2log (problem 3.3.6)

Percentage Accurate: 23.7% → 99.8%

Time: 8.8s

Alternatives: 6

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: 23.7% 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: 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]

Derivation

1. Initial program 23.1%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing
3. Step-by-step derivation

   1. diff-log N/A

      \[\leadsto \log \left(\frac{N + 1}{N}\right) \]

   2. div-inv N/A

      \[\leadsto \log \left(\left(N + 1\right) \cdot \frac{1}{N}\right) \]

   3. *-commutative N/A

      \[\leadsto \log \left(\frac{1}{N} \cdot \left(N + 1\right)\right) \]

   4. distribute-lft-in N/A

      \[\leadsto \log \left(\frac{1}{N} \cdot N + \frac{1}{N} \cdot 1\right) \]

   5. lft-mult-inverse N/A

      \[\leadsto \log \left(1 + \frac{1}{N} \cdot 1\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \log \left(1 + \frac{1}{N}\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{log1p}\left(\frac{1}{N}\right) \]

   8. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{log1p.f64}\left(\left(\frac{1}{N}\right)\right) \]

   9. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, N\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.1%

   \[\log \left(N + 1\right) - \log N \]
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 96.9%

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

   1. clear-num N/A

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

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

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

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

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

   6. unsub-neg N/A

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

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

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

   8. frac-2neg N/A

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

   9. remove-double-neg N/A

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

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

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

7. Applied egg-rr 96.9%

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

8. Taylor expanded in N around inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(N \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(N \cdot \left(1 + \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot N + \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right) \cdot N}\right)\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(N + \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)} \cdot N\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \color{blue}{\left(\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right) \cdot N\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \left(N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right)\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \left(N \cdot \left(\frac{1}{2} \cdot \frac{1}{N} + \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right)\right)\right) \]

   7. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \left(N \cdot \left(\frac{1}{2} \cdot \frac{1}{N}\right) + \color{blue}{N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \left(N \cdot \left(\frac{1}{N} \cdot \frac{1}{2}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \left(\left(N \cdot \frac{1}{N}\right) \cdot \frac{1}{2} + \color{blue}{N} \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)\right) \]

   10. rgt-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \left(1 \cdot \frac{1}{2} + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \left(\frac{1}{2} + \color{blue}{N} \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(N, \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right)\right)\right) \]

10. Simplified 97.3%

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

    1. associate-+r+ N/A

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

    2. *-commutative N/A

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

    3. associate-/r* N/A

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

    4. div-inv N/A

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

    5. associate-*l* N/A

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

    6. lft-mult-inverse N/A

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

    7. associate-/r/ N/A

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

    8. /-rgt-identity N/A

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

    9. frac-2neg N/A

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

    10. distribute-frac-neg N/A

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

    11. unsub-neg N/A

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

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

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

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

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

    14. distribute-frac-neg2 N/A

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

    15. distribute-frac-neg N/A

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

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

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

    17. +-commutative N/A

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

    18. distribute-neg-in N/A

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

    19. unsub-neg N/A

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

    20. --lowering--.f64 N/A

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

    21. metadata-eval N/A

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

    22. /-lowering-/.f64 97.3%

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

12. Applied egg-rr 97.3%

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

13. Final simplification 97.3%

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

Alternative 3: 95.7% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.1%

   \[\log \left(N + 1\right) - \log N \]
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 96.9%

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

   1. clear-num N/A

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

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

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

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

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

   6. unsub-neg N/A

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

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

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

   8. frac-2neg N/A

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

   9. remove-double-neg N/A

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

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

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

7. Applied egg-rr 96.9%

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

8. Taylor expanded in N around inf

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

10. Simplified 96.2%

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

Alternative 4: 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 23.1%

   \[\log \left(N + 1\right) - \log N \]
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 96.9%

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

   1. clear-num N/A

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

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

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

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

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

   6. unsub-neg N/A

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

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

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

   8. frac-2neg N/A

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

   9. remove-double-neg N/A

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

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

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

7. Applied egg-rr 96.9%

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

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. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. +-lowering-+.f64 93.9%

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

10. Simplified 93.9%

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

Alternative 5: 84.6% 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 23.1%

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

3. Taylor expanded in N around inf

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

   1. /-lowering-/.f64 85.2%

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

5. Simplified 85.2%

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

Alternative 6: 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 23.1%

   \[\log \left(N + 1\right) - \log N \]
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 96.9%

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

   1. clear-num N/A

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

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

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

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

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

   6. unsub-neg N/A

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

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

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

   8. frac-2neg N/A

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

   9. remove-double-neg N/A

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

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

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

7. Applied egg-rr 96.9%

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

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. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. +-lowering-+.f64 93.9%

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

10. Simplified 93.9%

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

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]

Reproduce

herbie shell --seed 2024164 
(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)))

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