neg-in 18.1%

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

   12. metadata-eval 18.1%

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

   13. sub-neg 18.1%

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

6. Applied egg-rr 18.1%

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

   1. unsub-neg 18.1%

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

8. Simplified 18.1%

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

9. Taylor expanded in x around inf 16.7%

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

10. Taylor expanded in x around 0 3.4%

    \[\leadsto \color{blue}{2} \]

11. Final simplification 3.4%

    \[\leadsto 2 \]
12. 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 2024041 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

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