3frac (problem 3.3.3)

Percentage Accurate: 69.9% → 99.8%

Time: 6.3s

Alternatives: 4

Speedup: 1.4×

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: 69.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.4%

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

   1. sub-neg 68.4%

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

   2. distribute-neg-frac 68.4%

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

   3. metadata-eval 68.4%

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

   4. metadata-eval 68.4%

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

   5. metadata-eval 68.4%

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

   6. associate-/r* 68.4%

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

   7. metadata-eval 68.4%

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

   8. neg-mul-1 68.4%

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

   9. +-commutative 68.4%

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

   10. associate-+l+ 68.3%

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

   11. +-commutative 68.3%

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

   12. neg-mul-1 68.3%

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

   13. metadata-eval 68.3%

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

   14. associate-/r* 68.3%

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

   15. metadata-eval 68.3%

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

   16. metadata-eval 68.3%

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

   17. +-commutative 68.3%

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

   18. +-commutative 68.3%

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

   19. sub-neg 68.3%

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

   20. metadata-eval 68.3%

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

3. Simplified 68.3%

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

   1. +-commutative 68.3%

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

   2. frac-add 21.6%

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

   3. frac-add 22.2%

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

   4. *-un-lft-identity 22.2%

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

   5. *-rgt-identity 22.2%

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

   6. +-commutative 22.2%

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

   7. +-commutative 22.2%

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

   8. +-commutative 22.2%

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

   9. +-commutative 22.2%

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

6. Applied egg-rr 22.2%

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

7. Taylor expanded in x around 0 99.7%

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

   1. expm1-log1p-u 99.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\left(x + 1\right) \cdot \left(x + -1\right)\right) \cdot x}\right)\right)} \]

   2. expm1-udef 67.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\left(x + 1\right) \cdot \left(x + -1\right)\right) \cdot x}\right)} - 1} \]

   3. *-commutative 67.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{x \cdot \left(\left(x + 1\right) \cdot \left(x + -1\right)\right)}}\right)} - 1 \]

   4. associate-/r* 67.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{x}}{\left(x + 1\right) \cdot \left(x + -1\right)}}\right)} - 1 \]

   5. metadata-eval 67.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{2}{x}}{\left(x + 1\right) \cdot \left(x + \color{blue}{\left(-1\right)}\right)}\right)} - 1 \]

   6. sub-neg 67.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{2}{x}}{\left(x + 1\right) \cdot \color{blue}{\left(x - 1\right)}}\right)} - 1 \]

   7. difference-of-sqr-1 67.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{2}{x}}{\color{blue}{x \cdot x - 1}}\right)} - 1 \]

   8. fma-neg 67.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{2}{x}}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)} - 1 \]

   9. metadata-eval 67.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)} - 1 \]

9. Applied egg-rr 67.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, -1\right)}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 99.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

    2. expm1-log1p 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.2% 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 68.4%

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

   1. sub-neg 68.4%

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

   2. distribute-neg-frac 68.4%

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

   3. metadata-eval 68.4%

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

   4. metadata-eval 68.4%

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

   5. metadata-eval 68.4%

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

   6. associate-/r* 68.4%

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

   7. metadata-eval 68.4%

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

   8. neg-mul-1 68.4%

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

   9. +-commutative 68.4%

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

   10. associate-+l+ 68.3%

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

   11. +-commutative 68.3%

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

   12. neg-mul-1 68.3%

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

   13. metadata-eval 68.3%

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

   14. associate-/r* 68.3%

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

   15. metadata-eval 68.3%

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

   16. metadata-eval 68.3%

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

   17. +-commutative 68.3%

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

   18. +-commutative 68.3%

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

   19. sub-neg 68.3%

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

   20. metadata-eval 68.3%

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

3. Simplified 68.3%

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

   1. +-commutative 68.3%

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

   2. frac-add 21.6%

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

   3. frac-add 22.2%

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

   4. *-un-lft-identity 22.2%

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

   5. *-rgt-identity 22.2%

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

   6. +-commutative 22.2%

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

   7. +-commutative 22.2%

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

   8. +-commutative 22.2%

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

   9. +-commutative 22.2%

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

6. Applied egg-rr 22.2%

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

7. Taylor expanded in x around 0 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 68.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.4%

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

   1. sub-neg 68.4%

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

   2. distribute-neg-frac 68.4%

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

   3. metadata-eval 68.4%

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

   4. metadata-eval 68.4%

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

   5. metadata-eval 68.4%

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

   6. associate-/r* 68.4%

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

   7. metadata-eval 68.4%

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

   8. neg-mul-1 68.4%

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

   9. +-commutative 68.4%

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

   10. associate-+l+ 68.3%

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

   11. +-commutative 68.3%

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

   12. neg-mul-1 68.3%

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

   13. metadata-eval 68.3%

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

   14. associate-/r* 68.3%

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

   15. metadata-eval 68.3%

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

   16. metadata-eval 68.3%

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

   17. +-commutative 68.3%

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

   18. +-commutative 68.3%

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

   19. sub-neg 68.3%

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

   20. metadata-eval 68.3%

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

3. Simplified 68.3%

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

5. Taylor expanded in x around inf 66.5%

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

6. Final simplification 66.5%

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

Alternative 4: 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 68.4%

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

   1. sub-neg 68.4%

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

   2. distribute-neg-frac 68.4%

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

   3. metadata-eval 68.4%

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

   4. metadata-eval 68.4%

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

   5. metadata-eval 68.4%

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

   6. associate-/r* 68.4%

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

   7. metadata-eval 68.4%

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

   8. neg-mul-1 68.4%

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

   9. +-commutative 68.4%

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

   10. associate-+l+ 68.3%

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

   11. +-commutative 68.3%

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

   12. neg-mul-1 68.3%

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

   13. metadata-eval 68.3%

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

   14. associate-/r* 68.3%

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

   15. metadata-eval 68.3%

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

   16. metadata-eval 68.3%

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

   17. +-commutative 68.3%

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

   18. +-commutative 68.3%

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

   19. sub-neg 68.3%

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

   20. metadata-eval 68.3%

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

3. Simplified 68.3%

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

5. Taylor expanded in x around 0 5.0%

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

6. Final simplification 5.0%

   \[\leadsto \frac{-2}{x} \]
7. 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 2024017 
(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))))