Asymptote A

Percentage Accurate: 78.1% → 99.9%

Time: 4.1s

Alternatives: 6

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (/ (/ 2.0 (- -1.0 x)) (+ -1.0 x)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 / ((-1.0d0) - x)) / ((-1.0d0) + x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[(2.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.9%

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

   1. frac-sub 76.7%

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

   2. associate-/r* 76.7%

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

   3. *-un-lft-identity 76.7%

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

   4. *-rgt-identity 76.7%

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

   5. associate--l- 76.7%

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

   6. +-commutative 76.7%

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

   7. +-commutative 76.7%

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

   8. sub-neg 76.7%

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

   9. metadata-eval 76.7%

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

3. Applied egg-rr 76.7%

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

   1. frac-2neg 76.7%

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

   2. div-inv 76.7%

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

   3. associate-+r+ 76.7%

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

   4. metadata-eval 76.7%

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

   5. +-commutative 76.7%

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

   6. distribute-neg-in 76.7%

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

   7. metadata-eval 76.7%

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

5. Applied egg-rr 76.7%

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

   1. associate-*r/ 76.7%

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

   2. *-rgt-identity 76.7%

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

   3. neg-sub0 76.7%

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

   4. associate-+l- 76.7%

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

   5. neg-sub0 76.7%

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

   6. associate-+r+ 99.9%

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

   7. neg-sub0 99.9%

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

   8. associate-+l- 99.9%

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

   9. +-inverses 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

   12. 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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (- (- 1.0 x) (/ 1.0 (+ -1.0 x))) (/ (/ -2.0 x) x)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.55], N[(N[(1.0 - x), $MachinePrecision] - N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 83.7%

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

   2. Taylor expanded in x around 0 64.9%

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

      1. neg-mul-1 64.9%

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

      2. unsub-neg 64.9%

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

   4. Simplified 64.9%

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

   if 1.55000000000000004 < x

   1. Initial program 51.6%

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

   2. Taylor expanded in x around inf 97.6%

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

      1. unpow2 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in x around 0 97.6%

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

      1. metadata-eval 97.6%

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

      2. +-inverses 97.6%

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

      3. associate--r+ 51.8%

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

      4. unpow2 51.8%

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

      5. associate-/r* 51.9%

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

      6. *-rgt-identity 51.9%

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

      7. associate-*r/ 51.9%

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

      8. associate-*l/ 51.8%

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

      9. *-rgt-identity 51.8%

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

      10. associate-*r/ 51.8%

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

      11. associate-*r* 51.8%

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

      12. unpow-1 51.8%

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

      13. unpow-1 51.8%

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

      14. pow-sqr 51.9%

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

      15. metadata-eval 51.9%

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

      16. *-commutative 51.9%

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

      17. associate--r+ 98.6%

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

      18. +-inverses 98.6%

          \[\leadsto {x}^{-2} \cdot \left(\color{blue}{0} - 2\right) \]

      19. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around 0 97.6%

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

      1. unpow2 97.6%

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

      2. associate-/r* 98.5%

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

   10. Simplified 98.5%

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

4. Final simplification 73.0%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (+ (/ 1.0 (+ x 1.0)) (- x -1.0)) (/ (/ -2.0 x) x)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (1.0d0 / (x + 1.0d0)) + (x - (-1.0d0))
    else
        tmp = ((-2.0d0) / x) / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.0], N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 83.7%

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

   2. Taylor expanded in x around 0 65.3%

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

      1. sub-neg 65.3%

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

      2. metadata-eval 65.3%

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

      3. +-commutative 65.3%

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

      4. neg-mul-1 65.3%

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

      5. unsub-neg 65.3%

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

   4. Simplified 65.3%

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

   if 1 < x

   1. Initial program 51.6%

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

   2. Taylor expanded in x around inf 97.6%

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

      1. unpow2 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in x around 0 97.6%

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

      1. metadata-eval 97.6%

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

      2. +-inverses 97.6%

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

      3. associate--r+ 51.8%

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

      4. unpow2 51.8%

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

      5. associate-/r* 51.9%

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

      6. *-rgt-identity 51.9%

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

      7. associate-*r/ 51.9%

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

      8. associate-*l/ 51.8%

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

      9. *-rgt-identity 51.8%

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

      10. associate-*r/ 51.8%

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

      11. associate-*r* 51.8%

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

      12. unpow-1 51.8%

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

      13. unpow-1 51.8%

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

      14. pow-sqr 51.9%

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

      15. metadata-eval 51.9%

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

      16. *-commutative 51.9%

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

      17. associate--r+ 98.6%

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

      18. +-inverses 98.6%

          \[\leadsto {x}^{-2} \cdot \left(\color{blue}{0} - 2\right) \]

      19. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around 0 97.6%

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

      1. unpow2 97.6%

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

      2. associate-/r* 98.5%

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

   10. Simplified 98.5%

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

4. Final simplification 73.3%

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

Alternative 4: 98.4% accurate, 1.6× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 (/ -2.0 (* x x))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 2.0d0
    else
        tmp = (-2.0d0) / (x * x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.0], 2.0, N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 83.7%

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

   2. Taylor expanded in x around 0 65.2%

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

   if 1 < x

   1. Initial program 51.6%

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

   2. Taylor expanded in x around inf 97.6%

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

      1. unpow2 97.6%

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

   4. Simplified 97.6%

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

4. Final simplification 73.0%

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

Alternative 5: 98.8% accurate, 1.6× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.0) 2.0 (/ (/ -2.0 x) x)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 2.0d0
    else
        tmp = ((-2.0d0) / x) / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.0], 2.0, N[(N[(-2.0 / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 83.7%

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

   2. Taylor expanded in x around 0 65.2%

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

   if 1 < x

   1. Initial program 51.6%

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

   2. Taylor expanded in x around inf 97.6%

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

      1. unpow2 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in x around 0 97.6%

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

      1. metadata-eval 97.6%

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

      2. +-inverses 97.6%

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

      3. associate--r+ 51.8%

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

      4. unpow2 51.8%

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

      5. associate-/r* 51.9%

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

      6. *-rgt-identity 51.9%

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

      7. associate-*r/ 51.9%

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

      8. associate-*l/ 51.8%

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

      9. *-rgt-identity 51.8%

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

      10. associate-*r/ 51.8%

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

      11. associate-*r* 51.8%

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

      12. unpow-1 51.8%

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

      13. unpow-1 51.8%

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

      14. pow-sqr 51.9%

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

      15. metadata-eval 51.9%

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

      16. *-commutative 51.9%

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

      17. associate--r+ 98.6%

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

      18. +-inverses 98.6%

          \[\leadsto {x}^{-2} \cdot \left(\color{blue}{0} - 2\right) \]

      19. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around 0 97.6%

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

      1. unpow2 97.6%

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

      2. associate-/r* 98.5%

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

   10. Simplified 98.5%

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

4. Final simplification 73.2%

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

Alternative 6: 52.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 2 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 2.0)

C

x = abs(x);
double code(double x) {
	return 2.0;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 2.0;
}

Python

x = abs(x)
def code(x):
	return 2.0

Julia

x = abs(x)
function code(x)
	return 2.0
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 2.0

Derivation

1. Initial program 75.9%

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

2. Taylor expanded in x around 0 50.0%

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

3. Final simplification 50.0%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023244 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))