3frac (problem 3.3.3)

Percentage Accurate: 69.5% → 98.8%

Time: 9.7s

Alternatives: 9

Speedup: 1.7×

Specification

\[\left|x\right| > 1\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {x}^{-3} \cdot 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x -3.0) 2.0))

C

double code(double x) {
	return pow(x, -3.0) * 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-3.0d0)) * 2.0d0
end function

Java

public static double code(double x) {
	return Math.pow(x, -3.0) * 2.0;
}

Python

def code(x):
	return math.pow(x, -3.0) * 2.0

Julia

function code(x)
	return Float64((x ^ -3.0) * 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -3.0) * 2.0;
end

Wolfram

code[x_] := N[(N[Power[x, -3.0], $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.8%

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

   1. clear-num 98.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{3}}{2}}} \]

   2. associate-/r/ 98.8%

      \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot 2} \]

   3. pow-flip 99.1%

      \[\leadsto \color{blue}{{x}^{\left(-3\right)}} \cdot 2 \]

   4. metadata-eval 99.1%

      \[\leadsto {x}^{\color{blue}{-3}} \cdot 2 \]

5. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{{x}^{-3} \cdot 2} \]

6. Final simplification 99.1%

   \[\leadsto {x}^{-3} \cdot 2 \]
7. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{x \cdot x} - \frac{2}{x}}{x \cdot \left(1 - x\right)} \end{array} \]

FPCore

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

C

double code(double x) {
	return ((2.0 / (x * x)) - (2.0 / x)) / (x * (1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.0d0 / (x * x)) - (2.0d0 / x)) / (x * (1.0d0 - x))
end function

Java

public static double code(double x) {
	return ((2.0 / (x * x)) - (2.0 / x)) / (x * (1.0 - x));
}

Python

def code(x):
	return ((2.0 / (x * x)) - (2.0 / x)) / (x * (1.0 - x))

Julia

function code(x)
	return Float64(Float64(Float64(2.0 / Float64(x * x)) - Float64(2.0 / x)) / Float64(x * Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 / (x * x)) - (2.0 / x)) / (x * (1.0 - x));
end

Wolfram

code[x_] := N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 16.7%

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

   2. /-rgt-identity 16.7%

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

   3. *-un-lft-identity 16.7%

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

   4. /-rgt-identity 16.7%

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

   5. +-commutative 16.7%

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

   6. *-commutative 16.7%

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

   7. +-commutative 16.7%

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

4. Applied egg-rr 16.7%

   \[\leadsto \color{blue}{\frac{x - \left(1 + x\right) \cdot 2}{x \cdot \left(1 + x\right)}} + \frac{1}{x - 1} \]
5. Step-by-step derivation

   1. associate-/l/ 72.9%

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

   2. +-commutative 72.9%

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

   3. frac-2neg 72.9%

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

   4. metadata-eval 72.9%

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

   5. frac-add 72.9%

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

6. Applied egg-rr 72.9%

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

7. Taylor expanded in x around inf 98.9%

   \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{x}^{2}} - 2 \cdot \frac{1}{x}}}{\left(\left(-x\right) + 1\right) \cdot x} \]
8. Step-by-step derivation

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

   3. associate-*r/ 98.9%

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

   4. metadata-eval 98.9%

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

9. Simplified 98.9%

   \[\leadsto \frac{\color{blue}{\frac{2}{{x}^{2}} - \frac{2}{x}}}{\left(\left(-x\right) + 1\right) \cdot x} \]
10. Step-by-step derivation

    1. unpow2 98.9%

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

11. Applied egg-rr 98.9%

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

12. Final simplification 98.9%

    \[\leadsto \frac{\frac{2}{x \cdot x} - \frac{2}{x}}{x \cdot \left(1 - x\right)} \]
13. Add Preprocessing

Alternative 3: 67.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 71.6%

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

4. Final simplification 71.6%

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

Alternative 4: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-2}{x}}{x \cdot \left(1 - x\right)} \end{array} \]

FPCore

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

C

double code(double x) {
	return (-2.0 / x) / (x * (1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-2.0d0) / x) / (x * (1.0d0 - x))
end function

Java

public static double code(double x) {
	return (-2.0 / x) / (x * (1.0 - x));
}

Python

def code(x):
	return (-2.0 / x) / (x * (1.0 - x))

Julia

function code(x)
	return Float64(Float64(-2.0 / x) / Float64(x * Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = (-2.0 / x) / (x * (1.0 - x));
end

Wolfram

code[x_] := N[(N[(-2.0 / x), $MachinePrecision] / N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 16.7%

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

   2. /-rgt-identity 16.7%

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

   3. *-un-lft-identity 16.7%

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

   4. /-rgt-identity 16.7%

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

   5. +-commutative 16.7%

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

   6. *-commutative 16.7%

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

   7. +-commutative 16.7%

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

4. Applied egg-rr 16.7%

   \[\leadsto \color{blue}{\frac{x - \left(1 + x\right) \cdot 2}{x \cdot \left(1 + x\right)}} + \frac{1}{x - 1} \]
5. Step-by-step derivation

   1. associate-/l/ 72.9%

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

   2. +-commutative 72.9%

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

   3. frac-2neg 72.9%

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

   4. metadata-eval 72.9%

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

   5. frac-add 72.9%

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

6. Applied egg-rr 72.9%

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

7. Taylor expanded in x around inf 98.9%

   \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{x}^{2}} - 2 \cdot \frac{1}{x}}}{\left(\left(-x\right) + 1\right) \cdot x} \]
8. Step-by-step derivation

   1. associate-*r/ 98.9%

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

   2. metadata-eval 98.9%

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

   3. associate-*r/ 98.9%

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

   4. metadata-eval 98.9%

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

9. Simplified 98.9%

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

10. Taylor expanded in x around inf 98.2%

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

11. Final simplification 98.2%

    \[\leadsto \frac{\frac{-2}{x}}{x \cdot \left(1 - x\right)} \]
12. Add Preprocessing

Alternative 5: 52.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x \cdot \left(1 - x\right)} \end{array} \]

FPCore

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

C

double code(double x) {
	return -2.0 / (x * (1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-2.0d0) / (x * (1.0d0 - x))
end function

Java

public static double code(double x) {
	return -2.0 / (x * (1.0 - x));
}

Python

def code(x):
	return -2.0 / (x * (1.0 - x))

Julia

function code(x)
	return Float64(-2.0 / Float64(x * Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = -2.0 / (x * (1.0 - x));
end

Wolfram

code[x_] := N[(-2.0 / N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-sub 16.7%

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

   2. /-rgt-identity 16.7%

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

   3. *-un-lft-identity 16.7%

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

   4. /-rgt-identity 16.7%

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

   5. +-commutative 16.7%

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

   6. *-commutative 16.7%

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

   7. +-commutative 16.7%

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

4. Applied egg-rr 16.7%

   \[\leadsto \color{blue}{\frac{x - \left(1 + x\right) \cdot 2}{x \cdot \left(1 + x\right)}} + \frac{1}{x - 1} \]
5. Step-by-step derivation

   1. associate-/l/ 72.9%

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

   2. +-commutative 72.9%

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

   3. frac-2neg 72.9%

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

   4. metadata-eval 72.9%

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

   5. frac-add 72.9%

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

6. Applied egg-rr 72.9%

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

7. Taylor expanded in x around 0 58.9%

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

8. Final simplification 58.9%

   \[\leadsto \frac{-2}{x \cdot \left(1 - x\right)} \]
9. Add Preprocessing

Alternative 6: 5.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 x))

C

double code(double x) {
	return -2.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-2.0d0) / x
end function

Java

public static double code(double x) {
	return -2.0 / x;
}

Python

def code(x):
	return -2.0 / x

Julia

function code(x)
	return Float64(-2.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = -2.0 / x;
end

Wolfram

code[x_] := N[(-2.0 / x), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 5.3%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]

4. Final simplification 5.3%

   \[\leadsto \frac{-2}{x} \]
5. Add Preprocessing

Alternative 7: 5.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 71.6%

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

4. Taylor expanded in x around 0 5.3%

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

5. Final simplification 5.3%

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

Alternative 8: 6.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 x))

C

double code(double x) {
	return 2.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / x
end function

Java

public static double code(double x) {
	return 2.0 / x;
}

Python

def code(x):
	return 2.0 / x

Julia

function code(x)
	return Float64(2.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / x;
end

Wolfram

code[x_] := N[(2.0 / x), $MachinePrecision]

Derivation

1. Initial program 72.9%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 72.9%

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

   2. cancel-sign-sub-inv 72.9%

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

   3. metadata-eval 72.9%

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

   4. div-inv 72.9%

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

   5. +-commutative 72.9%

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

   6. clear-num 72.9%

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

   7. frac-2neg 72.9%

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

   8. metadata-eval 72.9%

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

   9. frac-add 16.7%

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

4. Applied egg-rr 6.3%

   \[\leadsto \color{blue}{\frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot 0.5\right) \cdot -1}{\left(x \cdot 0.5\right) \cdot \left(-1 + \left(-x\right)\right)}} + \frac{1}{x - 1} \]
5. Step-by-step derivation

   1. *-commutative 6.3%

      \[\leadsto \frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot 0.5\right) \cdot -1}{\color{blue}{\left(-1 + \left(-x\right)\right) \cdot \left(x \cdot 0.5\right)}} + \frac{1}{x - 1} \]

   2. associate-*r* 6.3%

      \[\leadsto \frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot 0.5\right) \cdot -1}{\color{blue}{\left(\left(-1 + \left(-x\right)\right) \cdot x\right) \cdot 0.5}} + \frac{1}{x - 1} \]

   3. *-commutative 6.3%

      \[\leadsto \frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot 0.5\right) \cdot -1}{\color{blue}{\left(x \cdot \left(-1 + \left(-x\right)\right)\right)} \cdot 0.5} + \frac{1}{x - 1} \]

   4. distribute-rgt-in 6.3%

      \[\leadsto \frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot 0.5\right) \cdot -1}{\color{blue}{\left(-1 \cdot x + \left(-x\right) \cdot x\right)} \cdot 0.5} + \frac{1}{x - 1} \]

   5. neg-mul-1 6.3%

      \[\leadsto \frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot 0.5\right) \cdot -1}{\left(\color{blue}{\left(-x\right)} + \left(-x\right) \cdot x\right) \cdot 0.5} + \frac{1}{x - 1} \]

   6. *-rgt-identity 6.3%

      \[\leadsto \frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot 0.5\right) \cdot -1}{\left(\color{blue}{\left(-x\right) \cdot 1} + \left(-x\right) \cdot x\right) \cdot 0.5} + \frac{1}{x - 1} \]

   7. distribute-lft-in 6.3%

      \[\leadsto \frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot 0.5\right) \cdot -1}{\color{blue}{\left(\left(-x\right) \cdot \left(1 + x\right)\right)} \cdot 0.5} + \frac{1}{x - 1} \]

   8. distribute-lft-neg-in 6.3%

      \[\leadsto \frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot 0.5\right) \cdot -1}{\color{blue}{\left(-x \cdot \left(1 + x\right)\right)} \cdot 0.5} + \frac{1}{x - 1} \]

   9. associate-/r* 6.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(-1 + \left(-x\right)\right) + \left(x \cdot 0.5\right) \cdot -1}{-x \cdot \left(1 + x\right)}}{0.5}} + \frac{1}{x - 1} \]

6. Simplified 6.3%

   \[\leadsto \color{blue}{\frac{\frac{-1 + x \cdot -1.5}{x \cdot \left(-1 - x\right)}}{0.5}} + \frac{1}{x - 1} \]

7. Taylor expanded in x around 0 6.3%

   \[\leadsto \color{blue}{\frac{2}{x}} \]

8. Final simplification 6.3%

   \[\leadsto \frac{2}{x} \]
9. Add Preprocessing

Alternative 9: 3.4% accurate, 15.0× 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 72.9%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 3.3%

   \[\leadsto \color{blue}{\left(1 - 2 \cdot \frac{1}{x}\right)} + \frac{1}{x - 1} \]
4. Step-by-step derivation

   1. associate-*r/ 3.3%

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

   2. metadata-eval 3.3%

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

5. Simplified 3.3%

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

6. Taylor expanded in x around inf 3.3%

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

7. Final simplification 3.3%

   \[\leadsto 1 \]
8. Add Preprocessing

Developer target: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot \left(x \cdot x - 1\right)} \end{array} \]

FPCore

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

C

double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (x * ((x * x) - 1.0d0))
end function

Java

public static double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Python

def code(x):
	return 2.0 / (x * ((x * x) - 1.0))

Julia

function code(x)
	return Float64(2.0 / Float64(x * Float64(Float64(x * x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (x * ((x * x) - 1.0));
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024011 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))