Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 7

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. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.76 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \left(-1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.76) (not (<= x 1.0)))
   (/ (- (/ -1.0 x) x) (- x))
   (+ (/ x (+ 1.0 x)) (- -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.76) || !(x <= 1.0)) {
		tmp = ((-1.0 / x) - x) / -x;
	} else {
		tmp = (x / (1.0 + x)) + (-1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.76d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (((-1.0d0) / x) - x) / -x
    else
        tmp = (x / (1.0d0 + x)) + ((-1.0d0) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.76) || !(x <= 1.0)) {
		tmp = ((-1.0 / x) - x) / -x;
	} else {
		tmp = (x / (1.0 + x)) + (-1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.76) or not (x <= 1.0):
		tmp = ((-1.0 / x) - x) / -x
	else:
		tmp = (x / (1.0 + x)) + (-1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.76) || !(x <= 1.0))
		tmp = Float64(Float64(Float64(-1.0 / x) - x) / Float64(-x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + x)) + Float64(-1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.76) || ~((x <= 1.0)))
		tmp = ((-1.0 / x) - x) / -x;
	else
		tmp = (x / (1.0 + x)) + (-1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.76], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(N[(-1.0 / x), $MachinePrecision] - x), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.76000000000000001 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 98.9%

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

      1. frac-2neg 98.9%

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

      2. metadata-eval 98.9%

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

      3. frac-add 54.4%

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

      4. +-commutative 54.4%

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

      5. +-commutative 54.4%

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

   6. Applied egg-rr 54.4%

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

      1. associate-/r* 55.7%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around inf 98.9%

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

       1. sub-neg 98.9%

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

       2. distribute-neg-frac 98.9%

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

       3. metadata-eval 98.9%

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

       4. +-commutative 98.9%

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

       5. mul-1-neg 98.9%

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

       6. unsub-neg 98.9%

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

   11. Simplified 98.9%

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

   if -1.76000000000000001 < 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. neg-mul-1 99.3%

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

      4. +-commutative 99.3%

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

      5. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.76 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-1}{x} - x}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} + \left(-1 - x\right)\\ \end{array} \]

Alternative 3: 99.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(N[(-1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 98.7%

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

      1. frac-2neg 98.7%

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

      2. metadata-eval 98.7%

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

      3. frac-add 47.3%

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

      4. +-commutative 47.3%

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

      5. +-commutative 47.3%

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

   6. Applied egg-rr 47.3%

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

      1. associate-/r* 48.8%

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

   8. Simplified 98.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. neg-mul-1 99.3%

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

      4. +-commutative 99.3%

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

      5. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

   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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 99.0%

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

      1. frac-2neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. frac-add 62.5%

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

      4. +-commutative 62.5%

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

      5. +-commutative 62.5%

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

   6. Applied egg-rr 62.5%

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

      1. associate-/r* 63.7%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in x around inf 99.1%

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

       1. sub-neg 99.1%

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

       2. distribute-neg-frac 99.1%

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

       3. metadata-eval 99.1%

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

       4. +-commutative 99.1%

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

       5. mul-1-neg 99.1%

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

       6. unsub-neg 99.1%

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

   11. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 4: 99.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.75)) {
		tmp = ((-1.0 / x) - 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 = (((-1.0d0) / x) - 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 = ((-1.0 / x) - 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 = ((-1.0 / x) - 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(Float64(-1.0 / x) - x) / Float64(-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 = ((-1.0 / x) - 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[(N[(-1.0 / x), $MachinePrecision] - x), $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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 98.9%

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

      1. frac-2neg 98.9%

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

      2. metadata-eval 98.9%

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

      3. frac-add 54.4%

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

      4. +-commutative 54.4%

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

      5. +-commutative 54.4%

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

   6. Applied egg-rr 54.4%

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

      1. associate-/r* 55.7%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around inf 98.9%

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

       1. sub-neg 98.9%

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

       2. distribute-neg-frac 98.9%

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

       3. metadata-eval 98.9%

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

       4. +-commutative 98.9%

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

       5. mul-1-neg 98.9%

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

       6. unsub-neg 98.9%

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

   11. Simplified 98.9%

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

   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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.1%

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

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.92) (+ 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 <= 1.92) {
		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 <= 1.92d0) 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 <= 1.92) {
		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 <= 1.92:
		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 <= 1.92)
		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 <= 1.92)
		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, 1.92], 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 1.9199999999999999 < 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 98.8%

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

   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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.1%

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

Alternative 6: 99.0% accurate, 2.1× 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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around inf 98.8%

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

   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. metadata-eval 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. associate-/r* 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. associate-/r/ 100.0%

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

      15. metadata-eval 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. *-lft-identity 100.0%

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

      19. sub-neg 100.0%

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

      20. 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. Taylor expanded in x around 0 99.3%

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

   5. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.0%

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

Alternative 7: 49.6% 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. metadata-eval 100.0%

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

   4. metadata-eval 100.0%

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

   5. metadata-eval 100.0%

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

   6. associate-*l/ 100.0%

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

   7. associate-/r/ 100.0%

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

   8. metadata-eval 100.0%

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

   9. metadata-eval 100.0%

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

   10. metadata-eval 100.0%

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

   11. associate-/r* 100.0%

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

   12. associate-*r/ 100.0%

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

   13. associate-/l* 100.0%

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

   14. associate-/r/ 100.0%

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

   15. metadata-eval 100.0%

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

   16. metadata-eval 100.0%

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

   17. metadata-eval 100.0%

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

   18. *-lft-identity 100.0%

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

   19. sub-neg 100.0%

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

   20. 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. Taylor expanded in x around 0 48.7%

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

5. Taylor expanded in x around 0 47.8%

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

6. Final simplification 47.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023309 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))