Asymptote A

Percentage Accurate: 77.2% → 99.9%

Time: 5.3s

Alternatives: 6

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. frac-sub 78.3%

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

   2. associate-/r* 78.3%

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

   3. *-un-lft-identity 78.3%

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

   4. *-rgt-identity 78.3%

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

   5. associate--l- 78.2%

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

   6. +-commutative 78.2%

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

   7. +-commutative 78.2%

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

   8. sub-neg 78.2%

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

   9. metadata-eval 78.2%

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

3. Applied egg-rr 78.2%

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

4. Taylor expanded in x around 0 99.9%

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

   1. frac-2neg 99.9%

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

   2. div-inv 99.8%

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

   3. +-commutative 99.8%

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

   4. distribute-neg-frac 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. distribute-neg-in 99.8%

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

   9. metadata-eval 99.8%

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

   10. sub-neg 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.9%

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

   2. associate-*r/ 99.9%

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

   3. metadata-eval 99.9%

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

   4. metadata-eval 99.9%

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

   5. metadata-eval 99.9%

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

   6. +-inverses 99.9%

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

   7. sub-neg 99.9%

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

   8. associate-+r+ 78.3%

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

   9. +-commutative 78.3%

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

   10. distribute-neg-in 78.3%

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

   11. neg-sub0 78.3%

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

   12. distribute-neg-in 78.3%

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

   13. metadata-eval 78.3%

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

   14. remove-double-neg 78.3%

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

   15. +-commutative 78.3%

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

   16. associate--r- 78.3%

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

   17. neg-sub0 78.3%

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

   18. associate-/r* 78.2%

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

   19. distribute-rgt-in 78.2%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.55)))
   (/ (/ -2.0 x) x)
   (+ (- 1.0 x) (/ -1.0 (+ x -1.0)))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.55d0))) then
        tmp = ((-2.0d0) / x) / x
    else
        tmp = (1.0d0 - x) + ((-1.0d0) / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.55)) {
		tmp = (-2.0 / x) / x;
	} else {
		tmp = (1.0 - x) + (-1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.55):
		tmp = (-2.0 / x) / x
	else:
		tmp = (1.0 - x) + (-1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.55))
		tmp = Float64(Float64(-2.0 / x) / x);
	else
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.55]], $MachinePrecision]], N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.55000000000000004 < x

   1. Initial program 53.1%

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

   2. Taylor expanded in x around inf 97.4%

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

      1. unpow2 97.4%

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

   4. Simplified 97.4%

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

      1. associate-/r* 98.0%

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

      2. div-inv 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. associate-*r/ 98.0%

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

      2. *-rgt-identity 98.0%

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

   8. Simplified 98.0%

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

   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 98.3%

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

      1. neg-mul-1 98.3%

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

      2. unsub-neg 98.3%

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

   4. Simplified 98.3%

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

4. Final simplification 98.2%

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

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 53.1%

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

   2. Taylor expanded in x around inf 97.4%

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

      1. unpow2 97.4%

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

   4. Simplified 97.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.3%

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

      1. neg-mul-1 98.3%

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

      2. unsub-neg 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. unpow2 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 97.9%

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

Alternative 4: 99.0% accurate, 1.2× speedup

Math

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

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(Float64(-2.0 / 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[(N[(-2.0 / x), $MachinePrecision] / x), $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 53.1%

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

   2. Taylor expanded in x around inf 97.4%

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

      1. unpow2 97.4%

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

   4. Simplified 97.4%

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

      1. associate-/r* 98.0%

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

      2. div-inv 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. associate-*r/ 98.0%

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

      2. *-rgt-identity 98.0%

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

   8. Simplified 98.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.3%

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

      1. neg-mul-1 98.3%

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

      2. unsub-neg 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. unpow2 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 98.2%

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

Alternative 5: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. frac-sub 78.3%

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

   2. associate-/r* 78.3%

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

   3. *-un-lft-identity 78.3%

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

   4. *-rgt-identity 78.3%

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

   5. associate--l- 78.2%

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

   6. +-commutative 78.2%

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

   7. +-commutative 78.2%

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

   8. sub-neg 78.2%

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

   9. metadata-eval 78.2%

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

3. Applied egg-rr 78.2%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 50.4% 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 77.8%

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

2. Taylor expanded in x around 0 53.1%

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

3. Final simplification 53.1%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023175 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))