Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x + N[(1.0 / x), $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. 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. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. frac-add 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. +-commutative 100.0%

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

   5. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-rgt-identity 100.0%

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

   2. associate-+l+ 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*r/ 77.2%

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

   5. +-commutative 77.2%

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

8. Simplified 77.2%

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

9. Taylor expanded in x around 0 77.2%

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

10. Taylor expanded in x around 0 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{x}\\ \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{t_0}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 x))))
   (if (or (<= x -1.0) (not (<= x 1.0))) (/ t_0 x) (/ t_0 (/ -1.0 x)))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (1.0d0 / x)
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = t_0 / x
    else
        tmp = t_0 / ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	t_0 = x + (1.0 / x)
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = t_0 / x
	else:
		tmp = t_0 / (-1.0 / x)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = x + (1.0 / x);
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = t_0 / x;
	else
		tmp = t_0 / (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(t$95$0 / x), $MachinePrecision], N[(t$95$0 / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 100.0%

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

      2. frac-add 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 51.8%

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

      5. +-commutative 51.8%

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

   8. Simplified 51.8%

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

   9. Taylor expanded in x around 0 51.8%

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

   10. Taylor expanded in x around inf 98.6%

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

   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. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-add 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

   10. Taylor expanded in x around 0 98.3%

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

4. Final simplification 98.4%

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

Alternative 3: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (x + (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)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x + (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) || !(x <= 1.0)) {
		tmp = (x + (1.0 / x)) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(x + Float64(1.0 / 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 = (x + (1.0 / 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[(N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 100.0%

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

      2. frac-add 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 51.8%

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

      5. +-commutative 51.8%

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

   8. Simplified 51.8%

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

   9. Taylor expanded in x around 0 51.8%

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

   10. Taylor expanded in x around inf 98.6%

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

   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. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x + \frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
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 99.9%

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

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 98.5%

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

   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. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.3%

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

Alternative 5: 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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 6: 51.1% 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. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 52.5%

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

6. Final simplification 52.5%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024023 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))