Asymptote A

Percentage Accurate: 78.1% → 99.9%

Time: 9.3s

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: 78.1% 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 77.7%

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

   1. sub-neg 77.7%

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

   2. +-commutative 77.7%

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

   3. distribute-neg-frac2 77.7%

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

   4. neg-sub0 77.7%

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

   5. associate-+l- 77.7%

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

   6. neg-sub0 77.7%

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

   7. remove-double-neg 77.7%

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

   8. distribute-neg-in 77.7%

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

   9. sub-neg 77.7%

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

   10. distribute-neg-frac2 77.7%

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

   11. sub-neg 77.7%

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

   12. +-commutative 77.7%

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

   13. unsub-neg 77.7%

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

   14. sub-neg 77.7%

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

   15. +-commutative 77.7%

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

   16. unsub-neg 77.7%

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

   17. metadata-eval 77.7%

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

3. Simplified 77.7%

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

   1. frac-sub 78.4%

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

   2. *-rgt-identity 78.4%

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

   3. metadata-eval 78.4%

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

   4. div-inv 78.4%

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

   5. associate-/r* 78.4%

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

   6. metadata-eval 78.4%

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

   7. div-inv 78.4%

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

   8. *-un-lft-identity 78.4%

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

   9. associate--l- 81.3%

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

   10. div-inv 81.3%

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

   11. metadata-eval 81.3%

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

   12. *-rgt-identity 81.3%

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

   13. div-inv 81.3%

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

   14. metadata-eval 81.3%

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

   15. *-rgt-identity 81.3%

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

6. Applied egg-rr 81.3%

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

   1. div-sub 81.3%

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

   2. sub-neg 81.3%

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

   3. frac-2neg 81.3%

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

   4. metadata-eval 81.3%

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

   5. flip-- 81.3%

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

   6. metadata-eval 81.3%

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

   7. metadata-eval 81.3%

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

   8. +-commutative 81.3%

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

   9. distribute-neg-frac2 81.3%

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

   10. mul-1-neg 81.3%

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

   11. +-commutative 81.3%

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

   12. distribute-lft-in 81.3%

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

   13. metadata-eval 81.3%

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

   14. neg-mul-1 81.3%

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

   15. sub-neg 81.3%

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

   16. flip-+ 81.3%

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

   17. +-commutative 81.3%

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

   18. associate-+r- 81.3%

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

   19. +-commutative 81.3%

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

   20. associate--l+ 99.9%

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

8. Applied egg-rr 99.9%

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

   1. distribute-neg-frac2 99.9%

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

   2. +-inverses 99.9%

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

   3. metadata-eval 99.9%

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

   4. neg-sub0 99.9%

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

   5. associate--r- 99.9%

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

   6. metadata-eval 99.9%

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

   7. +-commutative 99.9%

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

   8. count-2 99.9%

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

   9. associate-*r/ 99.9%

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

   10. metadata-eval 99.9%

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

10. Simplified 99.9%

    \[\leadsto \frac{\color{blue}{\frac{2}{x + -1}}}{-1 - x} \]
11. Add Preprocessing

Alternative 2: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 86.6%

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

      1. sub-neg 86.6%

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

      2. +-commutative 86.6%

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

      3. distribute-neg-frac2 86.6%

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

      4. neg-sub0 86.6%

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

      5. associate-+l- 86.6%

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

      6. neg-sub0 86.6%

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

      7. remove-double-neg 86.6%

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

      8. distribute-neg-in 86.6%

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

      9. sub-neg 86.6%

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

      10. distribute-neg-frac2 86.6%

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

      11. sub-neg 86.6%

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

      12. +-commutative 86.6%

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

      13. unsub-neg 86.6%

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

      14. sub-neg 86.6%

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

      15. +-commutative 86.6%

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

      16. unsub-neg 86.6%

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

      17. metadata-eval 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 69.0%

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

   if 1 < x

   1. Initial program 51.6%

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

      1. sub-neg 51.6%

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

      2. +-commutative 51.6%

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

      3. distribute-neg-frac2 51.6%

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

      4. neg-sub0 51.6%

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

      5. associate-+l- 51.6%

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

      6. neg-sub0 51.6%

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

      7. remove-double-neg 51.6%

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

      8. distribute-neg-in 51.6%

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

      9. sub-neg 51.6%

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

      10. distribute-neg-frac2 51.6%

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

      11. sub-neg 51.6%

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

      12. +-commutative 51.6%

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

      13. unsub-neg 51.6%

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

      14. sub-neg 51.6%

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

      15. +-commutative 51.6%

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

      16. unsub-neg 51.6%

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

      17. metadata-eval 51.6%

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

   3. Simplified 51.6%

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

      1. frac-sub 52.9%

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

      2. *-rgt-identity 52.9%

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

      3. metadata-eval 52.9%

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

      4. div-inv 52.9%

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

      5. associate-/r* 52.9%

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

      6. metadata-eval 52.9%

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

      7. div-inv 52.9%

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

      8. *-un-lft-identity 52.9%

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

      9. associate--l- 59.4%

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

      10. div-inv 59.4%

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

      11. metadata-eval 59.4%

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

      12. *-rgt-identity 59.4%

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

      13. div-inv 59.4%

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

      14. metadata-eval 59.4%

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

      15. *-rgt-identity 59.4%

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

   6. Applied egg-rr 59.4%

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

      1. div-sub 59.4%

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

      2. sub-neg 59.4%

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

      3. frac-2neg 59.4%

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

      4. metadata-eval 59.4%

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

      5. flip-- 59.4%

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

      6. metadata-eval 59.4%

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

      7. metadata-eval 59.4%

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

      8. +-commutative 59.4%

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

      9. distribute-neg-frac2 59.4%

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

      10. mul-1-neg 59.4%

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

      11. +-commutative 59.4%

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

      12. distribute-lft-in 59.4%

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

      13. metadata-eval 59.4%

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

      14. neg-mul-1 59.4%

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

      15. sub-neg 59.4%

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

      16. flip-+ 59.4%

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

      17. +-commutative 59.4%

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

      18. associate-+r- 59.4%

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

      19. +-commutative 59.4%

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

      20. associate--l+ 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. distribute-neg-frac2 99.8%

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

      2. +-inverses 99.8%

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

      3. metadata-eval 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate--r- 99.8%

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

      6. metadata-eval 99.8%

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

      7. +-commutative 99.8%

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

      8. count-2 99.8%

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

      9. associate-*r/ 99.8%

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

      10. metadata-eval 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 98.6%

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

       1. neg-mul-1 98.6%

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

   13. Simplified 98.6%

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

   14. Taylor expanded in x around inf 99.2%

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

4. Final simplification 76.7%

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

Alternative 3: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. sub-neg 77.7%

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

   2. +-commutative 77.7%

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

   3. distribute-neg-frac2 77.7%

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

   4. neg-sub0 77.7%

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

   5. associate-+l- 77.7%

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

   6. neg-sub0 77.7%

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

   7. remove-double-neg 77.7%

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

   8. distribute-neg-in 77.7%

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

   9. sub-neg 77.7%

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

   10. distribute-neg-frac2 77.7%

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

   11. sub-neg 77.7%

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

   12. +-commutative 77.7%

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

   13. unsub-neg 77.7%

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

   14. sub-neg 77.7%

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

   15. +-commutative 77.7%

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

   16. unsub-neg 77.7%

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

   17. metadata-eval 77.7%

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

3. Simplified 77.7%

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

   1. sub-neg 77.7%

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

   2. distribute-neg-frac 77.7%

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

   3. metadata-eval 77.7%

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

6. Applied egg-rr 77.7%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 4: 52.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 86.6%

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

      1. sub-neg 86.6%

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

      2. +-commutative 86.6%

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

      3. distribute-neg-frac2 86.6%

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

      4. neg-sub0 86.6%

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

      5. associate-+l- 86.6%

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

      6. neg-sub0 86.6%

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

      7. remove-double-neg 86.6%

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

      8. distribute-neg-in 86.6%

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

      9. sub-neg 86.6%

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

      10. distribute-neg-frac2 86.6%

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

      11. sub-neg 86.6%

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

      12. +-commutative 86.6%

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

      13. unsub-neg 86.6%

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

      14. sub-neg 86.6%

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

      15. +-commutative 86.6%

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

      16. unsub-neg 86.6%

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

      17. metadata-eval 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 69.0%

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

   if 1 < x

   1. Initial program 51.6%

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

      1. sub-neg 51.6%

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

      2. +-commutative 51.6%

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

      3. distribute-neg-frac2 51.6%

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

      4. neg-sub0 51.6%

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

      5. associate-+l- 51.6%

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

      6. neg-sub0 51.6%

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

      7. remove-double-neg 51.6%

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

      8. distribute-neg-in 51.6%

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

      9. sub-neg 51.6%

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

      10. distribute-neg-frac2 51.6%

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

      11. sub-neg 51.6%

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

      12. +-commutative 51.6%

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

      13. unsub-neg 51.6%

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

      14. sub-neg 51.6%

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

      15. +-commutative 51.6%

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

      16. unsub-neg 51.6%

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

      17. metadata-eval 51.6%

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

   3. Simplified 51.6%

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

      1. sub-neg 51.6%

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

      2. distribute-neg-frac 51.6%

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

      3. metadata-eval 51.6%

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

   6. Applied egg-rr 51.6%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in x around inf 96.8%

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

   10. Taylor expanded in x around 0 6.7%

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

Alternative 5: 51.2% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 77.7%

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

   1. sub-neg 77.7%

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

   2. +-commutative 77.7%

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

   3. distribute-neg-frac2 77.7%

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

   4. neg-sub0 77.7%

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

   5. associate-+l- 77.7%

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

   6. neg-sub0 77.7%

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

   7. remove-double-neg 77.7%

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

   8. distribute-neg-in 77.7%

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

   9. sub-neg 77.7%

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

   10. distribute-neg-frac2 77.7%

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

   11. sub-neg 77.7%

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

   12. +-commutative 77.7%

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

   13. unsub-neg 77.7%

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

   14. sub-neg 77.7%

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

   15. +-commutative 77.7%

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

   16. unsub-neg 77.7%

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

   17. metadata-eval 77.7%

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

3. Simplified 77.7%

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

5. Taylor expanded in x around 0 52.2%

   \[\leadsto \color{blue}{2} \]
6. Add Preprocessing

Alternative 6: 10.8% 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 77.7%

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

   1. sub-neg 77.7%

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

   2. +-commutative 77.7%

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

   3. distribute-neg-frac2 77.7%

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

   4. neg-sub0 77.7%

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

   5. associate-+l- 77.7%

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

   6. neg-sub0 77.7%

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

   7. remove-double-neg 77.7%

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

   8. distribute-neg-in 77.7%

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

   9. sub-neg 77.7%

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

   10. distribute-neg-frac2 77.7%

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

   11. sub-neg 77.7%

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

   12. +-commutative 77.7%

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

   13. unsub-neg 77.7%

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

   14. sub-neg 77.7%

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

   15. +-commutative 77.7%

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

   16. unsub-neg 77.7%

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

   17. metadata-eval 77.7%

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

3. Simplified 77.7%

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

5. Taylor expanded in x around 0 51.6%

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

6. Taylor expanded in x around inf 10.9%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))