3frac (problem 3.3.3)

Percentage Accurate: 84.4% → 99.9%

Time: 6.4s

Alternatives: 5

Speedup: 1.4×

Specification

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: 84.4% 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: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. remove-double-neg 84.0%

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

   2. sub-neg 84.0%

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

   3. sub-neg 84.0%

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

   4. distribute-neg-frac 84.0%

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

   5. metadata-eval 84.0%

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

   6. metadata-eval 84.0%

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

   7. metadata-eval 84.0%

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

   8. associate-/r* 84.0%

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

   9. metadata-eval 84.0%

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

   10. neg-mul-1 84.0%

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

   11. associate--l+ 84.0%

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

   12. +-commutative 84.0%

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

   13. distribute-neg-frac 84.0%

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

   14. metadata-eval 84.0%

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

   15. metadata-eval 84.0%

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

   16. metadata-eval 84.0%

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

   17. associate-/r* 84.0%

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

   18. metadata-eval 84.0%

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

   19. neg-mul-1 84.0%

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

   20. sub0-neg 84.0%

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

   21. associate-+l- 84.0%

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

   22. neg-sub0 84.0%

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

3. Simplified 84.0%

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

   1. frac-sub 62.7%

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

   2. associate-/r* 83.9%

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

   3. *-rgt-identity 83.9%

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

5. Applied egg-rr 83.9%

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

   1. frac-add 83.9%

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

   2. /-rgt-identity 83.9%

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

   3. *-un-lft-identity 83.9%

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

   4. /-rgt-identity 83.9%

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

   5. div-sub 83.9%

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

   6. *-inverses 83.9%

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

   7. sub-neg 83.9%

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

   8. *-un-lft-identity 83.9%

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

   9. times-frac 83.9%

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

   10. metadata-eval 83.9%

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

   11. metadata-eval 83.9%

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

7. Applied egg-rr 83.9%

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

8. Taylor expanded in x around 0 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. remove-double-neg 84.0%

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

   2. sub-neg 84.0%

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

   3. sub-neg 84.0%

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

   4. distribute-neg-frac 84.0%

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

   5. metadata-eval 84.0%

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

   6. metadata-eval 84.0%

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

   7. metadata-eval 84.0%

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

   8. associate-/r* 84.0%

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

   9. metadata-eval 84.0%

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

   10. neg-mul-1 84.0%

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

   11. associate--l+ 84.0%

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

   12. +-commutative 84.0%

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

   13. distribute-neg-frac 84.0%

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

   14. metadata-eval 84.0%

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

   15. metadata-eval 84.0%

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

   16. metadata-eval 84.0%

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

   17. associate-/r* 84.0%

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

   18. metadata-eval 84.0%

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

   19. neg-mul-1 84.0%

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

   20. sub0-neg 84.0%

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

   21. associate-+l- 84.0%

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

   22. neg-sub0 84.0%

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

3. Simplified 84.0%

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

   1. frac-sub 62.7%

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

   2. associate-/r* 83.9%

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

   3. *-rgt-identity 83.9%

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

5. Applied egg-rr 83.9%

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

   1. associate-/l/ 62.7%

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

   2. frac-add 64.2%

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

   3. *-un-lft-identity 64.2%

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

7. Applied egg-rr 64.2%

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

8. Taylor expanded in x around 0 99.2%

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

9. Final simplification 99.2%

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

Alternative 3: 83.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. remove-double-neg 84.0%

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

   2. sub-neg 84.0%

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

   3. sub-neg 84.0%

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

   4. distribute-neg-frac 84.0%

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

   5. metadata-eval 84.0%

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

   6. metadata-eval 84.0%

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

   7. metadata-eval 84.0%

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

   8. associate-/r* 84.0%

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

   9. metadata-eval 84.0%

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

   10. neg-mul-1 84.0%

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

   11. associate--l+ 84.0%

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

   12. +-commutative 84.0%

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

   13. distribute-neg-frac 84.0%

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

   14. metadata-eval 84.0%

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

   15. metadata-eval 84.0%

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

   16. metadata-eval 84.0%

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

   17. associate-/r* 84.0%

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

   18. metadata-eval 84.0%

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

   19. neg-mul-1 84.0%

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

   20. sub0-neg 84.0%

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

   21. associate-+l- 84.0%

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

   22. neg-sub0 84.0%

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

3. Simplified 84.0%

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

4. Taylor expanded in x around 0 53.8%

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

   1. distribute-neg-in 53.8%

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

   2. metadata-eval 53.8%

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

   3. unsub-neg 53.8%

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

   4. associate-*r/ 53.8%

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

   5. metadata-eval 53.8%

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

6. Simplified 53.8%

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

7. Taylor expanded in x around 0 82.5%

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

8. Final simplification 82.5%

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

Alternative 4: 51.4% 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 84.0%

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

   1. remove-double-neg 84.0%

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

   2. sub-neg 84.0%

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

   3. sub-neg 84.0%

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

   4. distribute-neg-frac 84.0%

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

   5. metadata-eval 84.0%

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

   6. metadata-eval 84.0%

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

   7. metadata-eval 84.0%

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

   8. associate-/r* 84.0%

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

   9. metadata-eval 84.0%

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

   10. neg-mul-1 84.0%

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

   11. associate--l+ 84.0%

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

   12. +-commutative 84.0%

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

   13. distribute-neg-frac 84.0%

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

   14. metadata-eval 84.0%

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

   15. metadata-eval 84.0%

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

   16. metadata-eval 84.0%

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

   17. associate-/r* 84.0%

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

   18. metadata-eval 84.0%

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

   19. neg-mul-1 84.0%

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

   20. sub0-neg 84.0%

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

   21. associate-+l- 84.0%

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

   22. neg-sub0 84.0%

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

3. Simplified 84.0%

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

4. Taylor expanded in x around 0 54.6%

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

5. Final simplification 54.6%

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

Alternative 5: 3.3% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 84.0%

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

   1. remove-double-neg 84.0%

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

   2. sub-neg 84.0%

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

   3. sub-neg 84.0%

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

   4. distribute-neg-frac 84.0%

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

   5. metadata-eval 84.0%

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

   6. metadata-eval 84.0%

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

   7. metadata-eval 84.0%

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

   8. associate-/r* 84.0%

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

   9. metadata-eval 84.0%

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

   10. neg-mul-1 84.0%

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

   11. associate--l+ 84.0%

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

   12. +-commutative 84.0%

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

   13. distribute-neg-frac 84.0%

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

   14. metadata-eval 84.0%

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

   15. metadata-eval 84.0%

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

   16. metadata-eval 84.0%

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

   17. associate-/r* 84.0%

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

   18. metadata-eval 84.0%

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

   19. neg-mul-1 84.0%

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

   20. sub0-neg 84.0%

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

   21. associate-+l- 84.0%

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

   22. neg-sub0 84.0%

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

3. Simplified 84.0%

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

4. Taylor expanded in x around 0 53.8%

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

   1. distribute-neg-in 53.8%

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

   2. metadata-eval 53.8%

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

   3. unsub-neg 53.8%

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

   4. associate-*r/ 53.8%

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

   5. metadata-eval 53.8%

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

6. Simplified 53.8%

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

7. Taylor expanded in x around inf 3.2%

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

8. Final simplification 3.2%

   \[\leadsto -1 \]

Developer target: 99.6% 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 2023309 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

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

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