Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 3.4s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. frac-add 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. +-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. *-un-lft-identity 100.0%

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

   7. sub-neg 100.0%

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

   8. metadata-eval 100.0%

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

   9. sub-neg 100.0%

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

   10. metadata-eval 100.0%

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

   11. +-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. /-rgt-identity 100.0%

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

   3. +-commutative 100.0%

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

   4. associate-+l+ 100.0%

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

   5. +-commutative 100.0%

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

   6. +-commutative 100.0%

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

   7. /-rgt-identity 100.0%

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

   8. +-commutative 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 + \frac{\frac{2}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(-1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 1.0)))
   (+ 1.0 (/ (/ 2.0 x) x))
   (+ (/ x (+ x 1.0)) (- -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.0)) {
		tmp = 1.0 + ((2.0 / x) / x);
	} else {
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 1.0d0 + ((2.0d0 / x) / x)
    else
        tmp = (x / (x + 1.0d0)) + ((-1.0d0) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.0)) {
		tmp = 1.0 + ((2.0 / x) / x);
	} else {
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.5) or not (x <= 1.0):
		tmp = 1.0 + ((2.0 / x) / x)
	else:
		tmp = (x / (x + 1.0)) + (-1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 1.0))
		tmp = Float64(1.0 + Float64(Float64(2.0 / x) / x));
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(-1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 1.0)))
		tmp = 1.0 + ((2.0 / x) / x);
	else
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(1.0 + N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

      3. unpow2 99.5%

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

      4. associate-/r* 99.5%

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

   4. Simplified 99.5%

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

   if -1.5 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. neg-mul-1 99.4%

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

      3. metadata-eval 99.4%

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

      4. +-commutative 99.4%

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

      5. unsub-neg 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;1 + \frac{\frac{2}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(-1 - x\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 1.0 + ((2.0 / x) / x);
	} else {
		tmp = -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 <= 1.0d0))) then
        tmp = 1.0d0 + ((2.0d0 / x) / x)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

      3. unpow2 99.5%

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

      4. associate-/r* 99.5%

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

   4. Simplified 99.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

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

   3. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.4%

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

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 5: 49.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x + -1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 47.9%

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

3. Taylor expanded in x around 0 47.5%

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

4. Final simplification 47.5%

   \[\leadsto x + -1 \]

Alternative 6: 49.4% 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 100.0%

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

2. Taylor expanded in x around 0 47.9%

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

3. Taylor expanded in x around 0 47.0%

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

4. Final simplification 47.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))