Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{1}{x - 1} + \frac{x}{x + 1} \]
2. 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. *-commutative 100.0%

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

   2. associate-*r/ 77.6%

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

   3. associate-/r/ 77.5%

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

   4. associate-+l- 77.5%

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

   5. sub-neg 77.5%

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

   6. *-commutative 77.5%

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

   7. mul-1-neg 77.5%

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

   8. remove-double-neg 77.5%

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

   9. *-commutative 77.5%

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

8. Simplified 77.5%

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

10. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{1 + x} \cdot \frac{1 - \left(x + \frac{1 + x}{x}\right)}{1 - x}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 99.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{1 + x} \cdot \frac{1 - \left(x + \frac{1 + x}{x}\right)}{1 - x}\right)\right)} \]

    2. expm1-log1p 99.9%

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

    3. *-commutative 99.9%

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

    4. times-frac 76.9%

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

    5. associate-*l/ 77.5%

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

    6. associate-/r* 99.8%

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

    7. associate-/r/ 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.68 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x} + \frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -2 - \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.68) (not (<= x 1.0)))
   (+ (/ 1.0 x) (/ x (+ 1.0 x)))
   (* x (- (* x -2.0) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.68) || !(x <= 1.0)) {
		tmp = (1.0 / x) + (x / (1.0 + x));
	} else {
		tmp = x * ((x * -2.0) - (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.68d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (1.0d0 / x) + (x / (1.0d0 + x))
    else
        tmp = x * ((x * (-2.0d0)) - (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.68) || !(x <= 1.0)) {
		tmp = (1.0 / x) + (x / (1.0 + x));
	} else {
		tmp = x * ((x * -2.0) - (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.68) or not (x <= 1.0):
		tmp = (1.0 / x) + (x / (1.0 + x))
	else:
		tmp = x * ((x * -2.0) - (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.68) || !(x <= 1.0))
		tmp = Float64(Float64(1.0 / x) + Float64(x / Float64(1.0 + x)));
	else
		tmp = Float64(x * Float64(Float64(x * -2.0) - Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.68) || ~((x <= 1.0)))
		tmp = (1.0 / x) + (x / (1.0 + x));
	else
		tmp = x * ((x * -2.0) - (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.68], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] + N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * -2.0), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049 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 98.3%

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

   if -0.680000000000000049 < 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. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/r/ 99.8%

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

      4. associate-+l- 99.8%

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

      5. sub-neg 99.8%

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

      6. *-commutative 99.8%

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

      7. mul-1-neg 99.8%

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

      8. remove-double-neg 99.8%

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

      9. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.68 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x} + \frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -2 - \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(x \cdot -2 - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 1.0) (* x (- (* x -2.0) (/ 1.0 x))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x * ((x * -2.0) - (1.0 / x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = x * ((x * (-2.0d0)) - (1.0d0 / x))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x * ((x * -2.0) - (1.0 / x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.0:
		tmp = x * ((x * -2.0) - (1.0 / x))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(Float64(x * -2.0) - Float64(1.0 / x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x * ((x * -2.0) - (1.0 / x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], N[(x * N[(N[(x * -2.0), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $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. Taylor expanded in x around inf 98.3%

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

   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. 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. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/r/ 99.8%

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

      4. associate-+l- 99.8%

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

      5. sub-neg 99.8%

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

      6. *-commutative 99.8%

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

      7. mul-1-neg 99.8%

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

      8. remove-double-neg 99.8%

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

      9. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(x \cdot -2 - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 98.3%

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

   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 99.5%

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

4. Final simplification 98.9%

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

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 6: 50.3% 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 53.6%

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

6. Final simplification 53.6%

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

Reproduce

herbie shell --seed 2024017 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))