Asymptote A

Percentage Accurate: 77.4% → 99.9%

Time: 4.7s

Alternatives: 6

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% 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 77.0%

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

   1. frac-sub 77.8%

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

   2. associate-/r* 77.8%

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

   3. *-un-lft-identity 77.8%

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

   4. *-rgt-identity 77.8%

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

   5. associate--l- 77.8%

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

   6. +-commutative 77.8%

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

   7. +-commutative 77.8%

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

   8. sub-neg 77.8%

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

   9. metadata-eval 77.8%

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

3. Applied egg-rr 77.8%

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

   1. frac-2neg 77.8%

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

   2. div-inv 77.8%

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

   3. associate-+r+ 77.8%

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

   4. metadata-eval 77.8%

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

   5. associate--r+ 77.8%

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

   6. distribute-neg-in 77.8%

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

   7. metadata-eval 77.8%

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

5. Applied egg-rr 77.8%

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

   1. associate-*r/ 77.8%

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

   2. *-rgt-identity 77.8%

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

   3. associate--l- 77.8%

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

   4. +-commutative 77.8%

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

   5. associate--r+ 99.9%

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

   6. +-inverses 99.9%

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

   7. metadata-eval 99.9%

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

   8. metadata-eval 99.9%

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

   9. 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: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ -2.0 x) x)
   (if (<= x 1.55) (- (- 1.0 x) (/ 1.0 (+ -1.0 x))) (/ -2.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-2.0 / x) / x;
	} else if (x <= 1.55) {
		tmp = (1.0 - x) - (1.0 / (-1.0 + 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 if (x <= 1.55d0) then
        tmp = (1.0d0 - x) - (1.0d0 / ((-1.0d0) + 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 if (x <= 1.55) {
		tmp = (1.0 - x) - (1.0 / (-1.0 + x));
	} else {
		tmp = -2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (-2.0 / x) / x
	elif x <= 1.55:
		tmp = (1.0 - x) - (1.0 / (-1.0 + x))
	else:
		tmp = -2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(-2.0 / x) / x);
	elseif (x <= 1.55)
		tmp = Float64(Float64(1.0 - x) - Float64(1.0 / Float64(-1.0 + 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;
	elseif (x <= 1.55)
		tmp = (1.0 - x) - (1.0 / (-1.0 + x));
	else
		tmp = -2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.55], N[(N[(1.0 - x), $MachinePrecision] - N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 56.1%

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

      1. frac-sub 59.4%

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

      2. associate-/r* 59.4%

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

      3. *-un-lft-identity 59.4%

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

      4. *-rgt-identity 59.4%

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

      5. associate--l- 59.4%

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

      6. +-commutative 59.4%

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

      7. +-commutative 59.4%

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

      8. sub-neg 59.4%

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

      9. metadata-eval 59.4%

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

   3. Applied egg-rr 59.4%

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

      1. frac-2neg 59.4%

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

      2. div-inv 59.4%

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

      3. associate-+r+ 59.4%

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

      4. metadata-eval 59.4%

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

      5. associate--r+ 59.4%

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

      6. distribute-neg-in 59.4%

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

      7. metadata-eval 59.4%

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

   5. Applied egg-rr 59.4%

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

      1. associate-*r/ 59.4%

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

      2. *-rgt-identity 59.4%

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

      3. associate--l- 59.4%

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

      4. +-commutative 59.4%

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

      5. associate--r+ 99.9%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. 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. Taylor expanded in x around inf 99.4%

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

      1. unpow2 99.4%

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

      2. associate-/r* 99.5%

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

   10. Simplified 99.5%

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

   if -1 < x < 1.55000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

      1. neg-mul-1 98.7%

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

      2. unsub-neg 98.7%

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

   4. Simplified 98.7%

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

   if 1.55000000000000004 < x

   1. Initial program 46.4%

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

   2. Taylor expanded in x around inf 99.1%

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

      1. unpow2 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ -2.0 (* x x)) (+ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(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)) .or. (.not. (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) || !(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) or not (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) || !(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) || ~((x <= 1.0)))
		tmp = -2.0 / (x * x);
	else
		tmp = 2.0 + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], 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 or 1 < x

   1. Initial program 51.3%

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

   2. Taylor expanded in x around inf 99.2%

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

      1. unpow2 99.2%

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

   4. Simplified 99.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

      1. neg-mul-1 98.7%

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

      2. unsub-neg 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. unpow2 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 99.0%

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

Alternative 4: 98.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ -2.0 x) x)
   (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 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 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 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
	elif 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(Float64(-2.0 / x) / x);
	elseif (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;
	elseif (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[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision], 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 3 regimes

2. if x < -1

   1. Initial program 56.1%

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

      1. frac-sub 59.4%

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

      2. associate-/r* 59.4%

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

      3. *-un-lft-identity 59.4%

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

      4. *-rgt-identity 59.4%

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

      5. associate--l- 59.4%

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

      6. +-commutative 59.4%

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

      7. +-commutative 59.4%

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

      8. sub-neg 59.4%

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

      9. metadata-eval 59.4%

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

   3. Applied egg-rr 59.4%

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

      1. frac-2neg 59.4%

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

      2. div-inv 59.4%

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

      3. associate-+r+ 59.4%

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

      4. metadata-eval 59.4%

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

      5. associate--r+ 59.4%

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

      6. distribute-neg-in 59.4%

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

      7. metadata-eval 59.4%

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

   5. Applied egg-rr 59.4%

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

      1. associate-*r/ 59.4%

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

      2. *-rgt-identity 59.4%

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

      3. associate--l- 59.4%

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

      4. +-commutative 59.4%

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

      5. associate--r+ 99.9%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. 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. Taylor expanded in x around inf 99.4%

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

      1. unpow2 99.4%

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

      2. associate-/r* 99.5%

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

   10. Simplified 99.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

      1. neg-mul-1 98.7%

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

      2. unsub-neg 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. unpow2 98.7%

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

   7. Simplified 98.7%

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

   if 1 < x

   1. Initial program 46.4%

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

   2. Taylor expanded in x around inf 99.1%

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

      1. unpow2 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 99.0%

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

Alternative 5: 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 77.0%

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

2. Taylor expanded in x around 0 52.5%

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

3. Taylor expanded in x around inf 11.1%

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

4. Final simplification 11.1%

   \[\leadsto 1 \]

Alternative 6: 51.3% 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 77.0%

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

2. Taylor expanded in x around 0 53.3%

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

3. Final simplification 53.3%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023174 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))