Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. remove-double-neg 100.0%

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

   2. distribute-frac-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. sub-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. neg-mul-1 100.0%

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

   7. metadata-eval 100.0%

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

   8. *-commutative 100.0%

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

   9. associate-/l* 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. metadata-eval 100.0%

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

   13. associate-/l* 100.0%

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

   14. associate-*l/ 100.0%

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

   15. /-rgt-identity 100.0%

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

   16. distribute-rgt1-in 100.0%

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

   17. *-commutative 100.0%

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

   18. metadata-eval 100.0%

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

   19. neg-mul-1 100.0%

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

   20. unsub-neg 100.0%

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

   21. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. frac-sub 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. *-commutative 100.0%

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

   5. *-un-lft-identity 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

   1. neg-mul-1 100.0%

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

   2. +-commutative 100.0%

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

   3. unsub-neg 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 100.0%

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

    1. sub-neg 100.0%

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

    2. remove-double-neg 100.0%

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

    3. neg-sub0 100.0%

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

    4. sub-neg 100.0%

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

    5. metadata-eval 100.0%

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

    6. remove-double-neg 100.0%

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

    7. associate-+r+ 100.0%

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

    8. +-commutative 100.0%

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

    9. neg-mul-1 100.0%

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

    10. +-commutative 100.0%

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

    11. distribute-neg-in 100.0%

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

    12. metadata-eval 100.0%

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

    13. distribute-neg-frac 100.0%

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

    14. metadata-eval 100.0%

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

    15. associate-+r+ 100.0%

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

    16. +-commutative 100.0%

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

    17. +-commutative 100.0%

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

    18. associate-+l+ 100.0%

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

    19. metadata-eval 100.0%

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

    20. distribute-neg-in 100.0%

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

    21. sub-neg 100.0%

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

    22. +-commutative 100.0%

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

    23. associate-+r+ 100.0%

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

    24. metadata-eval 100.0%

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

    25. remove-double-neg 100.0%

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

    26. sub-neg 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

    \[\leadsto \frac{\frac{-1}{x} - x}{\frac{1}{x} - x} \]

Alternative 2: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-x}{\frac{1}{x} - x}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - x\right) - \frac{x}{-1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.6) (not (<= x 0.76)))
   (/ (- x) (- (/ 1.0 x) x))
   (- (- -1.0 x) (/ x (- -1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.6) || !(x <= 0.76)) {
		tmp = -x / ((1.0 / x) - x);
	} else {
		tmp = (-1.0 - x) - (x / (-1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.6d0)) .or. (.not. (x <= 0.76d0))) then
        tmp = -x / ((1.0d0 / x) - x)
    else
        tmp = ((-1.0d0) - x) - (x / ((-1.0d0) - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.6) || !(x <= 0.76)) {
		tmp = -x / ((1.0 / x) - x);
	} else {
		tmp = (-1.0 - x) - (x / (-1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.6) or not (x <= 0.76):
		tmp = -x / ((1.0 / x) - x)
	else:
		tmp = (-1.0 - x) - (x / (-1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.6) || !(x <= 0.76))
		tmp = Float64(Float64(-x) / Float64(Float64(1.0 / x) - x));
	else
		tmp = Float64(Float64(-1.0 - x) - Float64(x / Float64(-1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.6) || ~((x <= 0.76)))
		tmp = -x / ((1.0 / x) - x);
	else
		tmp = (-1.0 - x) - (x / (-1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.6], N[Not[LessEqual[x, 0.76]], $MachinePrecision]], N[((-x) / N[(N[(1.0 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 - x), $MachinePrecision] - N[(x / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001 or 0.76000000000000001 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. +-commutative 100.0%

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

      3. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

       1. sub-neg 100.0%

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

       2. remove-double-neg 100.0%

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

       3. neg-sub0 100.0%

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

       4. sub-neg 100.0%

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

       5. metadata-eval 100.0%

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

       6. remove-double-neg 100.0%

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

       7. associate-+r+ 100.0%

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

       8. +-commutative 100.0%

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

       9. neg-mul-1 100.0%

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

       10. +-commutative 100.0%

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

       11. distribute-neg-in 100.0%

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

       12. metadata-eval 100.0%

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

       13. distribute-neg-frac 100.0%

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

       14. metadata-eval 100.0%

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

       15. associate-+r+ 100.0%

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

       16. +-commutative 100.0%

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

       17. +-commutative 100.0%

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

       18. associate-+l+ 100.0%

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

       19. metadata-eval 100.0%

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

       20. distribute-neg-in 100.0%

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

       21. sub-neg 100.0%

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

       22. +-commutative 100.0%

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

       23. associate-+r+ 100.0%

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

       24. metadata-eval 100.0%

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

       25. remove-double-neg 100.0%

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

       26. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 98.6%

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

       1. neg-mul-1 98.6%

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

   14. Simplified 98.6%

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

   if -1.6000000000000001 < x < 0.76000000000000001

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. neg-mul-1 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-commutative 99.3%

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

      5. sub-neg 99.3%

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

   6. Simplified 99.3%

      \[\leadsto \color{blue}{\left(-1 - x\right)} - \frac{x}{-1 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-x}{\frac{1}{x} - x}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - x\right) - \frac{x}{-1 - x}\\ \end{array} \]

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-1 - x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} - t_0\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;\left(-1 - x\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{1}{x} - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (- -1.0 x))))
   (if (<= x -1.0)
     (- (/ 1.0 x) t_0)
     (if (<= x 0.76) (- (- -1.0 x) t_0) (/ (- x) (- (/ 1.0 x) x))))))

C

double code(double x) {
	double t_0 = x / (-1.0 - x);
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) - t_0;
	} else if (x <= 0.76) {
		tmp = (-1.0 - x) - t_0;
	} else {
		tmp = -x / ((1.0 / x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / ((-1.0d0) - x)
    if (x <= (-1.0d0)) then
        tmp = (1.0d0 / x) - t_0
    else if (x <= 0.76d0) then
        tmp = ((-1.0d0) - x) - t_0
    else
        tmp = -x / ((1.0d0 / x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x / (-1.0 - x);
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) - t_0;
	} else if (x <= 0.76) {
		tmp = (-1.0 - x) - t_0;
	} else {
		tmp = -x / ((1.0 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / (-1.0 - x)
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) - t_0
	elif x <= 0.76:
		tmp = (-1.0 - x) - t_0
	else:
		tmp = -x / ((1.0 / x) - x)
	return tmp

Julia

function code(x)
	t_0 = Float64(x / Float64(-1.0 - x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) - t_0);
	elseif (x <= 0.76)
		tmp = Float64(Float64(-1.0 - x) - t_0);
	else
		tmp = Float64(Float64(-x) / Float64(Float64(1.0 / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / (-1.0 - x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (1.0 / x) - t_0;
	elseif (x <= 0.76)
		tmp = (-1.0 - x) - t_0;
	else
		tmp = -x / ((1.0 / x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.76], N[(N[(-1.0 - x), $MachinePrecision] - t$95$0), $MachinePrecision], N[((-x) / N[(N[(1.0 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.7%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. neg-mul-1 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-commutative 99.3%

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

      5. sub-neg 99.3%

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

   6. Simplified 99.3%

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

   if 0.76000000000000001 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. +-commutative 100.0%

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

      3. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

       1. sub-neg 100.0%

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

       2. remove-double-neg 100.0%

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

       3. neg-sub0 100.0%

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

       4. sub-neg 100.0%

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

       5. metadata-eval 100.0%

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

       6. remove-double-neg 100.0%

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

       7. associate-+r+ 100.0%

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

       8. +-commutative 100.0%

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

       9. neg-mul-1 100.0%

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

       10. +-commutative 100.0%

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

       11. distribute-neg-in 100.0%

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

       12. metadata-eval 100.0%

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

       13. distribute-neg-frac 100.0%

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

       14. metadata-eval 100.0%

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

       15. associate-+r+ 100.0%

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

       16. +-commutative 100.0%

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

       17. +-commutative 100.0%

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

       18. associate-+l+ 100.0%

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

       19. metadata-eval 100.0%

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

       20. distribute-neg-in 100.0%

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

       21. sub-neg 100.0%

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

       22. +-commutative 100.0%

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

       23. associate-+r+ 100.0%

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

       24. metadata-eval 100.0%

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

       25. remove-double-neg 100.0%

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

       26. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 99.6%

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

       1. neg-mul-1 99.6%

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

   14. Simplified 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} - \frac{x}{-1 - x}\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;\left(-1 - x\right) - \frac{x}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{1}{x} - x}\\ \end{array} \]

Alternative 4: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.75 \lor \neg \left(x \leq 1.62\right):\\ \;\;\;\;\frac{-x}{\frac{1}{x} - x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.75) (not (<= x 1.62)))
   (/ (- x) (- (/ 1.0 x) x))
   (+ x (/ 1.0 (+ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.75) || !(x <= 1.62)) {
		tmp = -x / ((1.0 / x) - x);
	} else {
		tmp = x + (1.0 / (-1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.75d0)) .or. (.not. (x <= 1.62d0))) then
        tmp = -x / ((1.0d0 / x) - x)
    else
        tmp = x + (1.0d0 / ((-1.0d0) + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.75) || !(x <= 1.62)) {
		tmp = -x / ((1.0 / x) - x);
	} else {
		tmp = x + (1.0 / (-1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.75) or not (x <= 1.62):
		tmp = -x / ((1.0 / x) - x)
	else:
		tmp = x + (1.0 / (-1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.75) || !(x <= 1.62))
		tmp = Float64(Float64(-x) / Float64(Float64(1.0 / x) - x));
	else
		tmp = Float64(x + Float64(1.0 / Float64(-1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.75) || ~((x <= 1.62)))
		tmp = -x / ((1.0 / x) - x);
	else
		tmp = x + (1.0 / (-1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.75], N[Not[LessEqual[x, 1.62]], $MachinePrecision]], N[((-x) / N[(N[(1.0 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.75 or 1.6200000000000001 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-sub 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. +-commutative 100.0%

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

      3. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

       1. sub-neg 100.0%

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

       2. remove-double-neg 100.0%

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

       3. neg-sub0 100.0%

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

       4. sub-neg 100.0%

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

       5. metadata-eval 100.0%

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

       6. remove-double-neg 100.0%

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

       7. associate-+r+ 100.0%

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

       8. +-commutative 100.0%

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

       9. neg-mul-1 100.0%

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

       10. +-commutative 100.0%

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

       11. distribute-neg-in 100.0%

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

       12. metadata-eval 100.0%

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

       13. distribute-neg-frac 100.0%

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

       14. metadata-eval 100.0%

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

       15. associate-+r+ 100.0%

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

       16. +-commutative 100.0%

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

       17. +-commutative 100.0%

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

       18. associate-+l+ 100.0%

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

       19. metadata-eval 100.0%

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

       20. distribute-neg-in 100.0%

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

       21. sub-neg 100.0%

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

       22. +-commutative 100.0%

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

       23. associate-+r+ 100.0%

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

       24. metadata-eval 100.0%

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

       25. remove-double-neg 100.0%

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

       26. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 98.6%

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

       1. neg-mul-1 98.6%

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

   14. Simplified 98.6%

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

   if -0.75 < x < 1.6200000000000001

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

   6. Simplified 99.3%

      \[\leadsto \frac{1}{x + -1} - \color{blue}{\left(-x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.75 \lor \neg \left(x \leq 1.62\right):\\ \;\;\;\;\frac{-x}{\frac{1}{x} - x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \end{array} \]

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.9) (+ x (/ 1.0 (+ -1.0 x))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.9) {
		tmp = x + (1.0 / (-1.0 + x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.9d0) then
        tmp = x + (1.0d0 / ((-1.0d0) + x))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.9) {
		tmp = x + (1.0 / (-1.0 + x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.9:
		tmp = x + (1.0 / (-1.0 + x))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.9)
		tmp = Float64(x + Float64(1.0 / Float64(-1.0 + x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.9)
		tmp = x + (1.0 / (-1.0 + x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.9], N[(x + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.8999999999999999 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.5%

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

   if -1 < x < 1.8999999999999999

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

   6. Simplified 99.3%

      \[\leadsto \frac{1}{x + -1} - \color{blue}{\left(-x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{1}{x - 1} + \frac{x}{x + 1} \]

2. Final simplification 100.0%

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

Alternative 7: 98.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. metadata-eval 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-*l/ 100.0%

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

      15. /-rgt-identity 100.0%

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

      16. distribute-rgt1-in 100.0%

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

      17. *-commutative 100.0%

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

      18. metadata-eval 100.0%

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

      19. neg-mul-1 100.0%

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

      20. unsub-neg 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

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

Alternative 8: 50.0% accurate, 11.0× 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 100.0%

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

   1. remove-double-neg 100.0%

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

   2. distribute-frac-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. sub-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. neg-mul-1 100.0%

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

   7. metadata-eval 100.0%

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

   8. *-commutative 100.0%

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

   9. associate-/l* 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. metadata-eval 100.0%

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

   13. associate-/l* 100.0%

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

   14. associate-*l/ 100.0%

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

   15. /-rgt-identity 100.0%

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

   16. distribute-rgt1-in 100.0%

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

   17. *-commutative 100.0%

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

   18. metadata-eval 100.0%

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

   19. neg-mul-1 100.0%

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

   20. unsub-neg 100.0%

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

   21. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 55.8%

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

5. Final simplification 55.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))