2log (problem 3.3.6)

Percentage Accurate: 23.9% → 96.8%

Time: 11.2s

Alternatives: 12

Speedup: 68.3×

Specification

\[N > 1 \land N < 10^{+40}\]

Math

\[\begin{array}{l} \\ \log \left(N + 1\right) - \log N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = log((n + 1.0d0)) - log(n)
end function

Java

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}

Python

def code(N):
	return math.log((N + 1.0)) - math.log(N)

Julia

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end

MATLAB

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end

Wolfram

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]

Initial Program: 23.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(N + 1\right) - \log N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = log((n + 1.0d0)) - log(n)
end function

Java

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}

Python

def code(N):
	return math.log((N + 1.0)) - math.log(N)

Julia

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end

MATLAB

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end

Wolfram

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N - N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  1.0
  (-
   N
   (*
    N
    (/
     (+ -0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N))
     N)))))

C

double code(double N) {
	return 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / (n - (n * (((-0.5d0) + ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n)))
end function

Java

public static double code(double N) {
	return 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
}

Python

def code(N):
	return 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)))

Julia

function code(N)
	return Float64(1.0 / Float64(N - Float64(N * Float64(Float64(-0.5 + Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N))))
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
end

Wolfram

code[N_] := N[(1.0 / N[(N - N[(N * N[(N[(-0.5 + N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 98.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}} \]

8. Taylor expanded in N around -inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot N\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(N \cdot -1\right) \cdot \left(\color{blue}{-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}} - 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(N \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + -1\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 + \color{blue}{-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)\right)\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)\right)\right)\right)\right)\right) \]

   10. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 - \color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right)\right) \]

10. Simplified 98.2%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(-1 \cdot \left(-1 - \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}\right)\right)}} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(N \cdot -1\right) \cdot \color{blue}{\left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)}\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(N \cdot -1\right) \cdot \left(-1 + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)}\right)\right)\right) \]

    3. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(N \cdot -1\right) \cdot \left(-1 + -1 \cdot \color{blue}{\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}}\right)\right)\right) \]

    4. distribute-lft-in N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(N \cdot -1\right) \cdot -1 + \color{blue}{\left(N \cdot -1\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)}\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot N\right) \cdot -1 + \left(\color{blue}{N} \cdot -1\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right) \]

    6. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(N\right)\right) \cdot -1 + \left(\color{blue}{N} \cdot -1\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(N\right)\right) \cdot -1\right), \color{blue}{\left(\left(N \cdot -1\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(N\right)\right), -1\right), \left(\color{blue}{\left(N \cdot -1\right)} \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right) \]

    9. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(0 - N\right), -1\right), \left(\left(\color{blue}{N} \cdot -1\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), -1\right), \left(\left(\color{blue}{N} \cdot -1\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), -1\right), \left(\left(-1 \cdot N\right) \cdot \left(\color{blue}{-1} \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right) \]

    12. neg-mul-1 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), -1\right), \left(\left(\mathsf{neg}\left(N\right)\right) \cdot \left(\color{blue}{-1} \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), -1\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(N\right)\right), \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)}\right)\right)\right) \]

    14. neg-sub0 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), -1\right), \mathsf{*.f64}\left(\left(0 - N\right), \left(\color{blue}{-1} \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right) \]

    15. --lowering--.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), -1\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), \left(\color{blue}{-1} \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right) \]

    16. neg-mul-1 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), -1\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right)\right) \]

    17. distribute-neg-frac N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), -1\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), \left(\frac{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}\right)\right)}{\color{blue}{N}}\right)\right)\right)\right) \]

    18. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), -1\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, N\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}\right)\right)\right), \color{blue}{N}\right)\right)\right)\right) \]

12. Applied egg-rr 98.3%

    \[\leadsto \frac{1}{\color{blue}{\left(0 - N\right) \cdot -1 + \left(0 - N\right) \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}}} \]

13. Final simplification 98.3%

    \[\leadsto \frac{1}{N - N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}} \]
14. Add Preprocessing

Alternative 2: 96.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N \cdot \left(1 + \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}\right)} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  1.0
  (*
   N
   (+
    1.0
    (/ (- 0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N)) N)))))

C

double code(double N) {
	return 1.0 / (N * (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / (n * (1.0d0 + ((0.5d0 - ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n)))
end function

Java

public static double code(double N) {
	return 1.0 / (N * (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
}

Python

def code(N):
	return 1.0 / (N * (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)))

Julia

function code(N)
	return Float64(1.0 / Float64(N * Float64(1.0 + Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N))))
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / (N * (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
end

Wolfram

code[N_] := N[(1.0 / N[(N * N[(1.0 + N[(N[(0.5 - N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 98.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}} \]

8. Taylor expanded in N around -inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot N\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(N \cdot -1\right) \cdot \left(\color{blue}{-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}} - 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(N \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + -1\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 + \color{blue}{-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)\right)\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)\right)\right)\right)\right)\right) \]

   10. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 - \color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right)\right) \]

10. Simplified 98.2%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(-1 \cdot \left(-1 - \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}\right)\right)}} \]

11. Final simplification 98.2%

    \[\leadsto \frac{1}{N \cdot \left(1 + \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}\right)} \]
12. Add Preprocessing

Alternative 3: 96.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{1 + \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}}}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  (/
   1.0
   (+
    1.0
    (/ (- 0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N)) N)))
  N))

C

double code(double N) {
	return (1.0 / (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N))) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 / (1.0d0 + ((0.5d0 - ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n))) / n
end function

Java

public static double code(double N) {
	return (1.0 / (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N))) / N;
}

Python

def code(N):
	return (1.0 / (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N))) / N

Julia

function code(N)
	return Float64(Float64(1.0 / Float64(1.0 + Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N))) / N)
end

MATLAB

function tmp = code(N)
	tmp = (1.0 / (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N))) / N;
end

Wolfram

code[N_] := N[(N[(1.0 / N[(1.0 + N[(N[(0.5 - N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 98.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}} \]

8. Taylor expanded in N around -inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot N\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(N \cdot -1\right) \cdot \left(\color{blue}{-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}} - 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(N \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + -1\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 + \color{blue}{-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)\right)\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)\right)\right)\right)\right)\right) \]

   10. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 - \color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right)\right) \]

10. Simplified 98.2%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(-1 \cdot \left(-1 - \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}\right)\right)}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{1}{\left(-1 \cdot \left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right) \cdot \color{blue}{N}} \]

    2. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{-1 \cdot \left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)}}{\color{blue}{N}} \]

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

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{-1 \cdot \left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)}\right), \color{blue}{N}\right) \]

12. Applied egg-rr 98.2%

    \[\leadsto \color{blue}{\frac{\frac{1}{1 + \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}}}{N}} \]
13. Add Preprocessing

Alternative 4: 96.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{N}}{1 + \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  (/ 1.0 N)
  (+
   1.0
   (/ (- 0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N)) N))))

C

double code(double N) {
	return (1.0 / N) / (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 / n) / (1.0d0 + ((0.5d0 - ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n))
end function

Java

public static double code(double N) {
	return (1.0 / N) / (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N));
}

Python

def code(N):
	return (1.0 / N) / (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N))

Julia

function code(N)
	return Float64(Float64(1.0 / N) / Float64(1.0 + Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N)))
end

MATLAB

function tmp = code(N)
	tmp = (1.0 / N) / (1.0 + ((0.5 - ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N));
end

Wolfram

code[N_] := N[(N[(1.0 / N), $MachinePrecision] / N[(1.0 + N[(N[(0.5 - N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 98.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}} \]

8. Taylor expanded in N around -inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot N\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(N \cdot -1\right) \cdot \left(\color{blue}{-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}} - 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(N \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + -1\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 + \color{blue}{-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)\right)\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)\right)\right)\right)\right)\right) \]

   10. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 - \color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right)\right) \]

10. Simplified 98.2%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(-1 \cdot \left(-1 - \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}\right)\right)}} \]
11. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{N}}{\color{blue}{-1 \cdot \left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)}} \]

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

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{N}\right), \color{blue}{\left(-1 \cdot \left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)}\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \left(\color{blue}{-1} \cdot \left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right) \]

    4. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \left(\mathsf{neg}\left(\left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \left(\mathsf{neg}\left(\left(-1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right)\right)\right) \]

    6. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \left(\mathsf{neg}\left(\left(-1 + -1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right) \]

    7. distribute-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \left(\left(\mathsf{neg}\left(-1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)}\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \left(1 + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}}\right)\right)\right)\right) \]

    9. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)\right)\right)\right)\right)\right) \]

    10. distribute-neg-frac2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{\mathsf{neg}\left(N\right)}\right)\right)\right)\right) \]

    11. distribute-frac-neg N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \left(1 + \frac{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}\right)\right)}{\color{blue}{\mathsf{neg}\left(N\right)}}\right)\right) \]

    12. frac-2neg N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \left(1 + \frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{\color{blue}{N}}\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{N}\right)}\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, N\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}\right), \color{blue}{N}\right)\right)\right) \]

12. Applied egg-rr 98.1%

    \[\leadsto \color{blue}{\frac{\frac{1}{N}}{1 + \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}}} \]
13. Add Preprocessing

Alternative 5: 96.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{0.041666666666666664 + N \cdot \left(N \cdot \left(N + 0.5\right) + -0.08333333333333333\right)}{N \cdot N}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  1.0
  (/
   (+ 0.041666666666666664 (* N (+ (* N (+ N 0.5)) -0.08333333333333333)))
   (* N N))))

C

double code(double N) {
	return 1.0 / ((0.041666666666666664 + (N * ((N * (N + 0.5)) + -0.08333333333333333))) / (N * N));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / ((0.041666666666666664d0 + (n * ((n * (n + 0.5d0)) + (-0.08333333333333333d0)))) / (n * n))
end function

Java

public static double code(double N) {
	return 1.0 / ((0.041666666666666664 + (N * ((N * (N + 0.5)) + -0.08333333333333333))) / (N * N));
}

Python

def code(N):
	return 1.0 / ((0.041666666666666664 + (N * ((N * (N + 0.5)) + -0.08333333333333333))) / (N * N))

Julia

function code(N)
	return Float64(1.0 / Float64(Float64(0.041666666666666664 + Float64(N * Float64(Float64(N * Float64(N + 0.5)) + -0.08333333333333333))) / Float64(N * N)))
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / ((0.041666666666666664 + (N * ((N * (N + 0.5)) + -0.08333333333333333))) / (N * N));
end

Wolfram

code[N_] := N[(1.0 / N[(N[(0.041666666666666664 + N[(N * N[(N[(N * N[(N + 0.5), $MachinePrecision]), $MachinePrecision] + -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 98.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}} \]

8. Taylor expanded in N around -inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot N\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(N \cdot -1\right) \cdot \left(\color{blue}{-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}} - 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(N \cdot \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(-1 \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + -1\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 + \color{blue}{-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)\right)\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)\right)\right)\right)\right)\right) \]

   10. unsub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \left(-1 - \color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{*.f64}\left(-1, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right)\right) \]

10. Simplified 98.2%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(-1 \cdot \left(-1 - \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}\right)\right)}} \]

11. Taylor expanded in N around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{24} + N \cdot \left(N \cdot \left(\frac{1}{2} + N\right) - \frac{1}{12}\right)}{{N}^{2}}\right)}\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{24} + N \cdot \left(N \cdot \left(\frac{1}{2} + N\right) - \frac{1}{12}\right)\right), \color{blue}{\left({N}^{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(N \cdot \left(N \cdot \left(\frac{1}{2} + N\right) - \frac{1}{12}\right)\right)\right), \left({\color{blue}{N}}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \left(N \cdot \left(\frac{1}{2} + N\right) - \frac{1}{12}\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \left(N \cdot \left(\frac{1}{2} + N\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \left(N \cdot \left(\frac{1}{2} + N\right) + \frac{-1}{12}\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\left(N \cdot \left(\frac{1}{2} + N\right)\right), \frac{-1}{12}\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{*.f64}\left(N, \left(\frac{1}{2} + N\right)\right), \frac{-1}{12}\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{*.f64}\left(N, \left(N + \frac{1}{2}\right)\right), \frac{-1}{12}\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{*.f64}\left(N, \mathsf{+.f64}\left(N, \frac{1}{2}\right)\right), \frac{-1}{12}\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{*.f64}\left(N, \mathsf{+.f64}\left(N, \frac{1}{2}\right)\right), \frac{-1}{12}\right)\right)\right), \left(N \cdot \color{blue}{N}\right)\right)\right) \]

    11. *-lowering-*.f64 98.0%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{*.f64}\left(N, \mathsf{+.f64}\left(N, \frac{1}{2}\right)\right), \frac{-1}{12}\right)\right)\right), \mathsf{*.f64}\left(N, \color{blue}{N}\right)\right)\right) \]

13. Simplified 98.0%

    \[\leadsto \frac{1}{\color{blue}{\frac{0.041666666666666664 + N \cdot \left(N \cdot \left(N + 0.5\right) + -0.08333333333333333\right)}{N \cdot N}}} \]
14. Add Preprocessing

Alternative 6: 96.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ 1.0 (/ N (+ 1.0 (/ (- -0.5 (/ (+ (/ 0.25 N) -0.3333333333333333) N)) N)))))

C

double code(double N) {
	return 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / (n / (1.0d0 + (((-0.5d0) - (((0.25d0 / n) + (-0.3333333333333333d0)) / n)) / n)))
end function

Java

public static double code(double N) {
	return 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
}

Python

def code(N):
	return 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)))

Julia

function code(N)
	return Float64(1.0 / Float64(N / Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / N) + -0.3333333333333333) / N)) / N))))
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
end

Wolfram

code[N_] := N[(1.0 / N[(N / N[(1.0 + N[(N[(-0.5 - N[(N[(N[(0.25 / N), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 98.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}} \]
8. Add Preprocessing

Alternative 7: 96.4% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (+ 1.0 (/ (- -0.5 (/ (+ (/ 0.25 N) -0.3333333333333333) N)) N)) N))

C

double code(double N) {
	return (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 + (((-0.5d0) - (((0.25d0 / n) + (-0.3333333333333333d0)) / n)) / n)) / n
end function

Java

public static double code(double N) {
	return (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)) / N;
}

Python

def code(N):
	return (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)) / N

Julia

function code(N)
	return Float64(Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / N) + -0.3333333333333333) / N)) / N)) / N)
end

MATLAB

function tmp = code(N)
	tmp = (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)) / N;
end

Wolfram

code[N_] := N[(N[(1.0 + N[(N[(-0.5 - N[(N[(N[(0.25 / N), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N}} \]
6. Add Preprocessing

Alternative 8: 95.6% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N \cdot \left(1 + \frac{0.5 + \frac{-0.08333333333333333}{N}}{N}\right)} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ 1.0 (* N (+ 1.0 (/ (+ 0.5 (/ -0.08333333333333333 N)) N)))))

C

double code(double N) {
	return 1.0 / (N * (1.0 + ((0.5 + (-0.08333333333333333 / N)) / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / (n * (1.0d0 + ((0.5d0 + ((-0.08333333333333333d0) / n)) / n)))
end function

Java

public static double code(double N) {
	return 1.0 / (N * (1.0 + ((0.5 + (-0.08333333333333333 / N)) / N)));
}

Python

def code(N):
	return 1.0 / (N * (1.0 + ((0.5 + (-0.08333333333333333 / N)) / N)))

Julia

function code(N)
	return Float64(1.0 / Float64(N * Float64(1.0 + Float64(Float64(0.5 + Float64(-0.08333333333333333 / N)) / N))))
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / (N * (1.0 + ((0.5 + (-0.08333333333333333 / N)) / N)));
end

Wolfram

code[N_] := N[(1.0 / N[(N * N[(1.0 + N[(N[(0.5 + N[(-0.08333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 98.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}} \]

8. Taylor expanded in N around inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(N \cdot \left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right) \]

   2. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{N} - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{N} - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right)\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{N} - \frac{\color{blue}{\frac{1}{12}}}{{N}^{2}}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N} - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N} - \frac{\frac{1}{12}}{N \cdot \color{blue}{N}}\right)\right)\right)\right) \]

   7. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N} - \frac{\frac{\frac{1}{12}}{N}}{\color{blue}{N}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N} - \frac{\frac{\frac{1}{12} \cdot 1}{N}}{N}\right)\right)\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N} - \frac{\frac{1}{12} \cdot \frac{1}{N}}{N}\right)\right)\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{\color{blue}{N}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{12} \cdot \frac{1}{N}\right)\right)\right), N\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{12} \cdot \frac{1}{N}\right)\right)\right), N\right)\right)\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{12} \cdot 1}{N}\right)\right)\right), N\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{N}\right)\right)\right), N\right)\right)\right)\right) \]

   16. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{12}\right)}{N}\right)\right), N\right)\right)\right)\right) \]

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{12}\right)\right), N\right)\right), N\right)\right)\right)\right) \]

   18. metadata-eval 97.3%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{12}, N\right)\right), N\right)\right)\right)\right) \]

10. Simplified 97.3%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \frac{0.5 + \frac{-0.08333333333333333}{N}}{N}\right)}} \]
11. Add Preprocessing

Alternative 9: 95.1% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N}}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 N)) N)) N))

C

double code(double N) {
	return (1.0 + ((-0.5 + (0.3333333333333333 / N)) / N)) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / n)) / n)) / n
end function

Java

public static double code(double N) {
	return (1.0 + ((-0.5 + (0.3333333333333333 / N)) / N)) / N;
}

Python

def code(N):
	return (1.0 + ((-0.5 + (0.3333333333333333 / N)) / N)) / N

Julia

function code(N)
	return Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / N)) / N)) / N)
end

MATLAB

function tmp = code(N)
	tmp = (1.0 + ((-0.5 + (0.3333333333333333 / N)) / N)) / N;
end

Wolfram

code[N_] := N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{N}}{N}} \]
4. Step-by-step derivation

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

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

5. Simplified 97.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N}}{N}} \]
6. Add Preprocessing

Alternative 10: 93.2% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N + 0.5} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ 1.0 (+ N 0.5)))

C

double code(double N) {
	return 1.0 / (N + 0.5);
}

Fortran

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

Java

public static double code(double N) {
	return 1.0 / (N + 0.5);
}

Python

def code(N):
	return 1.0 / (N + 0.5)

Julia

function code(N)
	return Float64(1.0 / Float64(N + 0.5))
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / (N + 0.5);
end

Wolfram

code[N_] := N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 98.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}} \]

8. Taylor expanded in N around inf

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

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

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

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

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 94.5%

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

10. Simplified 94.5%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \frac{0.5}{N}\right)}} \]
11. Step-by-step derivation

    1. distribute-rgt-in N/A

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

    2. *-lft-identity N/A

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

    3. div-inv N/A

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

    4. associate-*l* N/A

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

    5. inv-pow N/A

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

    6. pow-plus N/A

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

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(N + \frac{1}{2} \cdot {N}^{0}\right)\right) \]

    8. metadata-eval N/A

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

    9. metadata-eval N/A

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

    10. +-lowering-+.f64 94.6%

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

12. Applied egg-rr 94.6%

    \[\leadsto \frac{1}{\color{blue}{N + 0.5}} \]
13. Add Preprocessing

Alternative 11: 84.4% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ 1.0 N))

C

double code(double N) {
	return 1.0 / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / n
end function

Java

public static double code(double N) {
	return 1.0 / N;
}

Python

def code(N):
	return 1.0 / N

Julia

function code(N)
	return Float64(1.0 / N)
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / N;
end

Wolfram

code[N_] := N[(1.0 / N), $MachinePrecision]

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{1}{N}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 85.1%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{N}\right) \]

5. Simplified 85.1%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
6. Add Preprocessing

Alternative 12: 9.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (N) :precision binary64 2.0)

C

double code(double N) {
	return 2.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 2.0d0
end function

Java

public static double code(double N) {
	return 2.0;
}

Python

def code(N):
	return 2.0

Julia

function code(N)
	return 2.0
end

MATLAB

function tmp = code(N)
	tmp = 2.0;
end

Wolfram

code[N_] := 2.0

Derivation

1. Initial program 23.3%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{N}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 98.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}} \]

8. Taylor expanded in N around inf

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

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

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

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

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 94.5%

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

10. Simplified 94.5%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \frac{0.5}{N}\right)}} \]

11. Taylor expanded in N around 0

    \[\leadsto \color{blue}{2} \]
12. Step-by-step derivation

13. Simplified 9.8%

    \[\leadsto \color{blue}{2} \]
14. Add Preprocessing

Developer Target 1: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{1}{N}\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (log1p (/ 1.0 N)))

C

double code(double N) {
	return log1p((1.0 / N));
}

Java

public static double code(double N) {
	return Math.log1p((1.0 / N));
}

Python

def code(N):
	return math.log1p((1.0 / N))

Julia

function code(N)
	return log1p(Float64(1.0 / N))
end

Wolfram

code[N_] := N[Log[1 + N[(1.0 / N), $MachinePrecision]], $MachinePrecision]

Developer Target 2: 26.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(1 + \frac{1}{N}\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (log (+ 1.0 (/ 1.0 N))))

C

double code(double N) {
	return log((1.0 + (1.0 / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = log((1.0d0 + (1.0d0 / n)))
end function

Java

public static double code(double N) {
	return Math.log((1.0 + (1.0 / N)));
}

Python

def code(N):
	return math.log((1.0 + (1.0 / N)))

Julia

function code(N)
	return log(Float64(1.0 + Float64(1.0 / N)))
end

MATLAB

function tmp = code(N)
	tmp = log((1.0 + (1.0 / N)));
end

Wolfram

code[N_] := N[Log[N[(1.0 + N[(1.0 / N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Developer Target 3: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{N} + \frac{-1}{2 \cdot {N}^{2}}\right) + \frac{1}{3 \cdot {N}^{3}}\right) + \frac{-1}{4 \cdot {N}^{4}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (+
  (+ (+ (/ 1.0 N) (/ -1.0 (* 2.0 (pow N 2.0)))) (/ 1.0 (* 3.0 (pow N 3.0))))
  (/ -1.0 (* 4.0 (pow N 4.0)))))

C

double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * pow(N, 2.0)))) + (1.0 / (3.0 * pow(N, 3.0)))) + (-1.0 / (4.0 * pow(N, 4.0)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (((1.0d0 / n) + ((-1.0d0) / (2.0d0 * (n ** 2.0d0)))) + (1.0d0 / (3.0d0 * (n ** 3.0d0)))) + ((-1.0d0) / (4.0d0 * (n ** 4.0d0)))
end function

Java

public static double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * Math.pow(N, 2.0)))) + (1.0 / (3.0 * Math.pow(N, 3.0)))) + (-1.0 / (4.0 * Math.pow(N, 4.0)));
}

Python

def code(N):
	return (((1.0 / N) + (-1.0 / (2.0 * math.pow(N, 2.0)))) + (1.0 / (3.0 * math.pow(N, 3.0)))) + (-1.0 / (4.0 * math.pow(N, 4.0)))

Julia

function code(N)
	return Float64(Float64(Float64(Float64(1.0 / N) + Float64(-1.0 / Float64(2.0 * (N ^ 2.0)))) + Float64(1.0 / Float64(3.0 * (N ^ 3.0)))) + Float64(-1.0 / Float64(4.0 * (N ^ 4.0))))
end

MATLAB

function tmp = code(N)
	tmp = (((1.0 / N) + (-1.0 / (2.0 * (N ^ 2.0)))) + (1.0 / (3.0 * (N ^ 3.0)))) + (-1.0 / (4.0 * (N ^ 4.0)));
end

Wolfram

code[N_] := N[(N[(N[(N[(1.0 / N), $MachinePrecision] + N[(-1.0 / N[(2.0 * N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(3.0 * N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(4.0 * N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024185 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (! :herbie-platform default (log1p (/ 1 N)))

  :alt
  (! :herbie-platform default (log (+ 1 (/ 1 N))))

  :alt
  (! :herbie-platform default (+ (/ 1 N) (/ -1 (* 2 (pow N 2))) (/ 1 (* 3 (pow N 3))) (/ -1 (* 4 (pow N 4)))))

  (- (log (+ N 1.0)) (log N)))