3frac (problem 3.3.3)

Percentage Accurate: 84.8% → 99.9%

Time: 9.0s

Alternatives: 10

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: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. remove-double-neg 80.7%

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

   2. sub-neg 80.7%

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

   3. sub-neg 80.7%

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

   4. distribute-neg-frac 80.7%

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

   5. metadata-eval 80.7%

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

   6. metadata-eval 80.7%

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

   7. metadata-eval 80.7%

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

   8. associate-/r* 80.7%

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

   9. metadata-eval 80.7%

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

   10. neg-mul-1 80.7%

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

   11. associate--l+ 80.7%

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

   12. +-commutative 80.7%

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

   13. distribute-neg-frac 80.7%

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

   14. metadata-eval 80.7%

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

   15. metadata-eval 80.7%

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

   16. metadata-eval 80.7%

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

   17. associate-/r* 80.7%

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

   18. metadata-eval 80.7%

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

   19. neg-mul-1 80.7%

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

   20. sub0-neg 80.7%

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

   21. associate-+l- 80.7%

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

   22. neg-sub0 80.7%

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

3. Simplified 80.7%

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

   1. frac-sub 55.5%

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

   2. div-inv 54.0%

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

   3. *-rgt-identity 54.0%

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

5. Applied egg-rr 54.0%

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

   1. associate-/r* 57.4%

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

7. Simplified 57.4%

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

   1. associate-*r/ 80.7%

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

   2. frac-add 79.4%

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

   3. *-un-lft-identity 79.4%

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

   4. div-inv 80.7%

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

9. Applied egg-rr 80.7%

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

10. Taylor expanded in x around 0 99.9%

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

    1. div-inv 99.9%

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

    2. *-un-lft-identity 99.9%

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

    3. times-frac 99.9%

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

    4. metadata-eval 99.9%

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

    5. +-commutative 99.9%

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

12. Applied egg-rr 99.9%

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

    1. associate-*r/ 99.9%

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

    2. metadata-eval 99.9%

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

    3. distribute-lft-neg-in 99.9%

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

    4. associate-*r/ 99.9%

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

    5. metadata-eval 99.9%

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

    6. distribute-neg-frac 99.9%

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

    7. metadata-eval 99.9%

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

    8. *-rgt-identity 99.9%

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

    9. times-frac 99.8%

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

    10. associate-*r/ 99.9%

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

    11. *-rgt-identity 99.9%

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

    12. associate-/l/ 99.9%

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

    13. distribute-lft1-in 99.9%

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

    14. fma-udef 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

    \[\leadsto \frac{\frac{-2}{\mathsf{fma}\left(x, x, x\right)}}{1 - x} \]

Alternative 2: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = (-1.0 / x) + (1.0 / (x + 1.0));
	} else if (x <= 1.0) {
		tmp = (1.0 - x) + ((-2.0 / x) + (-1.0 - x));
	} else {
		tmp = (1.0 / x) + (-1.0 / 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 / (x + 1.0d0))
    else if (x <= 1.0d0) then
        tmp = (1.0d0 - x) + (((-2.0d0) / x) + ((-1.0d0) - x))
    else
        tmp = (1.0d0 / x) + ((-1.0d0) / 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 / (x + 1.0));
	} else if (x <= 1.0) {
		tmp = (1.0 - x) + ((-2.0 / x) + (-1.0 - x));
	} else {
		tmp = (1.0 / x) + (-1.0 / x);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -0.650000000000000022

   1. Initial program 67.9%

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

      1. remove-double-neg 67.9%

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

      2. sub-neg 67.9%

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

      3. sub-neg 67.9%

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

      4. distribute-neg-frac 67.9%

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

      5. metadata-eval 67.9%

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

      6. metadata-eval 67.9%

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

      7. metadata-eval 67.9%

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

      8. associate-/r* 67.9%

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

      9. metadata-eval 67.9%

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

      10. neg-mul-1 67.9%

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

      11. associate--l+ 67.9%

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

      12. +-commutative 67.9%

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

      13. distribute-neg-frac 67.9%

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

      14. metadata-eval 67.9%

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

      15. metadata-eval 67.9%

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

      16. metadata-eval 67.9%

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

      17. associate-/r* 67.9%

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

      18. metadata-eval 67.9%

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

      19. neg-mul-1 67.9%

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

      20. sub0-neg 67.9%

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

      21. associate-+l- 67.9%

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

      22. neg-sub0 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in x around inf 66.8%

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

   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 99.2%

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

      1. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. sub-neg 99.3%

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

   9. Simplified 99.3%

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

   if 1 < x

   1. Initial program 61.8%

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

      1. remove-double-neg 61.8%

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

      2. sub-neg 61.8%

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

      3. sub-neg 61.8%

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

      4. distribute-neg-frac 61.8%

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

      5. metadata-eval 61.8%

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

      6. metadata-eval 61.8%

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

      7. metadata-eval 61.8%

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

      8. associate-/r* 61.8%

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

      9. metadata-eval 61.8%

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

      10. neg-mul-1 61.8%

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

      11. associate--l+ 61.8%

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

      12. +-commutative 61.8%

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

      13. distribute-neg-frac 61.8%

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

      14. metadata-eval 61.8%

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

      15. metadata-eval 61.8%

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

      16. metadata-eval 61.8%

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

      17. associate-/r* 61.8%

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

      18. metadata-eval 61.8%

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

      19. neg-mul-1 61.8%

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

      20. sub0-neg 61.8%

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

      21. associate-+l- 61.8%

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

      22. neg-sub0 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around inf 61.8%

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

   5. Taylor expanded in x around inf 61.8%

      \[\leadsto \color{blue}{\frac{1}{x}} + \frac{-1}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

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

Alternative 3: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -0.650000000000000022

   1. Initial program 67.9%

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

      1. remove-double-neg 67.9%

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

      2. sub-neg 67.9%

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

      3. sub-neg 67.9%

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

      4. distribute-neg-frac 67.9%

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

      5. metadata-eval 67.9%

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

      6. metadata-eval 67.9%

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

      7. metadata-eval 67.9%

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

      8. associate-/r* 67.9%

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

      9. metadata-eval 67.9%

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

      10. neg-mul-1 67.9%

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

      11. associate--l+ 67.9%

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

      12. +-commutative 67.9%

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

      13. distribute-neg-frac 67.9%

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

      14. metadata-eval 67.9%

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

      15. metadata-eval 67.9%

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

      16. metadata-eval 67.9%

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

      17. associate-/r* 67.9%

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

      18. metadata-eval 67.9%

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

      19. neg-mul-1 67.9%

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

      20. sub0-neg 67.9%

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

      21. associate-+l- 67.9%

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

      22. neg-sub0 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in x around inf 66.8%

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

      1. frac-2neg 66.8%

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

      2. metadata-eval 66.8%

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

      3. frac-add 66.8%

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

      4. *-un-lft-identity 66.8%

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

      5. *-commutative 66.8%

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

      6. *-un-lft-identity 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. +-commutative 66.8%

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

      2. unsub-neg 66.8%

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

      3. +-commutative 66.8%

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

      4. *-commutative 66.8%

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

      5. distribute-lft-neg-out 66.8%

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

      6. distribute-rgt-neg-in 66.8%

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

      7. distribute-neg-in 66.8%

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

      8. metadata-eval 66.8%

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

      9. unsub-neg 66.8%

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

   8. Simplified 66.8%

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

   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 99.2%

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

      1. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. sub-neg 99.3%

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

   9. Simplified 99.3%

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

   if 1 < x

   1. Initial program 61.8%

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

      1. remove-double-neg 61.8%

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

      2. sub-neg 61.8%

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

      3. sub-neg 61.8%

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

      4. distribute-neg-frac 61.8%

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

      5. metadata-eval 61.8%

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

      6. metadata-eval 61.8%

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

      7. metadata-eval 61.8%

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

      8. associate-/r* 61.8%

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

      9. metadata-eval 61.8%

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

      10. neg-mul-1 61.8%

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

      11. associate--l+ 61.8%

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

      12. +-commutative 61.8%

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

      13. distribute-neg-frac 61.8%

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

      14. metadata-eval 61.8%

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

      15. metadata-eval 61.8%

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

      16. metadata-eval 61.8%

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

      17. associate-/r* 61.8%

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

      18. metadata-eval 61.8%

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

      19. neg-mul-1 61.8%

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

      20. sub0-neg 61.8%

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

      21. associate-+l- 61.8%

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

      22. neg-sub0 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around inf 61.8%

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

   5. Taylor expanded in x around inf 61.8%

      \[\leadsto \color{blue}{\frac{1}{x}} + \frac{-1}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

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

Alternative 4: 83.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 / x));
	else
		tmp = Float64(Float64(-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 = (1.0 / x) + (-1.0 / x);
	else
		tmp = -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[(1.0 / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[((-x) - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 65.0%

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

      1. remove-double-neg 65.0%

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

      2. sub-neg 65.0%

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

      3. sub-neg 65.0%

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

      4. distribute-neg-frac 65.0%

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

      5. metadata-eval 65.0%

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

      6. metadata-eval 65.0%

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

      7. metadata-eval 65.0%

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

      8. associate-/r* 65.0%

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

      9. metadata-eval 65.0%

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

      10. neg-mul-1 65.0%

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

      11. associate--l+ 65.0%

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

      12. +-commutative 65.0%

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

      13. distribute-neg-frac 65.0%

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

      14. metadata-eval 65.0%

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

      15. metadata-eval 65.0%

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

      16. metadata-eval 65.0%

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

      17. associate-/r* 65.0%

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

      18. metadata-eval 65.0%

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

      19. neg-mul-1 65.0%

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

      20. sub0-neg 65.0%

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

      21. associate-+l- 65.0%

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

      22. neg-sub0 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in x around inf 64.4%

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

   5. Taylor expanded in x around inf 64.3%

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

   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 98.9%

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

      1. distribute-neg-in 98.9%

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

      2. metadata-eval 98.9%

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

      3. unsub-neg 98.9%

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

      4. associate-*r/ 98.9%

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

      5. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in x around 0 98.9%

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

      1. neg-mul-1 98.9%

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

      2. associate-*r/ 98.9%

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

      3. metadata-eval 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 79.8%

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

Alternative 5: 83.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (1.0 / x) + (-1.0 / 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 = (1.0d0 / x) + ((-1.0d0) / 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 = (1.0 / x) + (-1.0 / 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 = (1.0 / x) + (-1.0 / 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(1.0 / x) + Float64(-1.0 / 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 = (1.0 / x) + (-1.0 / 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[(1.0 / x), $MachinePrecision] + N[(-1.0 / 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 65.0%

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

      1. remove-double-neg 65.0%

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

      2. sub-neg 65.0%

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

      3. sub-neg 65.0%

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

      4. distribute-neg-frac 65.0%

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

      5. metadata-eval 65.0%

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

      6. metadata-eval 65.0%

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

      7. metadata-eval 65.0%

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

      8. associate-/r* 65.0%

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

      9. metadata-eval 65.0%

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

      10. neg-mul-1 65.0%

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

      11. associate--l+ 65.0%

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

      12. +-commutative 65.0%

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

      13. distribute-neg-frac 65.0%

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

      14. metadata-eval 65.0%

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

      15. metadata-eval 65.0%

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

      16. metadata-eval 65.0%

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

      17. associate-/r* 65.0%

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

      18. metadata-eval 65.0%

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

      19. neg-mul-1 65.0%

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

      20. sub0-neg 65.0%

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

      21. associate-+l- 65.0%

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

      22. neg-sub0 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in x around inf 64.4%

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

   5. Taylor expanded in x around inf 64.3%

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

   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 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 80.0%

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

Alternative 6: 84.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -0.65) {
		tmp = (-1.0 / x) + (1.0 / (x + 1.0));
	} else if (x <= 1.0) {
		tmp = (-2.0 * x) - (2.0 / x);
	} else {
		tmp = (1.0 / x) + (-1.0 / 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 / (x + 1.0d0))
    else if (x <= 1.0d0) then
        tmp = ((-2.0d0) * x) - (2.0d0 / x)
    else
        tmp = (1.0d0 / x) + ((-1.0d0) / 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 / (x + 1.0));
	} else if (x <= 1.0) {
		tmp = (-2.0 * x) - (2.0 / x);
	} else {
		tmp = (1.0 / x) + (-1.0 / x);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -0.650000000000000022

   1. Initial program 67.9%

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

      1. remove-double-neg 67.9%

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

      2. sub-neg 67.9%

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

      3. sub-neg 67.9%

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

      4. distribute-neg-frac 67.9%

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

      5. metadata-eval 67.9%

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

      6. metadata-eval 67.9%

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

      7. metadata-eval 67.9%

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

      8. associate-/r* 67.9%

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

      9. metadata-eval 67.9%

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

      10. neg-mul-1 67.9%

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

      11. associate--l+ 67.9%

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

      12. +-commutative 67.9%

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

      13. distribute-neg-frac 67.9%

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

      14. metadata-eval 67.9%

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

      15. metadata-eval 67.9%

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

      16. metadata-eval 67.9%

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

      17. associate-/r* 67.9%

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

      18. metadata-eval 67.9%

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

      19. neg-mul-1 67.9%

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

      20. sub0-neg 67.9%

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

      21. associate-+l- 67.9%

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

      22. neg-sub0 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in x around inf 66.8%

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

   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 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   if 1 < x

   1. Initial program 61.8%

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

      1. remove-double-neg 61.8%

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

      2. sub-neg 61.8%

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

      3. sub-neg 61.8%

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

      4. distribute-neg-frac 61.8%

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

      5. metadata-eval 61.8%

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

      6. metadata-eval 61.8%

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

      7. metadata-eval 61.8%

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

      8. associate-/r* 61.8%

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

      9. metadata-eval 61.8%

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

      10. neg-mul-1 61.8%

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

      11. associate--l+ 61.8%

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

      12. +-commutative 61.8%

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

      13. distribute-neg-frac 61.8%

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

      14. metadata-eval 61.8%

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

      15. metadata-eval 61.8%

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

      16. metadata-eval 61.8%

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

      17. associate-/r* 61.8%

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

      18. metadata-eval 61.8%

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

      19. neg-mul-1 61.8%

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

      20. sub0-neg 61.8%

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

      21. associate-+l- 61.8%

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

      22. neg-sub0 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around inf 61.8%

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

   5. Taylor expanded in x around inf 61.8%

      \[\leadsto \color{blue}{\frac{1}{x}} + \frac{-1}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

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

Alternative 7: 99.9% accurate, 1.4× speedup

Math

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

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

Derivation

1. Initial program 80.7%

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

   1. remove-double-neg 80.7%

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

   2. sub-neg 80.7%

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

   3. sub-neg 80.7%

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

   4. distribute-neg-frac 80.7%

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

   5. metadata-eval 80.7%

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

   6. metadata-eval 80.7%

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

   7. metadata-eval 80.7%

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

   8. associate-/r* 80.7%

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

   9. metadata-eval 80.7%

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

   10. neg-mul-1 80.7%

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

   11. associate--l+ 80.7%

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

   12. +-commutative 80.7%

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

   13. distribute-neg-frac 80.7%

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

   14. metadata-eval 80.7%

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

   15. metadata-eval 80.7%

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

   16. metadata-eval 80.7%

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

   17. associate-/r* 80.7%

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

   18. metadata-eval 80.7%

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

   19. neg-mul-1 80.7%

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

   20. sub0-neg 80.7%

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

   21. associate-+l- 80.7%

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

   22. neg-sub0 80.7%

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

3. Simplified 80.7%

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

   1. frac-sub 55.5%

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

   2. div-inv 54.0%

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

   3. *-rgt-identity 54.0%

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

5. Applied egg-rr 54.0%

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

   1. associate-/r* 57.4%

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

7. Simplified 57.4%

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

   1. associate-*r/ 80.7%

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

   2. frac-add 79.4%

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

   3. *-un-lft-identity 79.4%

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

   4. div-inv 80.7%

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

9. Applied egg-rr 80.7%

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

10. Taylor expanded in x around 0 99.9%

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

11. Final simplification 99.9%

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

Alternative 8: 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 80.7%

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

   1. remove-double-neg 80.7%

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

   2. sub-neg 80.7%

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

   3. sub-neg 80.7%

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

   4. distribute-neg-frac 80.7%

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

   5. metadata-eval 80.7%

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

   6. metadata-eval 80.7%

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

   7. metadata-eval 80.7%

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

   8. associate-/r* 80.7%

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

   9. metadata-eval 80.7%

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

   10. neg-mul-1 80.7%

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

   11. associate--l+ 80.7%

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

   12. +-commutative 80.7%

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

   13. distribute-neg-frac 80.7%

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

   14. metadata-eval 80.7%

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

   15. metadata-eval 80.7%

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

   16. metadata-eval 80.7%

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

   17. associate-/r* 80.7%

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

   18. metadata-eval 80.7%

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

   19. neg-mul-1 80.7%

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

   20. sub0-neg 80.7%

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

   21. associate-+l- 80.7%

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

   22. neg-sub0 80.7%

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

3. Simplified 80.7%

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

4. Taylor expanded in x around 0 47.1%

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

5. Final simplification 47.1%

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

Alternative 9: 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 80.7%

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

   1. remove-double-neg 80.7%

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

   2. sub-neg 80.7%

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

   3. sub-neg 80.7%

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

   4. distribute-neg-frac 80.7%

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

   5. metadata-eval 80.7%

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

   6. metadata-eval 80.7%

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

   7. metadata-eval 80.7%

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

   8. associate-/r* 80.7%

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

   9. metadata-eval 80.7%

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

   10. neg-mul-1 80.7%

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

   11. associate--l+ 80.7%

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

   12. +-commutative 80.7%

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

   13. distribute-neg-frac 80.7%

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

   14. metadata-eval 80.7%

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

   15. metadata-eval 80.7%

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

   16. metadata-eval 80.7%

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

   17. associate-/r* 80.7%

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

   18. metadata-eval 80.7%

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

   19. neg-mul-1 80.7%

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

   20. sub0-neg 80.7%

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

   21. associate-+l- 80.7%

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

   22. neg-sub0 80.7%

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

3. Simplified 80.7%

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

4. Taylor expanded in x around 0 46.3%

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

   1. distribute-neg-in 46.3%

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

   2. metadata-eval 46.3%

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

   3. unsub-neg 46.3%

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

   4. associate-*r/ 46.3%

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

   5. metadata-eval 46.3%

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

6. Simplified 46.3%

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

7. Taylor expanded in x around inf 3.3%

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

8. Final simplification 3.3%

   \[\leadsto -1 \]

Alternative 10: 3.3% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 80.7%

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

   1. remove-double-neg 80.7%

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

   2. sub-neg 80.7%

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

   3. sub-neg 80.7%

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

   4. distribute-neg-frac 80.7%

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

   5. metadata-eval 80.7%

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

   6. metadata-eval 80.7%

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

   7. metadata-eval 80.7%

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

   8. associate-/r* 80.7%

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

   9. metadata-eval 80.7%

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

   10. neg-mul-1 80.7%

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

   11. associate--l+ 80.7%

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

   12. +-commutative 80.7%

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

   13. distribute-neg-frac 80.7%

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

   14. metadata-eval 80.7%

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

   15. metadata-eval 80.7%

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

   16. metadata-eval 80.7%

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

   17. associate-/r* 80.7%

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

   18. metadata-eval 80.7%

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

   19. neg-mul-1 80.7%

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

   20. sub0-neg 80.7%

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

   21. associate-+l- 80.7%

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

   22. neg-sub0 80.7%

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

3. Simplified 80.7%

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

   1. frac-sub 55.5%

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

   2. *-rgt-identity 55.5%

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

5. Applied egg-rr 55.5%

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

6. Taylor expanded in x around inf 11.8%

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

7. Taylor expanded in x around 0 3.4%

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

8. Final simplification 3.4%

   \[\leadsto 2 \]

Developer target: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023320 
(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))))