3frac (problem 3.3.3)

Percentage Accurate: 84.1% → 99.5%

Time: 8.2s

Alternatives: 4

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: 84.1% 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: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. remove-double-neg 89.3%

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

   2. sub-neg 89.3%

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

   3. sub-neg 89.3%

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

   4. distribute-neg-frac 89.3%

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

   5. metadata-eval 89.3%

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

   6. metadata-eval 89.3%

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

   7. metadata-eval 89.3%

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

   8. associate-/r* 89.3%

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

   9. metadata-eval 89.3%

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

   10. neg-mul-1 89.3%

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

   11. associate--l+ 89.3%

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

   12. +-commutative 89.3%

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

   13. distribute-neg-frac 89.3%

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

   14. metadata-eval 89.3%

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

   15. metadata-eval 89.3%

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

   16. metadata-eval 89.3%

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

   17. associate-/r* 89.3%

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

   18. metadata-eval 89.3%

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

   19. neg-mul-1 89.3%

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

   20. sub0-neg 89.3%

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

   21. associate-+l- 89.3%

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

   22. neg-sub0 89.3%

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

3. Simplified 89.3%

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

   1. frac-sub 62.5%

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

   2. div-inv 61.1%

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

   3. *-rgt-identity 61.1%

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

5. Applied egg-rr 61.1%

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

6. Taylor expanded in x around 0 53.1%

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

   1. un-div-inv 53.1%

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

   2. frac-add 58.8%

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

   3. *-un-lft-identity 58.8%

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

   4. +-commutative 58.8%

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

   5. +-commutative 58.8%

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

8. Applied egg-rr 58.8%

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

9. Taylor expanded in x around 0 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 82.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-2}{x} - \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0)))
   (- (/ -2.0 x) (/ -2.0 x))
   (- (* -2.0 x) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-2.0 / x) - (-2.0 / x);
	} else {
		tmp = (-2.0 * x) - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = ((-2.0d0) / x) - ((-2.0d0) / x)
    else
        tmp = ((-2.0d0) * x) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (-2.0 / x) - (-2.0 / x);
	} else {
		tmp = (-2.0 * x) - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (-2.0 / x) - (-2.0 / x)
	else:
		tmp = (-2.0 * x) - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(-2.0 / x) - Float64(-2.0 / x));
	else
		tmp = Float64(Float64(-2.0 * x) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = (-2.0 / x) - (-2.0 / x);
	else
		tmp = (-2.0 * x) - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(-2.0 / x), $MachinePrecision] - N[(-2.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * x), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 78.2%

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

      1. remove-double-neg 78.2%

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

      2. sub-neg 78.2%

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

      3. sub-neg 78.2%

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

      4. distribute-neg-frac 78.2%

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

      5. metadata-eval 78.2%

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

      6. metadata-eval 78.2%

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

      7. metadata-eval 78.2%

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

      8. associate-/r* 78.2%

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

      9. metadata-eval 78.2%

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

      10. neg-mul-1 78.2%

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

      11. associate--l+ 78.2%

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

      12. +-commutative 78.2%

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

      13. distribute-neg-frac 78.2%

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

      14. metadata-eval 78.2%

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

      15. metadata-eval 78.2%

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

      16. metadata-eval 78.2%

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

      17. associate-/r* 78.2%

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

      18. metadata-eval 78.2%

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

      19. neg-mul-1 78.2%

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

      20. sub0-neg 78.2%

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

      21. associate-+l- 78.2%

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

      22. neg-sub0 78.2%

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

   3. Simplified 78.2%

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

      1. +-commutative 78.2%

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

      2. associate-+l- 78.1%

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

   5. Applied egg-rr 78.1%

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

   6. Taylor expanded in x around inf 75.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--l+ 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-/r* 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. sub0-neg 100.0%

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

      21. associate-+l- 100.0%

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

      22. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   6. Simplified 99.4%

      \[\leadsto \color{blue}{-2 \cdot x - \frac{2}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-2}{x} - \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \end{array} \]

Alternative 3: 83.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;\frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} - \frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.65)
   (+ (/ 1.0 (+ x 1.0)) (/ -1.0 x))
   (if (<= x 1.0) (- (* -2.0 x) (/ 2.0 x)) (- (/ -2.0 x) (/ -2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = (1.0 / (x + 1.0)) + (-1.0 / x);
	} else if (x <= 1.0) {
		tmp = (-2.0 * x) - (2.0 / x);
	} else {
		tmp = (-2.0 / x) - (-2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.65d0)) then
        tmp = (1.0d0 / (x + 1.0d0)) + ((-1.0d0) / x)
    else if (x <= 1.0d0) then
        tmp = ((-2.0d0) * x) - (2.0d0 / x)
    else
        tmp = ((-2.0d0) / x) - ((-2.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = (1.0 / (x + 1.0)) + (-1.0 / x);
	} else if (x <= 1.0) {
		tmp = (-2.0 * x) - (2.0 / x);
	} else {
		tmp = (-2.0 / x) - (-2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.65:
		tmp = (1.0 / (x + 1.0)) + (-1.0 / x)
	elif x <= 1.0:
		tmp = (-2.0 * x) - (2.0 / x)
	else:
		tmp = (-2.0 / x) - (-2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.65)
		tmp = Float64(Float64(1.0 / Float64(x + 1.0)) + Float64(-1.0 / x));
	elseif (x <= 1.0)
		tmp = Float64(Float64(-2.0 * x) - Float64(2.0 / x));
	else
		tmp = Float64(Float64(-2.0 / x) - Float64(-2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.65)
		tmp = (1.0 / (x + 1.0)) + (-1.0 / x);
	elseif (x <= 1.0)
		tmp = (-2.0 * x) - (2.0 / x);
	else
		tmp = (-2.0 / x) - (-2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.65], N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(-2.0 * x), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x), $MachinePrecision] - N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.650000000000000022

   1. Initial program 74.9%

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

      1. remove-double-neg 74.9%

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

      2. sub-neg 74.9%

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

      3. sub-neg 74.9%

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

      4. distribute-neg-frac 74.9%

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

      5. metadata-eval 74.9%

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

      6. metadata-eval 74.9%

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

      7. metadata-eval 74.9%

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

      8. associate-/r* 74.9%

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

      9. metadata-eval 74.9%

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

      10. neg-mul-1 74.9%

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

      11. associate--l+ 74.9%

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

      12. +-commutative 74.9%

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

      13. distribute-neg-frac 74.9%

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

      14. metadata-eval 74.9%

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

      15. metadata-eval 74.9%

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

      16. metadata-eval 74.9%

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

      17. associate-/r* 74.9%

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

      18. metadata-eval 74.9%

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

      19. neg-mul-1 74.9%

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

      20. sub0-neg 74.9%

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

      21. associate-+l- 74.9%

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

      22. neg-sub0 74.9%

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

   3. Simplified 74.9%

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

      1. frac-sub 22.6%

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

      2. div-inv 19.9%

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

      3. *-rgt-identity 19.9%

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

   5. Applied egg-rr 19.9%

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

   6. Taylor expanded in x around 0 19.9%

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

   7. Taylor expanded in x around inf 72.9%

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

   if -0.650000000000000022 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate--l+ 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. associate-/r* 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. sub0-neg 100.0%

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

      21. associate-+l- 100.0%

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

      22. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   if 1 < x

   1. Initial program 81.8%

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

      1. remove-double-neg 81.8%

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

      2. sub-neg 81.8%

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

      3. sub-neg 81.8%

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

      4. distribute-neg-frac 81.8%

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

      5. metadata-eval 81.8%

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

      6. metadata-eval 81.8%

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

      7. metadata-eval 81.8%

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

      8. associate-/r* 81.8%

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

      9. metadata-eval 81.8%

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

      10. neg-mul-1 81.8%

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

      11. associate--l+ 81.8%

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

      12. +-commutative 81.8%

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

      13. distribute-neg-frac 81.8%

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

      14. metadata-eval 81.8%

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

      15. metadata-eval 81.8%

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

      16. metadata-eval 81.8%

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

      17. associate-/r* 81.8%

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

      18. metadata-eval 81.8%

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

      19. neg-mul-1 81.8%

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

      20. sub0-neg 81.8%

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

      21. associate-+l- 81.8%

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

      22. neg-sub0 81.8%

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

   3. Simplified 81.8%

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

      1. +-commutative 81.8%

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

      2. associate-+l- 81.7%

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

   5. Applied egg-rr 81.7%

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

   6. Taylor expanded in x around inf 79.0%

      \[\leadsto \frac{-2}{x} - \color{blue}{\frac{-2}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;\frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x} - \frac{-2}{x}\\ \end{array} \]

Alternative 4: 51.8% 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 89.3%

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

   1. remove-double-neg 89.3%

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

   2. sub-neg 89.3%

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

   3. sub-neg 89.3%

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

   4. distribute-neg-frac 89.3%

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

   5. metadata-eval 89.3%

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

   6. metadata-eval 89.3%

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

   7. metadata-eval 89.3%

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

   8. associate-/r* 89.3%

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

   9. metadata-eval 89.3%

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

   10. neg-mul-1 89.3%

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

   11. associate--l+ 89.3%

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

   12. +-commutative 89.3%

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

   13. distribute-neg-frac 89.3%

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

   14. metadata-eval 89.3%

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

   15. metadata-eval 89.3%

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

   16. metadata-eval 89.3%

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

   17. associate-/r* 89.3%

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

   18. metadata-eval 89.3%

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

   19. neg-mul-1 89.3%

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

   20. sub0-neg 89.3%

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

   21. associate-+l- 89.3%

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

   22. neg-sub0 89.3%

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

3. Simplified 89.3%

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

4. Taylor expanded in x around 0 53.3%

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

5. Final simplification 53.3%

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

Developer target: 99.5% 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 2023313 
(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))))