Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 8

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + 1}{x}\\ \frac{x + \left(-1 + t_0\right)}{t_0 \cdot \left(x + -1\right)} \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = (x + 1.0) / x;
	return (x + (-1.0 + t_0)) / (t_0 * (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (x + 1.0d0) / x
    code = (x + ((-1.0d0) + t_0)) / (t_0 * (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	double t_0 = (x + 1.0) / x;
	return (x + (-1.0 + t_0)) / (t_0 * (x + -1.0));
}

Python

def code(x):
	t_0 = (x + 1.0) / x
	return (x + (-1.0 + t_0)) / (t_0 * (x + -1.0))

Julia

function code(x)
	t_0 = Float64(Float64(x + 1.0) / x)
	return Float64(Float64(x + Float64(-1.0 + t_0)) / Float64(t_0 * Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	t_0 = (x + 1.0) / x;
	tmp = (x + (-1.0 + t_0)) / (t_0 * (x + -1.0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]}, N[(N[(x + N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x + -1.0), $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. +-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. frac-add 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. *-commutative 100.0%

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

   6. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+l+ 100.0%

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

   3. +-commutative 100.0%

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

   4. *-rgt-identity 100.0%

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

   5. +-commutative 100.0%

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

   6. *-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \frac{x + \left(-1 + \frac{x + 1}{x}\right)}{\frac{x + 1}{x} \cdot \left(x + -1\right)} \]
10. Add Preprocessing

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.76 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.76) (not (<= x 1.6)))
   (/ x (+ x (/ -1.0 x)))
   (+ x (/ 1.0 (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.76) || !(x <= 1.6)) {
		tmp = x / (x + (-1.0 / 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 <= (-0.76d0)) .or. (.not. (x <= 1.6d0))) then
        tmp = x / (x + ((-1.0d0) / 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 <= -0.76) || !(x <= 1.6)) {
		tmp = x / (x + (-1.0 / x));
	} else {
		tmp = x + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.76) or not (x <= 1.6):
		tmp = x / (x + (-1.0 / x))
	else:
		tmp = x + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.76) || !(x <= 1.6))
		tmp = Float64(x / Float64(x + Float64(-1.0 / 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 <= -0.76) || ~((x <= 1.6)))
		tmp = x / (x + (-1.0 / x));
	else
		tmp = x + (1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.76], N[Not[LessEqual[x, 1.6]], $MachinePrecision]], N[(x / N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.76000000000000001 or 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

       1. expm1-log1p-u 100.0%

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

       2. expm1-udef 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. *-lft-identity 100.0%

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

       4. associate-*l/ 100.0%

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

       5. distribute-lft-in 100.0%

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

       6. lft-mult-inverse 100.0%

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

       7. *-rgt-identity 100.0%

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

       8. associate-+r+ 100.0%

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

       9. metadata-eval 100.0%

          \[\leadsto \frac{x + \left(\color{blue}{0} + \frac{1}{x}\right)}{x - \frac{1}{x}} \]

   13. Simplified 100.0%

       \[\leadsto \frac{x + \color{blue}{\left(0 + \frac{1}{x}\right)}}{x - \frac{1}{x}} \]

   14. Taylor expanded in x around inf 99.1%

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

   if -0.76000000000000001 < 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 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.76 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.6× speedup

Math

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

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. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.0%

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

   if -1 < x < 1.9199999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.9%

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

Alternative 4: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.55:\\ \;\;\;\;\frac{1}{x} + \frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.55)
   (+ (/ 1.0 x) (/ x (+ x 1.0)))
   (if (<= x 1.6) (+ x (/ 1.0 (+ x -1.0))) (/ x (+ x (/ -1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.55) {
		tmp = (1.0 / x) + (x / (x + 1.0));
	} else if (x <= 1.6) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = x / (x + (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.55d0)) then
        tmp = (1.0d0 / x) + (x / (x + 1.0d0))
    else if (x <= 1.6d0) then
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    else
        tmp = x / (x + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.55) {
		tmp = (1.0 / x) + (x / (x + 1.0));
	} else if (x <= 1.6) {
		tmp = x + (1.0 / (x + -1.0));
	} else {
		tmp = x / (x + (-1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.55:
		tmp = (1.0 / x) + (x / (x + 1.0))
	elif x <= 1.6:
		tmp = x + (1.0 / (x + -1.0))
	else:
		tmp = x / (x + (-1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.55)
		tmp = Float64(Float64(1.0 / x) + Float64(x / Float64(x + 1.0)));
	elseif (x <= 1.6)
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	else
		tmp = Float64(x / Float64(x + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.55)
		tmp = (1.0 / x) + (x / (x + 1.0));
	elseif (x <= 1.6)
		tmp = x + (1.0 / (x + -1.0));
	else
		tmp = x / (x + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.55], N[(N[(1.0 / x), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6], N[(x + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.55000000000000004

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.9%

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

   if -0.55000000000000004 < 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 98.7%

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

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-add 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. +-commutative 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 100.0%

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

       1. expm1-log1p-u 100.0%

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

       2. expm1-udef 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. *-lft-identity 100.0%

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

       4. associate-*l/ 100.0%

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

       5. distribute-lft-in 100.0%

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

       6. lft-mult-inverse 100.0%

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

       7. *-rgt-identity 100.0%

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

       8. associate-+r+ 100.0%

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

       9. metadata-eval 100.0%

          \[\leadsto \frac{x + \left(\color{blue}{0} + \frac{1}{x}\right)}{x - \frac{1}{x}} \]

   13. Simplified 100.0%

       \[\leadsto \frac{x + \color{blue}{\left(0 + \frac{1}{x}\right)}}{x - \frac{1}{x}} \]

   14. Taylor expanded in x around inf 99.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.55:\\ \;\;\;\;\frac{1}{x} + \frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \frac{-1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.0%

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

   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 98.7%

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

   6. Taylor expanded in x around 0 98.7%

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

4. Final simplification 98.8%

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

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x - N[(-1.0 / x), $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. 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. +-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. frac-add 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. *-commutative 100.0%

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

   6. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+l+ 100.0%

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

   3. +-commutative 100.0%

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

   4. *-rgt-identity 100.0%

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

   5. +-commutative 100.0%

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

   6. *-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 100.0%

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

    1. expm1-log1p-u 68.3%

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

    2. expm1-udef 68.3%

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

11. Applied egg-rr 68.3%

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

    1. expm1-def 68.3%

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

    2. expm1-log1p 100.0%

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

    3. *-lft-identity 100.0%

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

    4. associate-*l/ 100.0%

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

    5. distribute-lft-in 100.0%

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

    6. lft-mult-inverse 100.0%

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

    7. *-rgt-identity 100.0%

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

    8. associate-+r+ 100.0%

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

    9. metadata-eval 100.0%

       \[\leadsto \frac{x + \left(\color{blue}{0} + \frac{1}{x}\right)}{x - \frac{1}{x}} \]

13. Simplified 100.0%

    \[\leadsto \frac{x + \color{blue}{\left(0 + \frac{1}{x}\right)}}{x - \frac{1}{x}} \]

14. Taylor expanded in x around 0 100.0%

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

    1. metadata-eval 100.0%

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

    2. distribute-neg-frac 100.0%

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

    3. sub-neg 100.0%

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

16. Simplified 100.0%

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

17. Final simplification 100.0%

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

Alternative 8: 49.5% 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 54.1%

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

6. Taylor expanded in x around 0 53.5%

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

7. Final simplification 53.5%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))