Asymptote B

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

Alternatives: 7

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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

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

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. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 1.0 + (2.0 / (x * 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 <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 1.0d0 + (2.0d0 / (x * 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 <= -1.0) || !(x <= 1.0)) {
		tmp = 1.0 + (2.0 / (x * x));
	} else {
		tmp = x * ((x * -2.0) + (-1.0 / x));
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(1.0 + Float64(2.0 / Float64(x * 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 <= -1.0) || ~((x <= 1.0)))
		tmp = 1.0 + (2.0 / (x * x));
	else
		tmp = x * ((x * -2.0) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(1.0 + N[(2.0 / N[(x * 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 < -1 or 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{1}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto 1 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}} \]

      2. metadata-eval 99.9%

         \[\leadsto 1 + \frac{\color{blue}{2}}{{x}^{2}} \]

      3. unpow2 99.9%

         \[\leadsto 1 + \frac{2}{\color{blue}{x \cdot x}} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{2}{x \cdot x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. frac-add 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. associate-/r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 98.9%

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

4. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(1.0 + N[(2.0 / N[(x * 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. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{1}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto 1 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}} \]

      2. metadata-eval 99.9%

         \[\leadsto 1 + \frac{\color{blue}{2}}{{x}^{2}} \]

      3. unpow2 99.9%

         \[\leadsto 1 + \frac{2}{\color{blue}{x \cdot x}} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{2}{x \cdot x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. frac-add 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. associate-/r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 98.6%

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

4. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;1 + \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.45)))
   (+ 1.0 (/ 2.0 (* x x)))
   (+ x (/ 1.0 (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.45)) {
		tmp = 1.0 + (2.0 / (x * x));
	} else {
		tmp = x + (1.0 / (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.45d0))) then
        tmp = 1.0d0 + (2.0d0 / (x * x))
    else
        tmp = x + (1.0d0 / (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.45)) {
		tmp = 1.0 + (2.0 / (x * x));
	} else {
		tmp = x + (1.0 / (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.45):
		tmp = 1.0 + (2.0 / (x * x))
	else:
		tmp = x + (1.0 / (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.45))
		tmp = Float64(1.0 + Float64(2.0 / Float64(x * x)));
	else
		tmp = Float64(x + Float64(1.0 / Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.45)))
		tmp = 1.0 + (2.0 / (x * x));
	else
		tmp = x + (1.0 / (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.45]], $MachinePrecision]], N[(1.0 + N[(2.0 / N[(x * 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 < -1 or 1.44999999999999996 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{1}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto 1 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}} \]

      2. metadata-eval 99.9%

         \[\leadsto 1 + \frac{\color{blue}{2}}{{x}^{2}} \]

      3. unpow2 99.9%

         \[\leadsto 1 + \frac{2}{\color{blue}{x \cdot x}} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{2}{x \cdot x}} \]

   if -1 < x < 1.44999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;1 + \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{x + -1}\\ \end{array} \]

Alternative 5: 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. Taylor expanded in x around inf 99.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. frac-add 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-/r* 100.0%

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

      2. associate-/r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 98.6%

      \[\leadsto \color{blue}{-1} \]
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 6: 9.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -3.0)

C

double code(double x) {
	return -3.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -3.0d0
end function

Java

public static double code(double x) {
	return -3.0;
}

Python

def code(x):
	return -3.0

Julia

function code(x)
	return -3.0
end

MATLAB

function tmp = code(x)
	tmp = -3.0;
end

Wolfram

code[x_] := -3.0

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. frac-add 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. +-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. *-un-lft-identity 100.0%

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

   7. sub-neg 100.0%

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

   8. metadata-eval 100.0%

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

   9. sub-neg 100.0%

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

   10. metadata-eval 100.0%

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

   11. +-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. associate-/r/ 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 52.6%

   \[\leadsto \color{blue}{\left(-2 \cdot x - \frac{1}{x}\right)} \cdot x \]

8. Applied egg-rr 10.2%

   \[\leadsto \color{blue}{-3} \]

9. Final simplification 10.2%

   \[\leadsto -3 \]

Alternative 7: 50.8% 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. clear-num 100.0%

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

   2. frac-add 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. +-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. *-un-lft-identity 100.0%

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

   7. sub-neg 100.0%

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

   8. metadata-eval 100.0%

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

   9. sub-neg 100.0%

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

   10. metadata-eval 100.0%

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

   11. +-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-/r* 100.0%

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

   2. associate-/r/ 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

   6. +-commutative 99.7%

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

   7. +-commutative 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 52.8%

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

7. Final simplification 52.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023174 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1.0 (- x 1.0)) (/ x (+ x 1.0))))