Asymptote A

Percentage Accurate: 77.4% → 99.9%

Time: 6.0s

Alternatives: 5

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% 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 78.0%

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

   1. frac-sub 78.7%

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

   2. associate-/r* 78.8%

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

   3. *-un-lft-identity 78.8%

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

   4. *-rgt-identity 78.8%

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

   5. associate--l- 78.8%

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

   6. +-commutative 78.8%

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

   7. +-commutative 78.8%

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

   8. sub-neg 78.8%

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

   9. metadata-eval 78.8%

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

3. Applied egg-rr 78.8%

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

   1. div-sub 78.0%

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

   2. sub-neg 78.0%

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

   3. associate-+r+ 78.1%

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

   4. metadata-eval 78.1%

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

   5. +-commutative 78.1%

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

5. Applied egg-rr 78.1%

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

   1. sub-neg 78.1%

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

   2. div-sub 78.8%

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

   3. associate--r+ 99.9%

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

   4. +-inverses 99.9%

      \[\leadsto \frac{\frac{\color{blue}{0} - 2}{1 + x}}{x + -1} \]

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. associate-/r* 99.9%

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

   8. neg-mul-1 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. metadata-eval 99.9%

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

   11. unsub-neg 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 74.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(2.0 + Float64(2.0 * Float64(x * x)));
	else
		tmp = 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 + (2.0 * (x * x));
	else
		tmp = (-2.0 / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 87.4%

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

   2. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{2 + 2 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 67.9%

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

   4. Simplified 67.9%

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

   if 1 < x

   1. Initial program 46.0%

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

      1. frac-sub 48.5%

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

      2. associate-/r* 48.6%

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

      3. *-un-lft-identity 48.6%

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

      4. *-rgt-identity 48.6%

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

      5. associate--l- 48.6%

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

      6. +-commutative 48.6%

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

      7. +-commutative 48.6%

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

      8. sub-neg 48.6%

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

      9. metadata-eval 48.6%

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

   3. Applied egg-rr 48.6%

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

      1. div-sub 46.0%

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

      2. sub-neg 46.0%

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

      3. associate-+r+ 46.0%

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

      4. metadata-eval 46.0%

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

      5. +-commutative 46.0%

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

   5. Applied egg-rr 46.0%

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

      1. sub-neg 46.0%

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

      2. div-sub 48.6%

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

      3. associate--r+ 99.8%

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

      4. +-inverses 99.8%

         \[\leadsto \frac{\frac{\color{blue}{0} - 2}{1 + x}}{x + -1} \]

      5. metadata-eval 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-/r* 99.8%

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

      8. neg-mul-1 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. metadata-eval 99.8%

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

      11. unsub-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 97.1%

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

      1. unpow2 97.1%

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

      2. associate-/r* 97.8%

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

   10. Simplified 97.8%

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

4. Final simplification 74.7%

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

Alternative 3: 74.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 87.4%

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

   2. Taylor expanded in x around 0 67.7%

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

      1. neg-mul-1 67.7%

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

      2. unsub-neg 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in x around 0 67.6%

      \[\leadsto \color{blue}{2 + {x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 67.6%

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

   7. Simplified 67.6%

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

   if 1 < x

   1. Initial program 46.0%

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

   2. Taylor expanded in x around inf 97.1%

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

      1. unpow2 97.1%

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

   4. Simplified 97.1%

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

4. Final simplification 74.3%

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

Alternative 4: 74.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(2.0 + Float64(x * x));
	else
		tmp = 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 + (x * x);
	else
		tmp = (-2.0 / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 87.4%

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

   2. Taylor expanded in x around 0 67.7%

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

      1. neg-mul-1 67.7%

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

      2. unsub-neg 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in x around 0 67.6%

      \[\leadsto \color{blue}{2 + {x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 67.6%

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

   7. Simplified 67.6%

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

   if 1 < x

   1. Initial program 46.0%

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

      1. frac-sub 48.5%

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

      2. associate-/r* 48.6%

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

      3. *-un-lft-identity 48.6%

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

      4. *-rgt-identity 48.6%

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

      5. associate--l- 48.6%

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

      6. +-commutative 48.6%

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

      7. +-commutative 48.6%

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

      8. sub-neg 48.6%

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

      9. metadata-eval 48.6%

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

   3. Applied egg-rr 48.6%

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

      1. div-sub 46.0%

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

      2. sub-neg 46.0%

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

      3. associate-+r+ 46.0%

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

      4. metadata-eval 46.0%

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

      5. +-commutative 46.0%

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

   5. Applied egg-rr 46.0%

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

      1. sub-neg 46.0%

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

      2. div-sub 48.6%

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

      3. associate--r+ 99.8%

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

      4. +-inverses 99.8%

         \[\leadsto \frac{\frac{\color{blue}{0} - 2}{1 + x}}{x + -1} \]

      5. metadata-eval 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-/r* 99.8%

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

      8. neg-mul-1 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. metadata-eval 99.8%

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

      11. unsub-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 97.1%

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

      1. unpow2 97.1%

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

      2. associate-/r* 97.8%

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

   10. Simplified 97.8%

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

4. Final simplification 74.5%

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

Alternative 5: 50.8% 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 78.0%

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

2. Taylor expanded in x around 0 53.2%

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

3. Final simplification 53.2%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))