Asymptote A

Percentage Accurate: 78.2% → 99.9%

Time: 7.6s

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: 78.2% 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

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

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

Derivation

1. Initial program 74.8%

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

   1. sub-neg 74.8%

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

   2. +-commutative 74.8%

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

   3. distribute-neg-frac2 74.8%

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

   4. neg-sub0 74.8%

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

   5. associate-+l- 74.8%

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

   6. neg-sub0 74.8%

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

   7. remove-double-neg 74.8%

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

   8. distribute-neg-in 74.8%

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

   9. sub-neg 74.8%

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

   10. distribute-neg-frac2 74.8%

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

   11. sub-neg 74.8%

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

   12. +-commutative 74.8%

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

   13. unsub-neg 74.8%

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

   14. sub-neg 74.8%

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

   15. +-commutative 74.8%

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

   16. unsub-neg 74.8%

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

   17. metadata-eval 74.8%

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

3. Simplified 74.8%

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

   1. sub-neg 74.8%

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

   2. distribute-neg-frac 74.8%

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

   3. metadata-eval 74.8%

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

6. Applied egg-rr 74.8%

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

   1. metadata-eval 74.8%

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

   2. distribute-neg-frac 74.8%

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

   3. unsub-neg 74.8%

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

   4. *-rgt-identity 74.8%

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

   5. *-inverses 74.8%

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

   6. associate-/r* 50.9%

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

   7. *-commutative 50.9%

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

   8. *-lft-identity 50.9%

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

   9. associate-/r* 74.8%

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

   10. associate-*r/ 74.8%

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

   11. associate-*l/ 74.8%

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

   12. distribute-lft-out-- 74.8%

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

   13. *-inverses 74.8%

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

   14. div-sub 75.3%

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

   15. associate--r+ 78.6%

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

   16. *-commutative 78.6%

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

   17. times-frac 78.5%

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

8. Simplified 99.4%

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

   1. expm1-log1p-u 98.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)\right)} \]

   2. expm1-undefine 73.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} - 1} \]

10. Applied egg-rr 73.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} - 1} \]
11. Step-by-step derivation

    1. sub-neg 73.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right)} \]

    2. metadata-eval 73.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-1 - 1}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

    3. metadata-eval 73.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{-1 - \color{blue}{\left(1 + 0\right)}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

    4. +-inverses 73.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{-1 - \left(1 + \color{blue}{\left(x - x\right)}\right)}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

    5. associate--l+ 73.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{-1 - \color{blue}{\left(\left(1 + x\right) - x\right)}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

    6. associate-+l- 73.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(-1 - \left(1 + x\right)\right) + x}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

    7. metadata-eval 73.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\left(-1 - \left(1 + x\right)\right) + x}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \color{blue}{-1} \]

    8. +-commutative 73.3%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{\left(-1 - \left(1 + x\right)\right) + x}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)}} \]

    9. log1p-undefine 73.7%

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

    10. rem-exp-log 74.5%

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

    11. associate-+r+ 75.3%

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

    12. metadata-eval 75.3%

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

    13. +-lft-identity 75.3%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.76) 2.0 (/ 2.0 (* x (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = 2.0 / (x * (1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.76d0) then
        tmp = 2.0d0
    else
        tmp = 2.0d0 / (x * (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = 2.0 / (x * (1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.76:
		tmp = 2.0
	else:
		tmp = 2.0 / (x * (1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.76)
		tmp = 2.0;
	else
		tmp = Float64(2.0 / Float64(x * Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.76)
		tmp = 2.0;
	else
		tmp = 2.0 / (x * (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.76], 2.0, N[(2.0 / N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.76000000000000001

   1. Initial program 88.2%

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

      1. sub-neg 88.2%

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

      2. +-commutative 88.2%

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

      3. distribute-neg-frac2 88.2%

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

      4. neg-sub0 88.2%

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

      5. associate-+l- 88.2%

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

      6. neg-sub0 88.2%

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

      7. remove-double-neg 88.2%

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

      8. distribute-neg-in 88.2%

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

      9. sub-neg 88.2%

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

      10. distribute-neg-frac2 88.2%

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

      11. sub-neg 88.2%

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

      12. +-commutative 88.2%

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

      13. unsub-neg 88.2%

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

      14. sub-neg 88.2%

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

      15. +-commutative 88.2%

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

      16. unsub-neg 88.2%

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

      17. metadata-eval 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in x around 0 67.2%

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

   if 0.76000000000000001 < x

   1. Initial program 44.1%

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

      1. sub-neg 44.1%

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

      2. +-commutative 44.1%

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

      3. distribute-neg-frac2 44.1%

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

      4. neg-sub0 44.1%

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

      5. associate-+l- 44.1%

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

      6. neg-sub0 44.1%

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

      7. remove-double-neg 44.1%

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

      8. distribute-neg-in 44.1%

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

      9. sub-neg 44.1%

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

      10. distribute-neg-frac2 44.1%

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

      11. sub-neg 44.1%

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

      12. +-commutative 44.1%

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

      13. unsub-neg 44.1%

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

      14. sub-neg 44.1%

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

      15. +-commutative 44.1%

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

      16. unsub-neg 44.1%

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

      17. metadata-eval 44.1%

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

   3. Simplified 44.1%

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

      1. sub-neg 44.1%

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

      2. distribute-neg-frac 44.1%

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

      3. metadata-eval 44.1%

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

   6. Applied egg-rr 44.1%

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

      1. metadata-eval 44.1%

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

      2. distribute-neg-frac 44.1%

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

      3. unsub-neg 44.1%

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

      4. *-rgt-identity 44.1%

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

      5. *-inverses 44.1%

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

      6. associate-/r* 7.2%

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

      7. *-commutative 7.2%

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

      8. *-lft-identity 7.2%

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

      9. associate-/r* 44.1%

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

      10. associate-*r/ 44.1%

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

      11. associate-*l/ 44.1%

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

      12. distribute-lft-out-- 44.1%

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

      13. *-inverses 44.1%

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

      14. div-sub 44.6%

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

      15. associate--r+ 51.9%

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

      16. *-commutative 51.9%

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

      17. times-frac 51.9%

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

   8. Simplified 98.6%

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

      1. expm1-log1p-u 98.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)\right)} \]

      2. expm1-undefine 43.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} - 1} \]

   10. Applied egg-rr 43.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 43.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right)} \]

       2. metadata-eval 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-1 - 1}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

       3. metadata-eval 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-1 - \color{blue}{\left(1 + 0\right)}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

       4. +-inverses 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-1 - \left(1 + \color{blue}{\left(x - x\right)}\right)}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

       5. associate--l+ 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-1 - \color{blue}{\left(\left(1 + x\right) - x\right)}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

       6. associate-+l- 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(-1 - \left(1 + x\right)\right) + x}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

       7. metadata-eval 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\left(-1 - \left(1 + x\right)\right) + x}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \color{blue}{-1} \]

       8. +-commutative 43.6%

          \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{\left(-1 - \left(1 + x\right)\right) + x}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)}} \]

       9. log1p-undefine 43.6%

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

       10. rem-exp-log 43.6%

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

       11. associate-+r+ 44.6%

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

       12. metadata-eval 44.6%

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

       13. +-lft-identity 44.6%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in x around inf 99.2%

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

       1. div-inv 99.2%

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

       2. *-un-lft-identity 99.2%

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

       3. times-frac 99.2%

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

       4. metadata-eval 99.2%

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

   15. Applied egg-rr 99.2%

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

       1. associate-/l/ 98.0%

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

       2. associate-*r/ 98.0%

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

       3. metadata-eval 98.0%

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

       4. *-commutative 98.0%

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

   17. Simplified 98.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.76) 2.0 (/ (/ 2.0 x) (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = (2.0 / x) / (1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.76d0) then
        tmp = 2.0d0
    else
        tmp = (2.0d0 / x) / (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.76) {
		tmp = 2.0;
	} else {
		tmp = (2.0 / x) / (1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.76:
		tmp = 2.0
	else:
		tmp = (2.0 / x) / (1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.76)
		tmp = 2.0;
	else
		tmp = Float64(Float64(2.0 / x) / Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.76)
		tmp = 2.0;
	else
		tmp = (2.0 / x) / (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.76], 2.0, N[(N[(2.0 / x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.76000000000000001

   1. Initial program 88.2%

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

      1. sub-neg 88.2%

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

      2. +-commutative 88.2%

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

      3. distribute-neg-frac2 88.2%

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

      4. neg-sub0 88.2%

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

      5. associate-+l- 88.2%

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

      6. neg-sub0 88.2%

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

      7. remove-double-neg 88.2%

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

      8. distribute-neg-in 88.2%

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

      9. sub-neg 88.2%

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

      10. distribute-neg-frac2 88.2%

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

      11. sub-neg 88.2%

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

      12. +-commutative 88.2%

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

      13. unsub-neg 88.2%

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

      14. sub-neg 88.2%

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

      15. +-commutative 88.2%

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

      16. unsub-neg 88.2%

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

      17. metadata-eval 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in x around 0 67.2%

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

   if 0.76000000000000001 < x

   1. Initial program 44.1%

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

      1. sub-neg 44.1%

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

      2. +-commutative 44.1%

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

      3. distribute-neg-frac2 44.1%

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

      4. neg-sub0 44.1%

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

      5. associate-+l- 44.1%

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

      6. neg-sub0 44.1%

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

      7. remove-double-neg 44.1%

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

      8. distribute-neg-in 44.1%

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

      9. sub-neg 44.1%

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

      10. distribute-neg-frac2 44.1%

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

      11. sub-neg 44.1%

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

      12. +-commutative 44.1%

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

      13. unsub-neg 44.1%

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

      14. sub-neg 44.1%

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

      15. +-commutative 44.1%

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

      16. unsub-neg 44.1%

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

      17. metadata-eval 44.1%

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

   3. Simplified 44.1%

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

      1. sub-neg 44.1%

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

      2. distribute-neg-frac 44.1%

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

      3. metadata-eval 44.1%

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

   6. Applied egg-rr 44.1%

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

      1. metadata-eval 44.1%

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

      2. distribute-neg-frac 44.1%

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

      3. unsub-neg 44.1%

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

      4. *-rgt-identity 44.1%

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

      5. *-inverses 44.1%

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

      6. associate-/r* 7.2%

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

      7. *-commutative 7.2%

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

      8. *-lft-identity 7.2%

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

      9. associate-/r* 44.1%

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

      10. associate-*r/ 44.1%

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

      11. associate-*l/ 44.1%

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

      12. distribute-lft-out-- 44.1%

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

      13. *-inverses 44.1%

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

      14. div-sub 44.6%

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

      15. associate--r+ 51.9%

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

      16. *-commutative 51.9%

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

      17. times-frac 51.9%

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

   8. Simplified 98.6%

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

      1. expm1-log1p-u 98.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)\right)} \]

      2. expm1-undefine 43.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} - 1} \]

   10. Applied egg-rr 43.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 43.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right)} \]

       2. metadata-eval 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-1 - 1}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

       3. metadata-eval 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-1 - \color{blue}{\left(1 + 0\right)}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

       4. +-inverses 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-1 - \left(1 + \color{blue}{\left(x - x\right)}\right)}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

       5. associate--l+ 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-1 - \color{blue}{\left(\left(1 + x\right) - x\right)}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

       6. associate-+l- 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(-1 - \left(1 + x\right)\right) + x}}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \left(-1\right) \]

       7. metadata-eval 43.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\left(-1 - \left(1 + x\right)\right) + x}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)} + \color{blue}{-1} \]

       8. +-commutative 43.6%

          \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{\left(-1 - \left(1 + x\right)\right) + x}{\left(1 - x\right) \cdot \left(-1 - x\right)}\right)}} \]

       9. log1p-undefine 43.6%

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

       10. rem-exp-log 43.6%

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

       11. associate-+r+ 44.6%

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

       12. metadata-eval 44.6%

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

       13. +-lft-identity 44.6%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in x around inf 99.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.76:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{1 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% 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 74.8%

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

   1. sub-neg 74.8%

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

   2. +-commutative 74.8%

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

   3. distribute-neg-frac2 74.8%

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

   4. neg-sub0 74.8%

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

   5. associate-+l- 74.8%

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

   6. neg-sub0 74.8%

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

   7. remove-double-neg 74.8%

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

   8. distribute-neg-in 74.8%

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

   9. sub-neg 74.8%

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

   10. distribute-neg-frac2 74.8%

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

   11. sub-neg 74.8%

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

   12. +-commutative 74.8%

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

   13. unsub-neg 74.8%

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

   14. sub-neg 74.8%

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

   15. +-commutative 74.8%

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

   16. unsub-neg 74.8%

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

   17. metadata-eval 74.8%

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

3. Simplified 74.8%

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

   1. sub-neg 74.8%

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

   2. distribute-neg-frac 74.8%

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

   3. metadata-eval 74.8%

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

6. Applied egg-rr 74.8%

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

   1. metadata-eval 74.8%

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

   2. distribute-neg-frac 74.8%

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

   3. unsub-neg 74.8%

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

   4. *-rgt-identity 74.8%

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

   5. *-inverses 74.8%

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

   6. associate-/r* 50.9%

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

   7. *-commutative 50.9%

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

   8. *-lft-identity 50.9%

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

   9. associate-/r* 74.8%

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

   10. associate-*r/ 74.8%

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

   11. associate-*l/ 74.8%

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

   12. distribute-lft-out-- 74.8%

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

   13. *-inverses 74.8%

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

   14. div-sub 75.3%

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

   15. associate--r+ 78.6%

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

   16. *-commutative 78.6%

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

   17. times-frac 78.5%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 5: 10.9% 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 74.8%

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

   1. sub-neg 74.8%

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

   2. +-commutative 74.8%

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

   3. distribute-neg-frac2 74.8%

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

   4. neg-sub0 74.8%

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

   5. associate-+l- 74.8%

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

   6. neg-sub0 74.8%

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

   7. remove-double-neg 74.8%

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

   8. distribute-neg-in 74.8%

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

   9. sub-neg 74.8%

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

   10. distribute-neg-frac2 74.8%

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

   11. sub-neg 74.8%

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

   12. +-commutative 74.8%

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

   13. unsub-neg 74.8%

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

   14. sub-neg 74.8%

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

   15. +-commutative 74.8%

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

   16. unsub-neg 74.8%

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

   17. metadata-eval 74.8%

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

3. Simplified 74.8%

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

5. Taylor expanded in x around 0 47.3%

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

6. Taylor expanded in x around inf 10.2%

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

7. Final simplification 10.2%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 6: 51.7% 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 74.8%

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

   1. sub-neg 74.8%

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

   2. +-commutative 74.8%

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

   3. distribute-neg-frac2 74.8%

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

   4. neg-sub0 74.8%

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

   5. associate-+l- 74.8%

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

   6. neg-sub0 74.8%

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

   7. remove-double-neg 74.8%

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

   8. distribute-neg-in 74.8%

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

   9. sub-neg 74.8%

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

   10. distribute-neg-frac2 74.8%

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

   11. sub-neg 74.8%

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

   12. +-commutative 74.8%

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

   13. unsub-neg 74.8%

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

   14. sub-neg 74.8%

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

   15. +-commutative 74.8%

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

   16. unsub-neg 74.8%

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

   17. metadata-eval 74.8%

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

3. Simplified 74.8%

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

5. Taylor expanded in x around 0 47.5%

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

6. Final simplification 47.5%

   \[\leadsto 2 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))