Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 5

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, 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. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{1}{x + -1} + \frac{x}{1 + x} \]
4. Add Preprocessing

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{2}{x \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-1 + \left(x \cdot x\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 2.0 (* x x)))))
   (if (<= x -1.0) t_0 (if (<= x 1.0) (+ -1.0 (* (* x x) -2.0)) t_0))))

C

double code(double x) {
	double t_0 = 1.0 + (2.0 / (x * x));
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = -1.0 + ((x * x) * -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (2.0d0 / (x * x))
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = (-1.0d0) + ((x * x) * (-2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (2.0 / (x * x));
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = -1.0 + ((x * x) * -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (2.0 / (x * x))
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 1.0:
		tmp = -1.0 + ((x * x) * -2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(2.0 / Float64(x * x)))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(-1.0 + Float64(Float64(x * x) * -2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (2.0 / (x * x));
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = -1.0 + ((x * x) * -2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.0], N[(-1.0 + N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], t$95$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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(2 \cdot \frac{1}{{x}^{2}}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{2 \cdot 1}{\color{blue}{{x}^{2}}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{2}{{\color{blue}{x}}^{2}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 99.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

         \[\leadsto -2 \cdot {x}^{2} + -1 \]

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(-2 \cdot {x}^{2}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      7. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.6%

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

4. Final simplification 99.4%

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

Alternative 3: 99.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = -1.0 + ((x * x) * -2.0);
	} 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) + ((x * x) * (-2.0d0))
    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 + ((x * x) * -2.0);
	} 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 + ((x * x) * -2.0)
	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 = Float64(-1.0 + Float64(Float64(x * x) * -2.0));
	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 + ((x * x) * -2.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.6%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

         \[\leadsto -2 \cdot {x}^{2} + -1 \]

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(-2 \cdot {x}^{2}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      7. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.6%

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

4. Final simplification 99.1%

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

Alternative 4: 98.9% accurate, 1.0× 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 98.6%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 99.2%

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

Alternative 5: 50.7% 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. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 51.5%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))