Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 9

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(-1.0 / x), $MachinePrecision] - x), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - x), $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/ 76.1%

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

   3. associate-/l* 100.0%

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

8. Simplified 100.0%

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

   1. frac-2neg 100.0%

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

   2. div-inv 99.8%

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

10. Applied egg-rr 99.8%

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

12. Simplified 100.0%

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

13. Taylor expanded in x around 0 100.0%

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

    1. neg-mul-1 100.0%

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

    2. +-commutative 100.0%

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

    3. unsub-neg 100.0%

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

15. Simplified 100.0%

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

16. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(-1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.75) (not (<= x 1.0)))
   (/ (- (/ -1.0 x) x) (- x))
   (+ (/ x (+ x 1.0)) (- -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.75) || !(x <= 1.0)) {
		tmp = ((-1.0 / x) - 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 <= (-1.75d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (((-1.0d0) / x) - 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 <= -1.75) || !(x <= 1.0)) {
		tmp = ((-1.0 / x) - x) / -x;
	} else {
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.75) or not (x <= 1.0):
		tmp = ((-1.0 / x) - x) / -x
	else:
		tmp = (x / (x + 1.0)) + (-1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.75) || !(x <= 1.0))
		tmp = Float64(Float64(Float64(-1.0 / x) - x) / Float64(-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 <= -1.75) || ~((x <= 1.0)))
		tmp = ((-1.0 / x) - x) / -x;
	else
		tmp = (x / (x + 1.0)) + (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.75], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(N[(-1.0 / x), $MachinePrecision] - x), $MachinePrecision] / (-x)), $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 < -1.75 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. 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/ 53.3%

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

      3. associate-/l* 100.0%

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

   8. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. div-inv 99.7%

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

   10. Applied egg-rr 99.7%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 100.0%

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

       1. neg-mul-1 100.0%

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

       2. +-commutative 100.0%

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

       3. unsub-neg 100.0%

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

   15. Simplified 100.0%

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

   16. Taylor expanded in x around inf 97.9%

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

       1. mul-1-neg 97.9%

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

   18. Simplified 97.9%

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

   if -1.75 < 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 97.6%

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

      1. sub-neg 97.6%

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

      2. neg-mul-1 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

      5. unsub-neg 97.6%

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

   7. Simplified 97.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(-1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if 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 inf 98.6%

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

   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 97.6%

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

      1. sub-neg 97.6%

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

      2. neg-mul-1 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

      5. unsub-neg 97.6%

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

   7. Simplified 97.6%

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

   if 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. 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 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-rgt-identity 99.9%

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

      2. associate-*l/ 54.4%

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

      3. associate-/l* 100.0%

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

   8. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. div-inv 99.7%

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

   10. Applied egg-rr 99.7%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 100.0%

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

       1. neg-mul-1 100.0%

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

       2. +-commutative 100.0%

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

       3. unsub-neg 100.0%

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

   15. Simplified 100.0%

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

   16. Taylor expanded in x around inf 97.2%

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

       1. mul-1-neg 97.2%

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

   18. Simplified 97.2%

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

4. Final simplification 97.8%

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

Alternative 4: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.78\right):\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.78)))
   (/ (- (/ -1.0 x) x) (- x))
   (+ x (/ 1.0 (+ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.78)) {
		tmp = ((-1.0 / x) - x) / -x;
	} else {
		tmp = x + (1.0 / (-1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.78d0))) then
        tmp = (((-1.0d0) / x) - x) / -x
    else
        tmp = x + (1.0d0 / ((-1.0d0) + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.78)) {
		tmp = ((-1.0 / x) - x) / -x;
	} else {
		tmp = x + (1.0 / (-1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.78):
		tmp = ((-1.0 / x) - x) / -x
	else:
		tmp = x + (1.0 / (-1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.78))
		tmp = Float64(Float64(Float64(-1.0 / x) - x) / Float64(-x));
	else
		tmp = Float64(x + Float64(1.0 / Float64(-1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.78)))
		tmp = ((-1.0 / x) - x) / -x;
	else
		tmp = x + (1.0 / (-1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.78]], $MachinePrecision]], N[(N[(N[(-1.0 / x), $MachinePrecision] - x), $MachinePrecision] / (-x)), $MachinePrecision], N[(x + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.78000000000000003 < 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. 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/ 53.3%

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

      3. associate-/l* 100.0%

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

   8. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. div-inv 99.7%

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

   10. Applied egg-rr 99.7%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 100.0%

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

       1. neg-mul-1 100.0%

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

       2. +-commutative 100.0%

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

       3. unsub-neg 100.0%

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

   15. Simplified 100.0%

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

   16. Taylor expanded in x around inf 97.9%

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

       1. mul-1-neg 97.9%

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

   18. Simplified 97.9%

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

   if -1 < x < 1.78000000000000003

   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 97.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.78\right):\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.92:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.92) (+ x (/ 1.0 (+ -1.0 x))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.92) {
		tmp = x + (1.0 / (-1.0 + 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 <= 1.92d0) then
        tmp = x + (1.0d0 / ((-1.0d0) + 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 <= 1.92) {
		tmp = x + (1.0 / (-1.0 + x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.92:
		tmp = x + (1.0 / (-1.0 + x))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.92)
		tmp = Float64(x + Float64(1.0 / Float64(-1.0 + 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 <= 1.92)
		tmp = x + (1.0 / (-1.0 + x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.92], N[(x + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.9199999999999999 < 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. 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/ 53.3%

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

      3. associate-/l* 100.0%

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

   8. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. div-inv 99.7%

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

   10. Applied egg-rr 99.7%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 97.8%

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

   if -1 < x < 1.9199999999999999

   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 97.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.92:\\ \;\;\;\;x + \frac{1}{-1 + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% 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. 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/ 53.3%

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

      3. associate-/l* 100.0%

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

   8. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. div-inv 99.7%

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

   10. Applied egg-rr 99.7%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 97.8%

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

   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 97.6%

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

4. Final simplification 97.7%

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

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 8: 10.2% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

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. 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/ 76.1%

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

   3. associate-/l* 100.0%

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

8. Simplified 100.0%

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

   1. associate-/r/ 99.8%

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

   2. div-inv 99.7%

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

   3. associate-*r* 99.7%

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

10. Applied egg-rr 99.7%

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

11. Taylor expanded in x around inf 51.9%

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

    1. sub-neg 51.9%

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

    2. +-commutative 51.9%

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

    3. metadata-eval 51.9%

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

    4. associate-+l+ 51.9%

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

    5. associate-*r/ 51.9%

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

    6. metadata-eval 51.9%

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

13. Simplified 51.9%

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

15. Applied egg-rr 10.0%

    \[\leadsto \color{blue}{-2} \]

16. Final simplification 10.0%

    \[\leadsto -2 \]
17. Add Preprocessing

Alternative 9: 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. 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 48.4%

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

6. Final simplification 48.4%

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

Reproduce

herbie shell --seed 2024026 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))