3frac (problem 3.3.3)

Percentage Accurate: 85.3% → 99.7%

Time: 12.0s

Alternatives: 7

Speedup: 1.4×

Specification

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: 85.3% 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: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

3. Simplified 83.2%

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

   1. clear-num 83.2%

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

   2. frac-2neg 83.2%

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

   3. neg-mul-1 83.2%

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

   4. *-commutative 83.2%

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

   5. metadata-eval 83.2%

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

   6. metadata-eval 83.2%

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

   7. clear-num 83.2%

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

   8. frac-add 60.3%

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

   9. +-commutative 60.3%

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

   10. *-commutative 60.3%

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

   11. fma-def 60.3%

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

   12. metadata-eval 60.3%

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

   13. *-lft-identity 60.3%

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

   14. *-commutative 60.3%

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

   15. neg-mul-1 60.3%

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

   16. *-commutative 60.3%

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

   17. +-commutative 60.3%

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

   18. distribute-lft-in 60.3%

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

   19. *-commutative 60.3%

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

   20. neg-mul-1 60.3%

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

   21. *-commutative 60.3%

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

   22. neg-mul-1 60.3%

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

   23. remove-double-neg 60.3%

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

   24. *-rgt-identity 60.3%

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

   25. *-commutative 60.3%

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

5. Applied egg-rr 60.3%

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

   1. fma-udef 60.3%

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

   2. unsub-neg 60.3%

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

   3. +-commutative 60.3%

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

   4. distribute-rgt-in 60.3%

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

   5. metadata-eval 60.3%

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

7. Simplified 60.3%

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

   1. frac-add 61.4%

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

   2. frac-2neg 61.4%

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

   3. *-lft-identity 61.4%

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

   4. fma-def 60.4%

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

   5. *-commutative 60.4%

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

   6. +-commutative 60.4%

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

   7. fma-def 60.4%

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

   8. +-commutative 60.4%

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

   9. associate-*r* 60.4%

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

   10. *-commutative 60.4%

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

   11. +-commutative 60.4%

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

   12. distribute-rgt-out 60.4%

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

   13. *-commutative 60.4%

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

   14. fma-def 60.4%

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

   15. *-lft-identity 60.4%

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

9. Applied egg-rr 60.4%

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

11. Simplified 61.1%

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

    1. remove-double-neg 61.1%

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

    2. sub-neg 61.1%

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

    3. distribute-neg-in 61.1%

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

    4. sub-neg 61.1%

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

    5. +-commutative 61.1%

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

    6. *-rgt-identity 61.1%

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

    7. sqr-neg 61.1%

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

    8. distribute-lft-in 61.4%

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

    9. sub-neg 61.4%

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

    10. distribute-lft-neg-in 61.4%

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

    11. remove-double-neg 61.4%

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

    12. remove-double-neg 61.4%

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

    13. +-commutative 61.4%

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

    14. *-commutative 61.4%

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

13. Applied egg-rr 61.4%

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

14. Taylor expanded in x around 0 99.4%

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

15. Final simplification 99.4%

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

Alternative 2: 77.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-1.0d0) / (x * x)
    else if (x <= 1.0d0) then
        tmp = -x - (2.0d0 / x)
    else
        tmp = 1.0d0 / (x * x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0 / (x * x)
	elif x <= 1.0:
		tmp = -x - (2.0 / x)
	else:
		tmp = 1.0 / (x * x)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in x around inf 68.5%

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

   5. Taylor expanded in x around inf 53.2%

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

      1. unpow2 53.2%

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

   7. Simplified 53.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.0%

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

      1. distribute-neg-in 99.0%

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

      2. metadata-eval 99.0%

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

      3. +-commutative 99.0%

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

      4. distribute-lft-neg-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-*r/ 99.0%

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

      7. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in x around 0 99.0%

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

      1. neg-mul-1 99.0%

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

      2. associate-*r/ 99.0%

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

      3. metadata-eval 99.0%

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

   9. Simplified 99.0%

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

   if 1 < x

   1. Initial program 64.2%

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

   3. Simplified 64.2%

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

      1. flip-+ 24.2%

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

      2. associate-/r/ 17.0%

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

      3. metadata-eval 17.0%

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

   5. Applied egg-rr 17.0%

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

   6. Taylor expanded in x around inf 63.9%

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

   7. Taylor expanded in x around inf 47.4%

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

      1. unpow2 47.4%

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

   9. Simplified 47.4%

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

4. Final simplification 74.7%

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

Alternative 3: 76.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = -2.0 / 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) / (x * x)
    else
        tmp = (-2.0d0) / 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 / (x * x);
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = -1.0 / (x * x);
	else
		tmp = -2.0 / 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[(x * x), $MachinePrecision]), $MachinePrecision], N[(-2.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in x around inf 65.5%

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

   5. Taylor expanded in x around inf 49.0%

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

      1. unpow2 49.0%

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

   7. Simplified 49.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.0%

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

4. Final simplification 74.2%

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

Alternative 4: 77.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-1.0d0) / (x * x)
    else if (x <= 1.0d0) then
        tmp = (-2.0d0) / x
    else
        tmp = 1.0d0 / (x * x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0 / (x * x)
	elif x <= 1.0:
		tmp = -2.0 / x
	else:
		tmp = 1.0 / (x * x)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in x around inf 68.5%

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

   5. Taylor expanded in x around inf 53.2%

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

      1. unpow2 53.2%

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

   7. Simplified 53.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.0%

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

   if 1 < x

   1. Initial program 64.2%

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

   3. Simplified 64.2%

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

      1. flip-+ 24.2%

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

      2. associate-/r/ 17.0%

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

      3. metadata-eval 17.0%

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

   5. Applied egg-rr 17.0%

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

   6. Taylor expanded in x around inf 63.9%

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

   7. Taylor expanded in x around inf 47.4%

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

      1. unpow2 47.4%

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

   9. Simplified 47.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x}\\ \end{array} \]

Alternative 5: 83.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

3. Simplified 83.2%

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

4. Taylor expanded in x around 0 51.5%

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

   1. distribute-neg-in 51.5%

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

   2. metadata-eval 51.5%

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

   3. +-commutative 51.5%

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

   4. distribute-lft-neg-in 51.5%

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

   5. metadata-eval 51.5%

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

   6. associate-*r/ 51.5%

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

   7. metadata-eval 51.5%

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

6. Simplified 51.5%

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

7. Taylor expanded in x around 0 82.3%

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

8. Final simplification 82.3%

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

Alternative 6: 52.7% 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 83.2%

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

3. Simplified 83.2%

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

4. Taylor expanded in x around 0 52.4%

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

5. Final simplification 52.4%

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

Alternative 7: 3.3% 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 83.2%

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

3. Simplified 83.2%

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

4. Taylor expanded in x around 0 51.5%

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

   1. distribute-neg-in 51.5%

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

   2. metadata-eval 51.5%

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

   3. +-commutative 51.5%

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

   4. distribute-lft-neg-in 51.5%

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

   5. metadata-eval 51.5%

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

   6. associate-*r/ 51.5%

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

   7. metadata-eval 51.5%

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

6. Simplified 51.5%

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

7. Taylor expanded in x around inf 3.2%

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

8. Final simplification 3.2%

   \[\leadsto -1 \]

Developer target: 99.7% 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 2023297 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

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

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