Asymptote B

Percentage Accurate: 100.0% → 99.4%

Time: 2.7s

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: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;1 + \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + -1} + \left(x - x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.4)))
   (+ 1.0 (/ 2.0 (* x x)))
   (+ (/ 1.0 (+ x -1.0)) (- x (* x x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.4)) {
		tmp = 1.0 + (2.0 / (x * x));
	} else {
		tmp = (1.0 / (x + -1.0)) + (x - (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 <= 1.4d0))) then
        tmp = 1.0d0 + (2.0d0 / (x * x))
    else
        tmp = (1.0d0 / (x + (-1.0d0))) + (x - (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.4)) {
		tmp = 1.0 + (2.0 / (x * x));
	} else {
		tmp = (1.0 / (x + -1.0)) + (x - (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.4):
		tmp = 1.0 + (2.0 / (x * x))
	else:
		tmp = (1.0 / (x + -1.0)) + (x - (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.4))
		tmp = Float64(1.0 + Float64(2.0 / Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(x - Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.4)))
		tmp = 1.0 + (2.0 / (x * x));
	else
		tmp = (1.0 / (x + -1.0)) + (x - (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(1.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.3999999999999999 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if -1 < x < 1.3999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. unpow2 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;1 + \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + -1} + \left(x - x \cdot x\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

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

Julia

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

MATLAB

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

Wolfram

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

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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.9%

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

   3. Taylor expanded in x around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. unpow2 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. metadata-eval 99.9%

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

      6. +-commutative 99.9%

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

      7. *-commutative 99.9%

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

      8. cancel-sign-sub-inv 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 4: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], N[(-1.0 - N[(x * 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 100.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.9%

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

   3. Taylor expanded in x around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. unpow2 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. metadata-eval 99.9%

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

      6. +-commutative 99.9%

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

      7. *-commutative 99.9%

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

      8. cancel-sign-sub-inv 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 5: 99.0% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.0) 1.0 (if (<= x 1.0) -1.0 1.0)))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.0:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], -1.0, 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 100.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.9%

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

Alternative 6: 50.6% 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 51.5%

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

3. Final simplification 51.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023193 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))