Average Error: 29.6 → 0.1
Time: 10.5s
Precision: binary64
Cost: 26625
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \leq 978.1423904054542:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \leq 978.1423904054542:\\
\;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\

\end{array}
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (if (<= N 978.1423904054542)
   (+ (log (sqrt (/ (+ N 1.0) N))) (log (sqrt (/ (+ N 1.0) N))))
   (-
    (+ (/ 1.0 N) (/ 0.3333333333333333 (pow N 3.0)))
    (+ (/ 0.5 (* N N)) (/ 0.25 (pow N 4.0))))))
double code(double N) {
	return log(N + 1.0) - log(N);
}
↓
double code(double N) {
	double tmp;
	if (N <= 978.1423904054542) {
		tmp = log(sqrt((N + 1.0) / N)) + log(sqrt((N + 1.0) / N));
	} else {
		tmp = ((1.0 / N) + (0.3333333333333333 / pow(N, 3.0))) - ((0.5 / (N * N)) + (0.25 / pow(N, 4.0)));
	}
	return tmp;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error32.6
Cost61376
\[\sqrt[3]{\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)} \cdot \left(\sqrt[3]{\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)} \cdot \sqrt[3]{\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)}\right)\]
Alternative 2
Error29.7
Cost58944
\[\frac{{\log \left(1 + N\right)}^{3} - {\log N}^{3}}{\log N \cdot \log N + \log \left(1 + N\right) \cdot \left(\log N + \log \left(1 + N\right)\right)}\]
Alternative 3
Error56.3
Cost54208
\[\frac{\left({N}^{3} + N \cdot 0.3333333333333333\right) \cdot \left(\frac{0.5}{N \cdot N} - \frac{0.25}{{N}^{4}}\right) - {N}^{4} \cdot \left(\frac{0.25}{{N}^{4}} - \frac{\frac{0.0625}{{N}^{4}}}{{N}^{4}}\right)}{{N}^{4} \cdot \left(\frac{0.5}{N \cdot N} - \frac{0.25}{{N}^{4}}\right)}\]
Alternative 4
Error32.1
Cost53696
\[\left(\sqrt{\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}} + \sqrt{\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}}\right) \cdot \left(\sqrt{\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}} - \sqrt{\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}}\right)\]
Alternative 5
Error61.3
Cost51904
\[\left(\sqrt{\log \left(1 + N\right)} + \sqrt{\log N}\right) \cdot \left(\sqrt{\log \left(1 + N\right)} - \sqrt{\log N}\right)\]
Alternative 6
Error30.0
Cost45632
\[\sqrt[3]{\log \left(1 + N\right)} \cdot \left(\sqrt[3]{\log \left(1 + N\right)} \cdot \sqrt[3]{\log \left(1 + N\right)}\right) - \log N\]
Alternative 7
Error30.0
Cost39232
\[\log \left(\sqrt[3]{1 + N} \cdot \sqrt[3]{1 + N}\right) + \left(\log \left(\sqrt[3]{1 + N}\right) - \log N\right)\]
Alternative 8
Error29.7
Cost32704
\[\left(\log \left(\sqrt[3]{1 + N}\right) + \log \left(\sqrt[3]{1 + N}\right) \cdot 2\right) - \log N\]
Alternative 9
Error29.8
Cost32576
\[\sqrt{\log \left(1 + N\right)} \cdot \sqrt{\log \left(1 + N\right)} - \log N\]
Alternative 10
Error30.1
Cost32576
\[\left(\log \left(1 + N\right) - 2 \cdot \log \left(\sqrt[3]{N}\right)\right) - \log \left(\sqrt[3]{N}\right)\]
Alternative 11
Error29.6
Cost32576
\[\log \left(\sqrt{1 + N}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log N\right)\]
Alternative 12
Error52.0
Cost26880
\[\sqrt[3]{{\left(\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\right)}^{3}}\]
Alternative 13
Error31.0
Cost26496
\[\log \left(1 + {N}^{3}\right) - \left(\log N + \log \left(N \cdot N + \left(1 - N\right)\right)\right)\]
Alternative 14
Error30.9
Cost26496
\[\left(\log \left(1 + {N}^{3}\right) - \log \left(N \cdot N + \left(1 - N\right)\right)\right) - \log N\]
Alternative 15
Error29.7
Cost26304
\[\sqrt{\log \left(\frac{1 + N}{N}\right)} \cdot \sqrt{\log \left(\frac{1 + N}{N}\right)}\]
Alternative 16
Error29.5
Cost26304
\[\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\]
Alternative 17
Error29.8
Cost25984
\[\sqrt[3]{{\log \left(1 + N\right)}^{3}} - \log N\]
Alternative 18
Error30.0
Cost25920
\[e^{\log \log \left(1 + N\right)} - \log N\]
Alternative 19
Error62.5
Cost19904
\[\log \left(N \cdot N - 1\right) - \left(\log N + \log \left(N + -1\right)\right)\]
Alternative 20
Error31.9
Cost14016
\[\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\]
Alternative 21
Error29.6
Cost13120
\[\log \left(1 + N\right) - \log N\]
Alternative 22
Error31.9
Cost8064
\[\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.5}{N \cdot N} \cdot \frac{0.5}{N \cdot N}\right)\]
Alternative 23
Error31.9
Cost7936
\[\left(\frac{1}{N} + \frac{1}{N \cdot N} \cdot \frac{0.3333333333333333}{N}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\]
Alternative 24
Error31.9
Cost7808
\[\left(\frac{1}{N} + \frac{\frac{0.3333333333333333}{N \cdot N}}{N}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\]
Alternative 25
Error31.4
Cost7296
\[\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right)\]
Alternative 26
Error29.5
Cost6720
\[\log \left(\frac{1 + N}{N}\right)\]
Alternative 27
Error31.3
Cost6592
\[N - \log N\]
Alternative 28
Error32.1
Cost6528
\[-\log N\]
Alternative 29
Error31.9
Cost576
\[\frac{1}{N} - \frac{0.5}{N \cdot N}\]
Alternative 30
Error30.7
Cost192
\[\frac{1}{N}\]
Alternative 31
Error57.8
Cost64
\[1\]
Alternative 32
Error61.3
Cost64
\[0\]
Alternative 33
Error62.8
Cost64
\[-1\]

Error

Derivation

  1. Split input into 2 regimes

  2. if N < 978.142390405454194

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied diff-log_binary64_4960.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt_binary64_4260.1

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

    6. Applied log-prod_binary64_4900.1

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

    7. Simplified0.1

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

    if 978.142390405454194 < N

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 978.1423904054542:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))