3frac (problem 3.3.3)

Percentage Accurate: 69.4% → 99.8%

Time: 9.8s

Alternatives: 5

Speedup: 2.1×

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: 69.4% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{1}{x \cdot 0.5 + -0.5 \cdot \frac{1}{x}}}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (/ 1.0 (+ (* x 0.5) (* -0.5 (/ 1.0 x)))) x) x))

C

double code(double x) {
	return ((1.0 / ((x * 0.5) + (-0.5 * (1.0 / x)))) / x) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / ((x * 0.5d0) + ((-0.5d0) * (1.0d0 / x)))) / x) / x
end function

Java

public static double code(double x) {
	return ((1.0 / ((x * 0.5) + (-0.5 * (1.0 / x)))) / x) / x;
}

Python

def code(x):
	return ((1.0 / ((x * 0.5) + (-0.5 * (1.0 / x)))) / x) / x

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(Float64(x * 0.5) + Float64(-0.5 * Float64(1.0 / x)))) / x) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / ((x * 0.5) + (-0.5 * (1.0 / x)))) / x) / x;
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(N[(x * 0.5), $MachinePrecision] + N[(-0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 67.8%

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x + 1}\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + 1\right)\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} + \frac{1}{x - 1}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x\right)\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \left(\frac{1}{x - 1} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{2}{x}}\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{\color{blue}{x}}\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{\color{blue}{x}}\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{x}}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{x}\right)\right)\right) \]

   15. metadata-eval 67.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(-2, x\right)\right)\right) \]

3. Simplified 67.8%

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

5. Taylor expanded in x around -inf

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

   1. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot \left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2\right)}{\color{blue}{{x}^{3}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2\right)\right), \color{blue}{\left({x}^{3}\right)}\right) \]

7. Simplified 98.4%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2 + \frac{2 + \frac{2}{x \cdot x}}{x \cdot x}}{x}}{x}\right), \color{blue}{x}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 + \frac{2 + \frac{2}{x \cdot x}}{x \cdot x}}{x}\right), x\right), x\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + \frac{2 + \frac{2}{x \cdot x}}{x \cdot x}\right), x\right), x\right), x\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2 + \frac{2}{x \cdot x}}{x \cdot x}\right)\right), x\right), x\right), x\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(2 + \frac{2}{x \cdot x}\right), \left(x \cdot x\right)\right)\right), x\right), x\right), x\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2}{x \cdot x}\right)\right), \left(x \cdot x\right)\right)\right), x\right), x\right), x\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \left(x \cdot x\right)\right)\right), \left(x \cdot x\right)\right)\right), x\right), x\right), x\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot x\right)\right)\right), x\right), x\right), x\right) \]

   11. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right), x\right), x\right), x\right) \]

9. Applied egg-rr 99.7%

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

    1. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{x}{2 + \frac{2 + \frac{2}{x \cdot x}}{x \cdot x}}}\right), x\right), x\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{x}{2 + \frac{2 + \frac{2}{x \cdot x}}{x \cdot x}}\right)\right), x\right), x\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \left(2 + \frac{2 + \frac{2}{x \cdot x}}{x \cdot x}\right)\right)\right), x\right), x\right) \]

    4. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{2 + \frac{2}{x \cdot x}}{x \cdot x}\right)\right)\right)\right), x\right), x\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(2 + \frac{2}{x \cdot x}\right), \left(x \cdot x\right)\right)\right)\right)\right), x\right), x\right) \]

    6. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2}{x \cdot x}\right)\right), \left(x \cdot x\right)\right)\right)\right)\right), x\right), x\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \left(x \cdot x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right), x\right), x\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right), x\right), x\right) \]

    9. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), x\right), x\right) \]

11. Applied egg-rr 99.7%

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

12. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), x\right), x\right) \]
13. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right), x\right), x\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x\right)\right), x\right), x\right) \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x\right)\right)\right), x\right), x\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(x \cdot \frac{1}{2}\right), \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x\right)\right)\right), x\right), x\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x\right)\right)\right), x\right), x\right) \]

    6. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right), x\right), x\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\left(\frac{-1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right), x\right), x\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\frac{-1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot x\right)\right)\right)\right), x\right), x\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\frac{-1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot x\right)\right)\right)\right), x\right), x\right) \]

    10. associate-/r* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\frac{-1}{2} \cdot \left(\frac{\frac{1}{x}}{x} \cdot x\right)\right)\right)\right), x\right), x\right) \]

    11. associate-*l/ N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\frac{-1}{2} \cdot \frac{\frac{1}{x} \cdot x}{x}\right)\right)\right), x\right), x\right) \]

    12. lft-mult-inverse N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\frac{-1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right), x\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{1}{x}\right)\right)\right)\right), x\right), x\right) \]

    14. /-lowering-/.f64 99.8%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(1, x\right)\right)\right)\right), x\right), x\right) \]

14. Simplified 99.8%

    \[\leadsto \frac{\frac{\frac{1}{\color{blue}{x \cdot 0.5 + -0.5 \cdot \frac{1}{x}}}}{x}}{x} \]
15. Add Preprocessing

Alternative 2: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x + 1}\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + 1\right)\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} + \frac{1}{x - 1}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x\right)\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \left(\frac{1}{x - 1} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{2}{x}}\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{\color{blue}{x}}\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{\color{blue}{x}}\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{x}}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{x}\right)\right)\right) \]

   15. metadata-eval 67.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(-2, x\right)\right)\right) \]

3. Simplified 67.8%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 + 2 \cdot \frac{1}{{x}^{2}}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(2 \cdot \frac{1}{{x}^{2}}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2 \cdot 1}{{x}^{2}}\right)\right), \left({x}^{3}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2}{{x}^{2}}\right)\right), \left({x}^{3}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \left({x}^{2}\right)\right)\right), \left({x}^{3}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \left(x \cdot x\right)\right)\right), \left({x}^{3}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({x}^{3}\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   12. *-lowering-*.f64 98.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

7. Simplified 98.2%

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

   1. associate-*r* N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 + \frac{2}{x \cdot x}}{x \cdot x}\right), \color{blue}{x}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + \frac{2}{x \cdot x}\right), \left(x \cdot x\right)\right), x\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2}{x \cdot x}\right)\right), \left(x \cdot x\right)\right), x\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \left(x \cdot x\right)\right)\right), \left(x \cdot x\right)\right), x\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot x\right)\right), x\right) \]

   8. *-lowering-*.f64 99.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), x\right) \]

9. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\frac{2 + \frac{2}{x \cdot x}}{x \cdot x}}{x}} \]
10. Add Preprocessing

Alternative 3: 98.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x + 1}\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + 1\right)\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} + \frac{1}{x - 1}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x\right)\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \left(\frac{1}{x - 1} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{2}{x}}\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{\color{blue}{x}}\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{\color{blue}{x}}\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{x}}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{x}\right)\right)\right) \]

   15. metadata-eval 67.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(-2, x\right)\right)\right) \]

3. Simplified 67.8%

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

5. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. unpow2 N/A

      \[\leadsto \frac{2}{{x}^{2} \cdot x} \]

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x}\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot 1}{{x}^{2}}\right), x\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{{x}^{2}}\right), x\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left({x}^{2}\right)\right), x\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(x \cdot x\right)\right), x\right) \]

   11. *-lowering-*.f64 99.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right), x\right) \]

7. Simplified 99.1%

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

   1. associate-/r* N/A

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

   2. associate-/l/ N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{x}\right), \color{blue}{\left(x \cdot x\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, x\right), \left(\color{blue}{x} \cdot x\right)\right) \]

   5. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, x\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

9. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{x}}{x \cdot x}} \]
10. Add Preprocessing

Alternative 4: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x + 1}\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + 1\right)\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} + \frac{1}{x - 1}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x\right)\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \left(\frac{1}{x - 1} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{2}{x}}\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{\color{blue}{x}}\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{\color{blue}{x}}\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{x}}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{x}\right)\right)\right) \]

   15. metadata-eval 67.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(-2, x\right)\right)\right) \]

3. Simplified 67.8%

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

5. Taylor expanded in x around inf

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

   1. unpow3 N/A

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

   2. unpow2 N/A

      \[\leadsto \frac{2}{{x}^{2} \cdot x} \]

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x}\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot 1}{{x}^{2}}\right), x\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{{x}^{2}}\right), x\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left({x}^{2}\right)\right), x\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(x \cdot x\right)\right), x\right) \]

   11. *-lowering-*.f64 99.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right), x\right) \]

7. Simplified 99.1%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   4. *-lowering-*.f64 97.9%

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

9. Applied egg-rr 97.9%

   \[\leadsto \color{blue}{\frac{2}{x \cdot \left(x \cdot x\right)}} \]
10. Add Preprocessing

Alternative 5: 5.1% 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 67.8%

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{x + 1}\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + 1\right)\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)} + \frac{1}{x - 1}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x\right)\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\frac{2}{x}\right)\right) + \frac{1}{x - 1}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \left(\frac{1}{x - 1} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{x - 1}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{2}{x}\right)\right)}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x - 1\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{2}{x}}\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{\color{blue}{x}}\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{2}{\color{blue}{x}}\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\mathsf{neg}\left(\frac{2}{x}\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{x}}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{x}\right)\right)\right) \]

   15. metadata-eval 67.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, x\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right), \mathsf{/.f64}\left(-2, x\right)\right)\right) \]

3. Simplified 67.8%

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

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 4.9%

      \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{x}\right) \]

7. Simplified 4.9%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]
8. Add Preprocessing

Developer Target 1: 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 2024158 
(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))))