Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 4

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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + x}{x}\\ \frac{t_0 + \left(x + -1\right)}{t_0 \cdot \left(x + -1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 x) x))) (/ (+ t_0 (+ x -1.0)) (* t_0 (+ x -1.0)))))

C

double code(double x) {
	double t_0 = (1.0 + x) / x;
	return (t_0 + (x + -1.0)) / (t_0 * (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (1.0d0 + x) / x
    code = (t_0 + (x + (-1.0d0))) / (t_0 * (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	double t_0 = (1.0 + x) / x;
	return (t_0 + (x + -1.0)) / (t_0 * (x + -1.0));
}

Python

def code(x):
	t_0 = (1.0 + x) / x
	return (t_0 + (x + -1.0)) / (t_0 * (x + -1.0))

Julia

function code(x)
	t_0 = Float64(Float64(1.0 + x) / x)
	return Float64(Float64(t_0 + Float64(x + -1.0)) / Float64(t_0 * Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	t_0 = (1.0 + x) / x;
	tmp = (t_0 + (x + -1.0)) / (t_0 * (x + -1.0));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

3. Simplified 100.0%

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

   1. +-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. frac-add 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. *-commutative 100.0%

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

   6. *-un-lft-identity 100.0%

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

   7. sub-neg 100.0%

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

   8. metadata-eval 100.0%

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

   9. sub-neg 100.0%

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

   10. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

      3. unpow2 99.7%

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

      4. associate-/r* 99.7%

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

   6. Simplified 99.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{-1} \]
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 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: 50.8% 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. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 51.1%

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

5. Taylor expanded in x around 0 50.3%

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

6. Final simplification 50.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))