Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.75)) {
		tmp = (x + (1.0 / x)) / x;
	} else {
		tmp = x + (1.0 / (x + -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 <= 1.75d0))) then
        tmp = (x + (1.0d0 / x)) / x
    else
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.75)) {
		tmp = (x + (1.0 / x)) / x;
	} else {
		tmp = x + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.75 < 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+ 100.0%

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

      3. associate-*l/ 48.7%

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

      4. +-commutative 48.7%

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

   8. Simplified 48.7%

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

   9. Taylor expanded in x around 0 48.7%

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

   10. Taylor expanded in x around inf 99.2%

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

   if -1 < x < 1.75

   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 99.6%

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

4. Final simplification 99.4%

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

Alternative 3: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.55:\\ \;\;\;\;\frac{x}{1 + x} + \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.55)
   (+ (/ x (+ 1.0 x)) (/ 1.0 x))
   (if (<= x 1.75) (+ x (/ 1.0 (+ x -1.0))) (/ (+ x (/ 1.0 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.55) {
		tmp = (x / (1.0 + x)) + (1.0 / x);
	} else if (x <= 1.75) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = (x + (1.0 / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.55d0)) then
        tmp = (x / (1.0d0 + x)) + (1.0d0 / x)
    else if (x <= 1.75d0) then
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    else
        tmp = (x + (1.0d0 / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.55) {
		tmp = (x / (1.0 + x)) + (1.0 / x);
	} else if (x <= 1.75) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = (x + (1.0 / x)) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.55:
		tmp = (x / (1.0 + x)) + (1.0 / x)
	elif x <= 1.75:
		tmp = x + (1.0 / (x + -1.0))
	else:
		tmp = (x + (1.0 / x)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.55)
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(1.0 / x));
	elseif (x <= 1.75)
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(Float64(x + Float64(1.0 / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.55)
		tmp = (x / (1.0 + x)) + (1.0 / x);
	elseif (x <= 1.75)
		tmp = x + (1.0 / (x + -1.0));
	else
		tmp = (x + (1.0 / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.55], N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.75], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.55000000000000004

   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 99.2%

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

   if -0.55000000000000004 < x < 1.75

   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 99.6%

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

   if 1.75 < 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+ 100.0%

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

      3. associate-*l/ 49.1%

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

      4. +-commutative 49.1%

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

   8. Simplified 49.1%

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

   9. Taylor expanded in x around 0 49.1%

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

   10. Taylor expanded in x around inf 99.3%

       \[\leadsto \frac{x + \frac{1}{x}}{\color{blue}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.9) (+ x (/ 1.0 (+ x -1.0))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.9) {
		tmp = x + (1.0 / (x + -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 <= 1.9d0) then
        tmp = x + (1.0d0 / (x + (-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 <= 1.9) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.9:
		tmp = x + (1.0 / (x + -1.0))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.9)
		tmp = Float64(x + Float64(1.0 / Float64(x + -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 <= 1.9)
		tmp = x + (1.0 / (x + -1.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.9], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.8999999999999999 < 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. Taylor expanded in x around inf 99.2%

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

   if -1 < x < 1.8999999999999999

   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 99.6%

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

Alternative 5: 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. Taylor expanded in x around inf 99.2%

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

   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 99.6%

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

Alternative 6: 49.7% 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 52.5%

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

Reproduce

herbie shell --seed 2024188 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))