3frac (problem 3.3.3)

Percentage Accurate: 85.3% → 99.5%

Time: 7.8s

Alternatives: 8

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: 85.3% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. sub-neg 86.2%

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

   2. distribute-neg-frac 86.2%

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

   3. metadata-eval 86.2%

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

   4. metadata-eval 86.2%

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

   5. metadata-eval 86.2%

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

   6. associate-/r* 86.2%

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

   7. metadata-eval 86.2%

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

   8. neg-mul-1 86.2%

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

   9. +-commutative 86.2%

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

   10. associate-+l+ 86.2%

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

   11. +-commutative 86.2%

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

   12. neg-mul-1 86.2%

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

   13. metadata-eval 86.2%

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

   14. associate-/r* 86.2%

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

   15. metadata-eval 86.2%

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

   16. metadata-eval 86.2%

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

   17. +-commutative 86.2%

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

   18. +-commutative 86.2%

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

3. Simplified 86.2%

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

   1. +-commutative 86.2%

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

   2. frac-add 55.3%

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

   3. frac-add 57.8%

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

   4. *-un-lft-identity 57.8%

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

   5. *-commutative 57.8%

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

   6. neg-mul-1 57.8%

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

   7. distribute-neg-in 57.8%

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

   8. metadata-eval 57.8%

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

5. Applied egg-rr 57.8%

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

6. Taylor expanded in x around 0 99.9%

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

7. Taylor expanded in x around 0 99.9%

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

   1. mul-1-neg 99.9%

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

   2. unsub-neg 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

    \[\leadsto \frac{-2}{x - {x}^{3}} \]

Alternative 2: 74.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 91.3%

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

      1. sub-neg 91.3%

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

      2. distribute-neg-frac 91.3%

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

      3. metadata-eval 91.3%

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

      4. metadata-eval 91.3%

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

      5. metadata-eval 91.3%

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

      6. associate-/r* 91.3%

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

      7. metadata-eval 91.3%

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

      8. neg-mul-1 91.3%

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

      9. +-commutative 91.3%

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

      10. associate-+l+ 91.3%

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

      11. +-commutative 91.3%

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

      12. neg-mul-1 91.3%

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

      13. metadata-eval 91.3%

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

      14. associate-/r* 91.3%

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

      15. metadata-eval 91.3%

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

      16. metadata-eval 91.3%

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

      17. +-commutative 91.3%

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

      18. +-commutative 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in x around 0 62.7%

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

      1. associate-*r/ 62.7%

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

      2. metadata-eval 62.7%

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

   6. Simplified 62.7%

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

   if 1 < x

   1. Initial program 71.3%

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

      1. sub-neg 71.3%

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

      2. distribute-neg-frac 71.3%

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

      3. metadata-eval 71.3%

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

      4. metadata-eval 71.3%

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

      5. metadata-eval 71.3%

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

      6. associate-/r* 71.3%

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

      7. metadata-eval 71.3%

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

      8. neg-mul-1 71.3%

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

      9. +-commutative 71.3%

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

      10. associate-+l+ 71.2%

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

      11. +-commutative 71.2%

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

      12. neg-mul-1 71.2%

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

      13. metadata-eval 71.2%

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

      14. associate-/r* 71.2%

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

      15. metadata-eval 71.2%

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

      16. metadata-eval 71.2%

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

      17. +-commutative 71.2%

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

      18. +-commutative 71.2%

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

   3. Simplified 71.2%

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

      1. +-commutative 71.2%

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

      2. frac-add 13.5%

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

      3. frac-add 18.6%

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

      4. *-un-lft-identity 18.6%

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

      5. *-commutative 18.6%

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

      6. neg-mul-1 18.6%

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

      7. distribute-neg-in 18.6%

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

      8. metadata-eval 18.6%

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

   5. Applied egg-rr 18.6%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. frac-2neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. div-inv 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 69.0%

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

      7. sqr-neg 69.0%

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

      8. sqrt-unprod 69.0%

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

      9. add-sqr-sqrt 69.0%

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

      10. *-commutative 69.0%

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

      11. associate-/r* 69.0%

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

      12. sub-neg 69.0%

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

      13. distribute-rgt-in 69.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 97.2%

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

      16. sqr-neg 97.2%

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

      17. sqrt-unprod 97.1%

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

      18. add-sqr-sqrt 97.2%

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

      19. distribute-rgt-in 97.2%

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

      20. pow2 97.2%

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

   8. Applied egg-rr 97.2%

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

      1. associate-*r/ 97.2%

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

      2. associate-*r/ 97.2%

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

      3. metadata-eval 97.2%

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

      4. +-commutative 97.2%

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

   10. Simplified 97.2%

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

   11. Taylor expanded in x around inf 97.2%

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

       1. +-commutative 97.2%

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

       2. unpow2 97.2%

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

       3. distribute-rgt-out 97.2%

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

   13. Simplified 97.2%

       \[\leadsto \frac{\frac{2}{x}}{\color{blue}{x \cdot \left(x + 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

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(-2.0 / Float64(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[(-2.0 / N[(x * N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.2%

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

   1. sub-neg 86.2%

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

   2. distribute-neg-frac 86.2%

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

   3. metadata-eval 86.2%

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

   4. metadata-eval 86.2%

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

   5. metadata-eval 86.2%

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

   6. associate-/r* 86.2%

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

   7. metadata-eval 86.2%

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

   8. neg-mul-1 86.2%

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

   9. +-commutative 86.2%

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

   10. associate-+l+ 86.2%

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

   11. +-commutative 86.2%

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

   12. neg-mul-1 86.2%

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

   13. metadata-eval 86.2%

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

   14. associate-/r* 86.2%

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

   15. metadata-eval 86.2%

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

   16. metadata-eval 86.2%

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

   17. +-commutative 86.2%

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

   18. +-commutative 86.2%

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

3. Simplified 86.2%

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

   1. +-commutative 86.2%

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

   2. frac-add 55.3%

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

   3. frac-add 57.8%

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

   4. *-un-lft-identity 57.8%

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

   5. *-commutative 57.8%

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

   6. neg-mul-1 57.8%

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

   7. distribute-neg-in 57.8%

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

   8. metadata-eval 57.8%

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

5. Applied egg-rr 57.8%

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

6. Taylor expanded in x around 0 99.9%

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

7. Final simplification 99.9%

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

Alternative 4: 66.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (-2.0 / x) - 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) then
        tmp = ((-2.0d0) / x) - 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) {
		tmp = (-2.0 / x) - x;
	} else {
		tmp = (2.0 / x) + (-2.0 / x);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(-2.0 / x) - 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)
		tmp = (-2.0 / x) - x;
	else
		tmp = (2.0 / x) + (-2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 91.3%

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

      1. associate-+l- 91.3%

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

      2. sub-neg 91.3%

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

      3. +-commutative 91.3%

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

      4. sub-neg 91.3%

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

      5. distribute-neg-in 91.3%

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

      6. distribute-neg-frac 91.3%

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

      7. metadata-eval 91.3%

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

      8. remove-double-neg 91.3%

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

      9. sub-neg 91.3%

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

      10. metadata-eval 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in x around 0 62.5%

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

   5. Taylor expanded in x around 0 62.2%

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

      1. mul-1-neg 62.2%

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

      2. associate-*r/ 62.2%

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

      3. metadata-eval 62.2%

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

   7. Simplified 62.2%

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

   8. Taylor expanded in x around 0 62.2%

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

      1. sub-neg 62.2%

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

      2. associate-*r/ 62.2%

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

      3. metadata-eval 62.2%

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

      4. distribute-neg-frac 62.2%

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

      5. metadata-eval 62.2%

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

      6. +-commutative 62.2%

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

      7. mul-1-neg 62.2%

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

      8. sub-neg 62.2%

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

   10. Simplified 62.2%

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

   if 1 < x

   1. Initial program 71.3%

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

      1. sub-neg 71.3%

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

      2. distribute-neg-frac 71.3%

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

      3. metadata-eval 71.3%

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

      4. metadata-eval 71.3%

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

      5. metadata-eval 71.3%

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

      6. associate-/r* 71.3%

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

      7. metadata-eval 71.3%

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

      8. neg-mul-1 71.3%

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

      9. +-commutative 71.3%

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

      10. associate-+l+ 71.2%

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

      11. +-commutative 71.2%

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

      12. neg-mul-1 71.2%

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

      13. metadata-eval 71.2%

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

      14. associate-/r* 71.2%

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

      15. metadata-eval 71.2%

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

      16. metadata-eval 71.2%

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

      17. +-commutative 71.2%

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

      18. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in x around inf 69.5%

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

4. Final simplification 64.0%

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

Alternative 5: 67.0% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 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) {
		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) 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) {
		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:
		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)
		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)
		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, 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 2 regimes

2. if x < 1

   1. Initial program 91.3%

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

      1. sub-neg 91.3%

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

      2. distribute-neg-frac 91.3%

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

      3. metadata-eval 91.3%

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

      4. metadata-eval 91.3%

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

      5. metadata-eval 91.3%

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

      6. associate-/r* 91.3%

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

      7. metadata-eval 91.3%

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

      8. neg-mul-1 91.3%

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

      9. +-commutative 91.3%

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

      10. associate-+l+ 91.3%

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

      11. +-commutative 91.3%

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

      12. neg-mul-1 91.3%

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

      13. metadata-eval 91.3%

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

      14. associate-/r* 91.3%

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

      15. metadata-eval 91.3%

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

      16. metadata-eval 91.3%

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

      17. +-commutative 91.3%

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

      18. +-commutative 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in x around 0 62.7%

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

      1. associate-*r/ 62.7%

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

      2. metadata-eval 62.7%

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

   6. Simplified 62.7%

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

   if 1 < x

   1. Initial program 71.3%

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

      1. sub-neg 71.3%

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

      2. distribute-neg-frac 71.3%

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

      3. metadata-eval 71.3%

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

      4. metadata-eval 71.3%

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

      5. metadata-eval 71.3%

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

      6. associate-/r* 71.3%

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

      7. metadata-eval 71.3%

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

      8. neg-mul-1 71.3%

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

      9. +-commutative 71.3%

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

      10. associate-+l+ 71.2%

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

      11. +-commutative 71.2%

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

      12. neg-mul-1 71.2%

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

      13. metadata-eval 71.2%

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

      14. associate-/r* 71.2%

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

      15. metadata-eval 71.2%

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

      16. metadata-eval 71.2%

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

      17. +-commutative 71.2%

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

      18. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in x around inf 69.5%

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

4. Final simplification 64.4%

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

Alternative 6: 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 86.2%

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

   1. sub-neg 86.2%

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

   2. distribute-neg-frac 86.2%

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

   3. metadata-eval 86.2%

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

   4. metadata-eval 86.2%

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

   5. metadata-eval 86.2%

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

   6. associate-/r* 86.2%

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

   7. metadata-eval 86.2%

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

   8. neg-mul-1 86.2%

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

   9. +-commutative 86.2%

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

   10. associate-+l+ 86.2%

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

   11. +-commutative 86.2%

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

   12. neg-mul-1 86.2%

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

   13. metadata-eval 86.2%

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

   14. associate-/r* 86.2%

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

   15. metadata-eval 86.2%

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

   16. metadata-eval 86.2%

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

   17. +-commutative 86.2%

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

   18. +-commutative 86.2%

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

3. Simplified 86.2%

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

4. Taylor expanded in x around 0 48.5%

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

5. Final simplification 48.5%

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

Alternative 7: 2.9% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 86.2%

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

   1. associate-+l- 86.2%

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

   2. sub-neg 86.2%

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

   3. +-commutative 86.2%

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

   4. sub-neg 86.2%

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

   5. distribute-neg-in 86.2%

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

   6. distribute-neg-frac 86.2%

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

   7. metadata-eval 86.2%

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

   8. remove-double-neg 86.2%

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

   9. sub-neg 86.2%

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

   10. metadata-eval 86.2%

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

3. Simplified 86.2%

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

4. Taylor expanded in x around 0 47.6%

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

5. Taylor expanded in x around 0 46.9%

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

   1. mul-1-neg 46.9%

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

   2. associate-*r/ 46.9%

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

   3. metadata-eval 46.9%

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

7. Simplified 46.9%

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

8. Taylor expanded in x around inf 3.0%

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

   1. mul-1-neg 3.0%

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

10. Simplified 3.0%

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

11. Final simplification 3.0%

    \[\leadsto -x \]

Alternative 8: 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 86.2%

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

   1. associate-+l- 86.2%

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

   2. sub-neg 86.2%

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

   3. +-commutative 86.2%

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

   4. sub-neg 86.2%

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

   5. distribute-neg-in 86.2%

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

   6. distribute-neg-frac 86.2%

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

   7. metadata-eval 86.2%

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

   8. remove-double-neg 86.2%

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

   9. sub-neg 86.2%

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

   10. metadata-eval 86.2%

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

3. Simplified 86.2%

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

4. Taylor expanded in x around 0 47.6%

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

5. Taylor expanded in x around inf 3.5%

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

6. Final simplification 3.5%

   \[\leadsto 1 \]

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 2023319 
(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))))