Average Error: 29.7 → 0.1
Time: 4.7s
Precision: binary64
Cost: 71873
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0003029094019879608:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \log \left(\sqrt[3]{N + 1}\right) - \log \left(\sqrt{N}\right)\right) + \log \left(\frac{{\left(\sqrt[3]{\sqrt{N + 1}}\right)}^{2}}{\sqrt{N}}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0003029094019879608:\\
\;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \log \left(\sqrt[3]{N + 1}\right) - \log \left(\sqrt{N}\right)\right) + \log \left(\frac{{\left(\sqrt[3]{\sqrt{N + 1}}\right)}^{2}}{\sqrt{N}}\right)\\

\end{array}
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0003029094019879608)
   (-
    (+ (/ 1.0 N) (/ 0.3333333333333333 (pow N 3.0)))
    (+ (/ 0.5 (* N N)) (/ 0.25 (pow N 4.0))))
   (+
    (- (* 2.0 (log (cbrt (+ N 1.0)))) (log (sqrt N)))
    (log (/ (pow (cbrt (sqrt (+ N 1.0))) 2.0) (sqrt N))))))
double code(double N) {
	return log(N + 1.0) - log(N);
}
↓
double code(double N) {
	double tmp;
	if ((log(N + 1.0) - log(N)) <= 0.0003029094019879608) {
		tmp = ((1.0 / N) + (0.3333333333333333 / pow(N, 3.0))) - ((0.5 / (N * N)) + (0.25 / pow(N, 4.0)));
	} else {
		tmp = ((2.0 * log(cbrt(N + 1.0))) - log(sqrt(N))) + log(pow(cbrt(sqrt(N + 1.0)), 2.0) / sqrt(N));
	}
	return tmp;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.1
Cost45953
\[\begin{array}{l} \mathbf{if}\;N \leq 929.9693967107072:\\ \;\;\;\;\left(2 \cdot \log \left(\sqrt[3]{N + 1}\right) - \log \left(\sqrt{N}\right)\right) + \log \left(\frac{\sqrt[3]{N + 1}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \end{array}\]
Alternative 2
Error0.0
Cost39681
\[\begin{array}{l} \mathbf{if}\;N \leq 929.9693967107072:\\ \;\;\;\;\log \left(\frac{\sqrt[3]{N + 1}}{\sqrt{N}}\right) + \left(2 \cdot \left(\log \left(N + 1\right) \cdot 0.3333333333333333\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \end{array}\]
Alternative 3
Error0.0
Cost14337
\[\begin{array}{l} \mathbf{if}\;N \leq 1274.9763547642451:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \end{array}\]
Alternative 4
Error0.1
Cost20673
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 1.0169354759170801 \cdot 10^{-05}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array}\]
Alternative 5
Error0.1
Cost7041
\[\begin{array}{l} \mathbf{if}\;N \leq 158463.51741184632:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot N}\\ \end{array}\]
Alternative 6
Error0.8
Cost6849
\[\begin{array}{l} \mathbf{if}\;N \leq 0.686945551724479:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot N}\\ \end{array}\]
Alternative 7
Error27.5
Cost897
\[\begin{array}{l} \mathbf{if}\;N \leq 0.9058357690173492:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot N}\\ \end{array}\]
Alternative 8
Error27.9
Cost513
\[\begin{array}{l} \mathbf{if}\;N \leq 1.0062406200728051:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array}\]
Alternative 9
Error57.8
Cost64
\[1\]

Error

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 3.02909401988e-4

    1. Initial program 59.3

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

    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

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

    if 3.02909401988e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

    1. Initial program 0.1

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

    3. Applied diff-log_binary64_5110.1

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

    5. Applied add-sqr-sqrt_binary64_4410.1

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

    6. Applied add-cube-cbrt_binary64_4540.1

      \[\leadsto \log \left(\frac{\color{blue}{\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) \cdot \sqrt[3]{N + 1}}}{\sqrt{N} \cdot \sqrt{N}}\right)\]

    7. Applied times-frac_binary64_4250.1

      \[\leadsto \log \color{blue}{\left(\frac{\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}}{\sqrt{N}} \cdot \frac{\sqrt[3]{N + 1}}{\sqrt{N}}\right)}\]

    8. Applied log-prod_binary64_5050.1

      \[\leadsto \color{blue}{\log \left(\frac{\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt[3]{N + 1}}{\sqrt{N}}\right)}\]

    9. Simplified0.1

      \[\leadsto \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{N + 1}\right) - \log \left(\sqrt{N}\right)\right)} + \log \left(\frac{\sqrt[3]{N + 1}}{\sqrt{N}}\right)\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt_binary64_4410.1

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

    12. Applied cbrt-prod_binary64_4500.1

      \[\leadsto \left(2 \cdot \log \left(\sqrt[3]{N + 1}\right) - \log \left(\sqrt{N}\right)\right) + \log \left(\frac{\color{blue}{\sqrt[3]{\sqrt{N + 1}} \cdot \sqrt[3]{\sqrt{N + 1}}}}{\sqrt{N}}\right)\]
    13. Using strategy rm

    14. Applied pow2_binary64_5000.1

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

    15. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0003029094019879608:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \log \left(\sqrt[3]{N + 1}\right) - \log \left(\sqrt{N}\right)\right) + \log \left(\frac{{\left(\sqrt[3]{\sqrt{N + 1}}\right)}^{2}}{\sqrt{N}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))