3frac (problem 3.3.3)

Percentage Accurate: 69.3% → 99.8%

Time: 8.4s

Alternatives: 5

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: 69.3% 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: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.9%

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

3. Simplified 66.9%

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

   1. *-un-lft-identity 66.9%

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

   2. +-commutative 66.9%

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

   3. associate-+l+ 66.9%

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

6. Applied egg-rr 66.9%

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

   1. *-lft-identity 66.9%

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

   2. +-commutative 66.9%

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

8. Simplified 66.9%

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

   1. +-commutative 66.9%

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

   2. frac-add 17.5%

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

   3. frac-add 20.2%

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

   4. *-un-lft-identity 20.2%

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

10. Applied egg-rr 20.2%

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

11. Taylor expanded in x around 0 98.6%

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

    1. associate-/r* 99.8%

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

    2. div-inv 99.7%

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

    3. *-commutative 99.7%

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

    4. metadata-eval 99.7%

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

    5. sub-neg 99.7%

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

    6. difference-of-sqr-1 99.8%

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

    7. fma-neg 99.8%

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

    8. metadata-eval 99.8%

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

13. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{1}{x}} \]
14. Add Preprocessing

Alternative 2: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

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

3. Simplified 66.9%

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

   1. *-un-lft-identity 66.9%

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

   2. +-commutative 66.9%

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

   3. associate-+l+ 66.9%

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

6. Applied egg-rr 66.9%

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

   1. *-lft-identity 66.9%

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

   2. +-commutative 66.9%

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

8. Simplified 66.9%

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

   1. +-commutative 66.9%

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

   2. frac-add 17.5%

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

   3. frac-add 20.2%

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

   4. *-un-lft-identity 20.2%

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

10. Applied egg-rr 20.2%

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

    1. *-rgt-identity 20.2%

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

    2. distribute-lft-in 20.2%

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

    3. *-rgt-identity 20.2%

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

    4. *-rgt-identity 20.2%

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

    5. *-rgt-identity 20.2%

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

12. Applied egg-rr 20.2%

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

13. Taylor expanded in x around 0 98.6%

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

14. Final simplification 98.6%

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

Alternative 3: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

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

3. Simplified 66.9%

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

   1. *-un-lft-identity 66.9%

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

   2. +-commutative 66.9%

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

   3. associate-+l+ 66.9%

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

6. Applied egg-rr 66.9%

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

   1. *-lft-identity 66.9%

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

   2. +-commutative 66.9%

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

8. Simplified 66.9%

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

   1. +-commutative 66.9%

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

   2. frac-add 17.5%

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

   3. frac-add 20.2%

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

   4. *-un-lft-identity 20.2%

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

10. Applied egg-rr 20.2%

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

11. Taylor expanded in x around 0 98.6%

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

12. Final simplification 98.6%

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

Alternative 4: 97.0% accurate, 1.7× speedup

Math

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

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(x * Float64(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[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.9%

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

3. Simplified 66.9%

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

   1. *-un-lft-identity 66.9%

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

   2. +-commutative 66.9%

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

   3. associate-+l+ 66.9%

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

6. Applied egg-rr 66.9%

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

   1. *-lft-identity 66.9%

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

   2. +-commutative 66.9%

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

8. Simplified 66.9%

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

   1. +-commutative 66.9%

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

   2. frac-add 17.5%

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

   3. frac-add 20.2%

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

   4. *-un-lft-identity 20.2%

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

10. Applied egg-rr 20.2%

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

11. Taylor expanded in x around 0 98.6%

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

12. Taylor expanded in x around inf 95.4%

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

13. Final simplification 95.4%

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

Alternative 5: 67.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 66.9%

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

3. Simplified 66.9%

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

   1. *-un-lft-identity 66.9%

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

   2. +-commutative 66.9%

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

   3. associate-+l+ 66.9%

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

6. Applied egg-rr 66.9%

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

   1. *-lft-identity 66.9%

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

   2. +-commutative 66.9%

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

8. Simplified 66.9%

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

9. Taylor expanded in x around inf 64.1%

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

10. Taylor expanded in x around 0 64.1%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 99.1% 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 2024144 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64
  :pre (> (fabs x) 1.0)

  :alt
  (! :herbie-platform default (/ 2 (* x (- (* x x) 1))))

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