Asymptote A

Percentage Accurate: 77.6% → 99.9%

Time: 6.8s

Alternatives: 6

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{x + -1}}{-1 - x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. sub-neg 77.6%

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

   2. +-commutative 77.6%

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

   3. distribute-neg-frac2 77.6%

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

   4. neg-sub0 77.6%

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

   5. associate-+l- 77.6%

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

   6. neg-sub0 77.6%

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

   7. remove-double-neg 77.6%

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

   8. distribute-neg-in 77.6%

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

   9. sub-neg 77.6%

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

   10. distribute-neg-frac2 77.6%

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

   11. sub-neg 77.6%

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

   12. +-commutative 77.6%

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

   13. unsub-neg 77.6%

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

   14. sub-neg 77.6%

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

   15. +-commutative 77.6%

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

   16. unsub-neg 77.6%

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

   17. metadata-eval 77.6%

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

3. Simplified 77.6%

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

   1. frac-sub 78.9%

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

   2. *-rgt-identity 78.9%

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

   3. metadata-eval 78.9%

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

   4. div-inv 78.9%

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

   5. associate-/r* 78.9%

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

   6. metadata-eval 78.9%

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

   7. div-inv 78.9%

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

   8. *-un-lft-identity 78.9%

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

   9. associate--l- 81.7%

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

   10. div-inv 81.7%

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

   11. metadata-eval 81.7%

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

   12. *-rgt-identity 81.7%

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

   13. div-inv 81.7%

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

   14. metadata-eval 81.7%

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

   15. *-rgt-identity 81.7%

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

6. Applied egg-rr 81.7%

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

   1. +-commutative 81.7%

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

   2. associate-+l- 99.9%

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

8. Applied egg-rr 99.9%

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

   1. div-inv 99.9%

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

   2. +-inverses 99.9%

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

   3. metadata-eval 99.9%

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

   4. metadata-eval 99.9%

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

   5. div-inv 99.9%

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

   6. *-un-lft-identity 99.9%

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

10. Applied egg-rr 99.9%

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

    1. *-lft-identity 99.9%

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

    2. associate-/r* 99.1%

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

    3. metadata-eval 99.1%

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

    4. distribute-neg-frac 99.1%

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

    5. distribute-frac-neg2 99.1%

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

    6. distribute-lft-neg-in 99.1%

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

    7. associate-/r* 99.9%

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

    8. neg-sub0 99.9%

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

    9. associate--r- 99.9%

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

    10. metadata-eval 99.9%

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

    11. +-commutative 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

    \[\leadsto \frac{\frac{2}{x + -1}}{-1 - x} \]
14. Add Preprocessing

Alternative 2: 63.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{-2}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 5.5e+102) (/ -2.0 (+ x -1.0)) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e+102) {
		tmp = -2.0 / (x + -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.5d+102) then
        tmp = (-2.0d0) / (x + (-1.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e+102) {
		tmp = -2.0 / (x + -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e+102:
		tmp = -2.0 / (x + -1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e+102)
		tmp = Float64(-2.0 / Float64(x + -1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e+102)
		tmp = -2.0 / (x + -1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e+102], N[(-2.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 5.49999999999999981e102

   1. Initial program 80.0%

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

      1. sub-neg 80.0%

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

      2. +-commutative 80.0%

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

      3. distribute-neg-frac2 80.0%

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

      4. neg-sub0 80.0%

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

      5. associate-+l- 80.0%

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

      6. neg-sub0 80.0%

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

      7. remove-double-neg 80.0%

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

      8. distribute-neg-in 80.0%

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

      9. sub-neg 80.0%

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

      10. distribute-neg-frac2 80.0%

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

      11. sub-neg 80.0%

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

      12. +-commutative 80.0%

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

      13. unsub-neg 80.0%

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

      14. sub-neg 80.0%

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

      15. +-commutative 80.0%

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

      16. unsub-neg 80.0%

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

      17. metadata-eval 80.0%

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

   3. Simplified 80.0%

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

      1. sub-neg 80.0%

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

      2. distribute-neg-frac 80.0%

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

      3. metadata-eval 80.0%

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

   6. Applied egg-rr 80.0%

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

      1. metadata-eval 80.0%

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

      2. distribute-neg-frac 80.0%

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

      3. unsub-neg 80.0%

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

      4. *-rgt-identity 80.0%

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

      5. *-inverses 80.0%

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

      6. associate-/r* 64.6%

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

      7. *-commutative 64.6%

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

      8. *-lft-identity 64.6%

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

      9. associate-/r* 80.0%

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

      10. associate-*r/ 80.0%

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

      11. associate-*l/ 80.0%

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

      12. distribute-lft-out-- 80.0%

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

      13. *-inverses 80.0%

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

      14. div-sub 81.6%

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

      15. associate--r+ 84.3%

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

      16. *-commutative 84.3%

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

      17. times-frac 84.3%

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

   8. Simplified 99.5%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in x around 0 60.9%

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

   if 5.49999999999999981e102 < x

   1. Initial program 68.7%

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

      1. sub-neg 68.7%

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

      2. +-commutative 68.7%

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

      3. distribute-neg-frac2 68.7%

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

      4. neg-sub0 68.7%

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

      5. associate-+l- 68.7%

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

      6. neg-sub0 68.7%

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

      7. remove-double-neg 68.7%

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

      8. distribute-neg-in 68.7%

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

      9. sub-neg 68.7%

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

      10. distribute-neg-frac2 68.7%

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

      11. sub-neg 68.7%

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

      12. +-commutative 68.7%

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

      13. unsub-neg 68.7%

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

      14. sub-neg 68.7%

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

      15. +-commutative 68.7%

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

      16. unsub-neg 68.7%

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

      17. metadata-eval 68.7%

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

   3. Simplified 68.7%

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

      1. frac-sub 68.7%

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

      2. *-rgt-identity 68.7%

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

      3. metadata-eval 68.7%

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

      4. div-inv 68.7%

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

      5. associate-/r* 68.7%

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

      6. metadata-eval 68.7%

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

      7. div-inv 68.7%

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

      8. *-un-lft-identity 68.7%

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

      9. associate--l- 72.3%

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

      10. div-inv 72.3%

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

      11. metadata-eval 72.3%

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

      12. *-rgt-identity 72.3%

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

      13. div-inv 72.3%

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

      14. metadata-eval 72.3%

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

      15. *-rgt-identity 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. +-commutative 72.3%

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

      2. associate-+l- 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. div-inv 99.9%

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

      2. +-inverses 99.9%

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

      3. metadata-eval 99.9%

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

      4. metadata-eval 99.9%

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

      5. div-inv 99.9%

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

      6. *-un-lft-identity 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-lft-identity 99.9%

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

       2. associate-/r* 97.7%

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

       3. metadata-eval 97.7%

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

       4. distribute-neg-frac 97.7%

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

       5. distribute-frac-neg2 97.7%

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

       6. distribute-lft-neg-in 97.7%

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

       7. associate-/r* 99.9%

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

       8. neg-sub0 99.9%

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

       9. associate--r- 99.9%

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

       10. metadata-eval 99.9%

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

       11. +-commutative 99.9%

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

   12. Simplified 99.9%

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

       1. frac-2neg 99.9%

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

       2. metadata-eval 99.9%

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

       3. +-commutative 99.9%

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

       4. distribute-neg-in 99.9%

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

       5. metadata-eval 99.9%

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

       6. sub-neg 99.9%

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

       7. associate-/l/ 97.7%

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

       8. metadata-eval 97.7%

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

       9. *-commutative 97.7%

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

       10. sub-div 97.7%

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

       11. associate-/r* 97.8%

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

       12. add-sqr-sqrt 97.8%

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

       13. *-commutative 97.8%

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

       14. associate-/r* 99.8%

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

       15. sqrt-unprod 72.3%

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

   14. Applied egg-rr 68.7%

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

       1. +-inverses 68.7%

          \[\leadsto \color{blue}{0} \]

   16. Simplified 68.7%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{-2}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. sub-neg 77.6%

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

   2. +-commutative 77.6%

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

   3. distribute-neg-frac2 77.6%

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

   4. neg-sub0 77.6%

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

   5. associate-+l- 77.6%

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

   6. neg-sub0 77.6%

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

   7. remove-double-neg 77.6%

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

   8. distribute-neg-in 77.6%

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

   9. sub-neg 77.6%

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

   10. distribute-neg-frac2 77.6%

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

   11. sub-neg 77.6%

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

   12. +-commutative 77.6%

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

   13. unsub-neg 77.6%

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

   14. sub-neg 77.6%

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

   15. +-commutative 77.6%

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

   16. unsub-neg 77.6%

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

   17. metadata-eval 77.6%

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

3. Simplified 77.6%

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

   1. sub-neg 77.6%

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

   2. distribute-neg-frac 77.6%

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

   3. metadata-eval 77.6%

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

6. Applied egg-rr 77.6%

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

   1. metadata-eval 77.6%

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

   2. distribute-neg-frac 77.6%

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

   3. unsub-neg 77.6%

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

   4. *-rgt-identity 77.6%

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

   5. *-inverses 77.6%

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

   6. associate-/r* 52.5%

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

   7. *-commutative 52.5%

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

   8. *-lft-identity 52.5%

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

   9. associate-/r* 77.6%

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

   10. associate-*r/ 77.6%

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

   11. associate-*l/ 77.6%

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

   12. distribute-lft-out-- 77.6%

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

   13. *-inverses 77.6%

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

   14. div-sub 78.9%

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

   15. associate--r+ 81.7%

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

   16. *-commutative 81.7%

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

   17. times-frac 81.7%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

   \[\leadsto \frac{-2}{\left(-1 - x\right) \cdot \left(1 - x\right)} \]
10. Add Preprocessing

Alternative 4: 23.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) 1.0 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	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
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 85.8%

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

      1. sub-neg 85.8%

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

      2. +-commutative 85.8%

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

      3. distribute-neg-frac2 85.8%

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

      4. neg-sub0 85.8%

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

      5. associate-+l- 85.8%

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

      6. neg-sub0 85.8%

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

      7. remove-double-neg 85.8%

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

      8. distribute-neg-in 85.8%

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

      9. sub-neg 85.8%

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

      10. distribute-neg-frac2 85.8%

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

      11. sub-neg 85.8%

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

      12. +-commutative 85.8%

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

      13. unsub-neg 85.8%

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

      14. sub-neg 85.8%

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

      15. +-commutative 85.8%

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

      16. unsub-neg 85.8%

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

      17. metadata-eval 85.8%

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

   3. Simplified 85.8%

      \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 65.6%

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

   6. Taylor expanded in x around inf 13.2%

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

   if 1 < x

   1. Initial program 57.2%

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

      1. sub-neg 57.2%

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

      2. +-commutative 57.2%

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

      3. distribute-neg-frac2 57.2%

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

      4. neg-sub0 57.2%

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

      5. associate-+l- 57.2%

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

      6. neg-sub0 57.2%

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

      7. remove-double-neg 57.2%

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

      8. distribute-neg-in 57.2%

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

      9. sub-neg 57.2%

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

      10. distribute-neg-frac2 57.2%

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

      11. sub-neg 57.2%

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

      12. +-commutative 57.2%

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

      13. unsub-neg 57.2%

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

      14. sub-neg 57.2%

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

      15. +-commutative 57.2%

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

      16. unsub-neg 57.2%

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

      17. metadata-eval 57.2%

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

   3. Simplified 57.2%

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

      1. frac-sub 58.5%

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

      2. *-rgt-identity 58.5%

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

      3. metadata-eval 58.5%

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

      4. div-inv 58.5%

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

      5. associate-/r* 58.4%

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

      6. metadata-eval 58.4%

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

      7. div-inv 58.4%

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

      8. *-un-lft-identity 58.4%

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

      9. associate--l- 63.8%

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

      10. div-inv 63.8%

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

      11. metadata-eval 63.8%

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

      12. *-rgt-identity 63.8%

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

      13. div-inv 63.8%

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

      14. metadata-eval 63.8%

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

      15. *-rgt-identity 63.8%

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

   6. Applied egg-rr 63.8%

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

      1. +-commutative 63.8%

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

      2. associate-+l- 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

      2. +-inverses 99.7%

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

      3. metadata-eval 99.7%

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

      4. metadata-eval 99.7%

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

      5. div-inv 99.7%

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

      6. *-un-lft-identity 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. associate-/r* 98.2%

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

       3. metadata-eval 98.2%

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

       4. distribute-neg-frac 98.2%

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

       5. distribute-frac-neg2 98.2%

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

       6. distribute-lft-neg-in 98.2%

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

       7. associate-/r* 99.7%

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

       8. neg-sub0 99.7%

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

       9. associate--r- 99.7%

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

       10. metadata-eval 99.7%

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

       11. +-commutative 99.7%

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

   12. Simplified 99.7%

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

       1. frac-2neg 99.7%

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

       2. metadata-eval 99.7%

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

       3. +-commutative 99.7%

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

       4. distribute-neg-in 99.7%

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

       5. metadata-eval 99.7%

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

       6. sub-neg 99.7%

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

       7. associate-/l/ 98.2%

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

       8. metadata-eval 98.2%

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

       9. *-commutative 98.2%

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

       10. sub-div 98.2%

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

       11. associate-/r* 98.2%

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

       12. add-sqr-sqrt 98.2%

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

       13. *-commutative 98.2%

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

       14. associate-/r* 99.7%

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

       15. sqrt-unprod 74.9%

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

   14. Applied egg-rr 51.9%

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

       1. +-inverses 51.9%

          \[\leadsto \color{blue}{0} \]

   16. Simplified 51.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], 2.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 85.8%

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

      1. sub-neg 85.8%

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

      2. +-commutative 85.8%

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

      3. distribute-neg-frac2 85.8%

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

      4. neg-sub0 85.8%

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

      5. associate-+l- 85.8%

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

      6. neg-sub0 85.8%

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

      7. remove-double-neg 85.8%

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

      8. distribute-neg-in 85.8%

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

      9. sub-neg 85.8%

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

      10. distribute-neg-frac2 85.8%

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

      11. sub-neg 85.8%

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

      12. +-commutative 85.8%

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

      13. unsub-neg 85.8%

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

      14. sub-neg 85.8%

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

      15. +-commutative 85.8%

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

      16. unsub-neg 85.8%

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

      17. metadata-eval 85.8%

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

   3. Simplified 85.8%

      \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.2%

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

   if 1 < x

   1. Initial program 57.2%

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

      1. sub-neg 57.2%

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

      2. +-commutative 57.2%

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

      3. distribute-neg-frac2 57.2%

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

      4. neg-sub0 57.2%

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

      5. associate-+l- 57.2%

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

      6. neg-sub0 57.2%

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

      7. remove-double-neg 57.2%

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

      8. distribute-neg-in 57.2%

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

      9. sub-neg 57.2%

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

      10. distribute-neg-frac2 57.2%

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

      11. sub-neg 57.2%

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

      12. +-commutative 57.2%

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

      13. unsub-neg 57.2%

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

      14. sub-neg 57.2%

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

      15. +-commutative 57.2%

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

      16. unsub-neg 57.2%

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

      17. metadata-eval 57.2%

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

   3. Simplified 57.2%

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

      1. frac-sub 58.5%

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

      2. *-rgt-identity 58.5%

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

      3. metadata-eval 58.5%

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

      4. div-inv 58.5%

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

      5. associate-/r* 58.4%

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

      6. metadata-eval 58.4%

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

      7. div-inv 58.4%

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

      8. *-un-lft-identity 58.4%

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

      9. associate--l- 63.8%

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

      10. div-inv 63.8%

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

      11. metadata-eval 63.8%

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

      12. *-rgt-identity 63.8%

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

      13. div-inv 63.8%

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

      14. metadata-eval 63.8%

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

      15. *-rgt-identity 63.8%

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

   6. Applied egg-rr 63.8%

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

      1. +-commutative 63.8%

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

      2. associate-+l- 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

      2. +-inverses 99.7%

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

      3. metadata-eval 99.7%

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

      4. metadata-eval 99.7%

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

      5. div-inv 99.7%

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

      6. *-un-lft-identity 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. associate-/r* 98.2%

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

       3. metadata-eval 98.2%

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

       4. distribute-neg-frac 98.2%

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

       5. distribute-frac-neg2 98.2%

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

       6. distribute-lft-neg-in 98.2%

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

       7. associate-/r* 99.7%

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

       8. neg-sub0 99.7%

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

       9. associate--r- 99.7%

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

       10. metadata-eval 99.7%

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

       11. +-commutative 99.7%

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

   12. Simplified 99.7%

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

       1. frac-2neg 99.7%

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

       2. metadata-eval 99.7%

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

       3. +-commutative 99.7%

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

       4. distribute-neg-in 99.7%

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

       5. metadata-eval 99.7%

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

       6. sub-neg 99.7%

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

       7. associate-/l/ 98.2%

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

       8. metadata-eval 98.2%

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

       9. *-commutative 98.2%

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

       10. sub-div 98.2%

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

       11. associate-/r* 98.2%

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

       12. add-sqr-sqrt 98.2%

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

       13. *-commutative 98.2%

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

       14. associate-/r* 99.7%

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

       15. sqrt-unprod 74.9%

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

   14. Applied egg-rr 51.9%

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

       1. +-inverses 51.9%

          \[\leadsto \color{blue}{0} \]

   16. Simplified 51.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 77.6%

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

   1. sub-neg 77.6%

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

   2. +-commutative 77.6%

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

   3. distribute-neg-frac2 77.6%

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

   4. neg-sub0 77.6%

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

   5. associate-+l- 77.6%

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

   6. neg-sub0 77.6%

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

   7. remove-double-neg 77.6%

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

   8. distribute-neg-in 77.6%

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

   9. sub-neg 77.6%

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

   10. distribute-neg-frac2 77.6%

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

   11. sub-neg 77.6%

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

   12. +-commutative 77.6%

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

   13. unsub-neg 77.6%

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

   14. sub-neg 77.6%

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

   15. +-commutative 77.6%

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

   16. unsub-neg 77.6%

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

   17. metadata-eval 77.6%

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

3. Simplified 77.6%

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

   1. frac-sub 78.9%

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

   2. *-rgt-identity 78.9%

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

   3. metadata-eval 78.9%

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

   4. div-inv 78.9%

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

   5. associate-/r* 78.9%

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

   6. metadata-eval 78.9%

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

   7. div-inv 78.9%

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

   8. *-un-lft-identity 78.9%

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

   9. associate--l- 81.7%

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

   10. div-inv 81.7%

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

   11. metadata-eval 81.7%

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

   12. *-rgt-identity 81.7%

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

   13. div-inv 81.7%

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

   14. metadata-eval 81.7%

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

   15. *-rgt-identity 81.7%

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

6. Applied egg-rr 81.7%

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

   1. +-commutative 81.7%

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

   2. associate-+l- 99.9%

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

8. Applied egg-rr 99.9%

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

   1. div-inv 99.9%

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

   2. +-inverses 99.9%

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

   3. metadata-eval 99.9%

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

   4. metadata-eval 99.9%

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

   5. div-inv 99.9%

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

   6. *-un-lft-identity 99.9%

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

10. Applied egg-rr 99.9%

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

    1. *-lft-identity 99.9%

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

    2. associate-/r* 99.1%

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

    3. metadata-eval 99.1%

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

    4. distribute-neg-frac 99.1%

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

    5. distribute-frac-neg2 99.1%

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

    6. distribute-lft-neg-in 99.1%

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

    7. associate-/r* 99.9%

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

    8. neg-sub0 99.9%

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

    9. associate--r- 99.9%

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

    10. metadata-eval 99.9%

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

    11. +-commutative 99.9%

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

12. Simplified 99.9%

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

    1. frac-2neg 99.9%

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

    2. metadata-eval 99.9%

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

    3. +-commutative 99.9%

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

    4. distribute-neg-in 99.9%

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

    5. metadata-eval 99.9%

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

    6. sub-neg 99.9%

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

    7. associate-/l/ 99.1%

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

    8. metadata-eval 99.1%

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

    9. *-commutative 99.1%

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

    10. sub-div 99.1%

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

    11. associate-/r* 99.1%

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

    12. add-sqr-sqrt 52.2%

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

    13. *-commutative 52.2%

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

    14. associate-/r* 53.0%

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

    15. sqrt-unprod 43.0%

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

14. Applied egg-rr 29.8%

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

    1. +-inverses 29.8%

       \[\leadsto \color{blue}{0} \]

16. Simplified 29.8%

    \[\leadsto \color{blue}{0} \]

17. Final simplification 29.8%

    \[\leadsto 0 \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))