3frac (problem 3.3.3)

Percentage Accurate: 69.9% → 99.5%

Time: 10.5s

Alternatives: 7

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: 69.9% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {x}^{-3} \cdot \left(2 - \frac{-2}{x \cdot x}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

3. Taylor expanded in x around inf

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

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

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

   2. remove-double-neg N/A

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

   3. sub-neg N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

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

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.0

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

5. Simplified 99.0%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. distribute-neg-in N/A

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

   10. metadata-eval N/A

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

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

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

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

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

   15. *-lowering-*.f64 99.7

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

7. Applied egg-rr 99.7%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. cube-mult N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

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

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

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

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

   8. cube-mult N/A

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

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

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

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

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. frac-times N/A

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

   15. unsub-neg N/A

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

   16. --lowering--.f64 N/A

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

   17. frac-times N/A

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

   18. metadata-eval N/A

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

   19. /-lowering-/.f64 N/A

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

   20. *-lowering-*.f64 99.0

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

9. Applied egg-rr 99.0%

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

    1. inv-pow N/A

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

    2. cube-unmult N/A

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

    3. pow-pow N/A

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

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

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

    5. metadata-eval 99.9

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

11. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{{x}^{-3}} \cdot \left(2 - \frac{-2}{x \cdot x}\right) \]
12. Add Preprocessing

Alternative 2: 99.3% accurate, 0.9× speedup

Math

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

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(Float64(2.0 + Float64(2.0 / Float64(x * x))) / 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[(N[(2.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 70.0%

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

3. Taylor expanded in x around inf

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

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

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

   2. remove-double-neg N/A

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

   3. sub-neg N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

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

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.0

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

5. Simplified 99.0%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. distribute-neg-in N/A

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

   10. metadata-eval N/A

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

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

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

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

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

   15. *-lowering-*.f64 99.7

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

7. Applied egg-rr 99.7%

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

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \frac{2}{x \cdot x}}{x \cdot x} \cdot \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

3. Taylor expanded in x around inf

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

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

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

   2. remove-double-neg N/A

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

   3. sub-neg N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

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

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.0

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

5. Simplified 99.0%

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

   1. div-inv N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-in N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r* N/A

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

   7. times-frac N/A

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

   8. distribute-neg-frac N/A

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{2 + \frac{2}{x \cdot x}}{x \cdot x} \cdot \frac{1}{x}} \]
8. Add Preprocessing

Alternative 4: 98.6% accurate, 1.1× speedup

Math

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

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(2.0 - Float64(-2.0 / Float64(x * x))) / Float64(x * Float64(x * x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

3. Taylor expanded in x around inf

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

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

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

   2. remove-double-neg N/A

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

   3. sub-neg N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

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

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.0

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

5. Simplified 99.0%

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

Alternative 5: 98.8% accurate, 1.6× 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 70.0%

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

3. Taylor expanded in x around inf

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

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

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 98.2

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

5. Simplified 98.2%

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

   1. associate-/r* N/A

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

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

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

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

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

   4. *-lowering-*.f64 98.9

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

7. Applied egg-rr 98.9%

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

Alternative 6: 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 70.0%

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

3. Taylor expanded in x around inf

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

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

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 98.2

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

5. Simplified 98.2%

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

Alternative 7: 5.1% accurate, 3.8× 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 70.0%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 4.9

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

5. Simplified 4.9%

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

Developer Target 1: 99.1% accurate, 1.8× 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 2024199 
(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))))