Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (* x (/ 1.0 x))
   (if (<= x 1.6) (+ x (/ 1.0 (+ x -1.0))) (+ 1.0 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * (1.0 / x);
	} else if (x <= 1.6) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 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)) then
        tmp = x * (1.0d0 / x)
    else if (x <= 1.6d0) then
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    else
        tmp = 1.0d0 + (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * (1.0 / x);
	} else if (x <= 1.6) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = 1.0 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = x * (1.0 / x)
	elif x <= 1.6:
		tmp = x + (1.0 / (x + -1.0))
	else:
		tmp = 1.0 + (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x * Float64(1.0 / x));
	elseif (x <= 1.6)
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(1.0 + Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x * (1.0 / x);
	elseif (x <= 1.6)
		tmp = x + (1.0 / (x + -1.0));
	else
		tmp = 1.0 + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(x * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(1.0 / x), $MachinePrecision]), $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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-add 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 50.5%

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

      2. associate-/r/ 50.4%

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

      3. *-commutative 50.4%

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

      4. mul-1-neg 50.4%

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

      5. unsub-neg 50.4%

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

      6. associate--r+ 50.4%

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

   8. Simplified 50.4%

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

   9. Taylor expanded in x around inf 99.7%

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

   if -1 < x < 1.6000000000000001

   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 99.2%

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

   if 1.6000000000000001 < 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 96.1%

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

   6. Taylor expanded in x around inf 96.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. 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-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-add 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 50.6%

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

      2. associate-/r/ 50.5%

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

      3. *-commutative 50.5%

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

      4. mul-1-neg 50.5%

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

      5. unsub-neg 50.5%

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

      6. associate--r+ 50.5%

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

   8. Simplified 50.5%

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

   9. Taylor expanded in x around inf 97.8%

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

   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 99.2%

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

4. Final simplification 98.5%

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

Alternative 4: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(x * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], -1.0, N[(1.0 + N[(1.0 / x), $MachinePrecision]), $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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. frac-add 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*l/ 50.5%

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

      2. associate-/r/ 50.4%

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

      3. *-commutative 50.4%

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

      4. mul-1-neg 50.4%

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

      5. unsub-neg 50.4%

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

      6. associate--r+ 50.4%

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

   8. Simplified 50.4%

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

   9. Taylor expanded in x around inf 99.7%

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

   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 99.2%

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

   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. Taylor expanded in x around inf 96.1%

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

   6. Taylor expanded in x around inf 96.2%

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

4. Final simplification 98.6%

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

Alternative 5: 51.3% 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 53.4%

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

6. Final simplification 53.4%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))