Average Error: 29.5 → 0.1
Time: 3.3s
Precision: binary64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 1.9136352946702573 \cdot 10^{-07}:\\ \;\;\;\;\frac{1}{N} \cdot \left(1 + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) + \left(\log \left(\sqrt[3]{N + 1}\right) - \log N\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \leq 1.9136352946702573 \cdot 10^{-07}:\\
\;\;\;\;\frac{1}{N} \cdot \left(1 + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) + \left(\log \left(\sqrt[3]{N + 1}\right) - \log 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)) 1.9136352946702573e-07)
   (* (/ 1.0 N) (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 N)) N)))
   (+
    (log (* (cbrt (+ N 1.0)) (cbrt (+ N 1.0))))
    (- (log (cbrt (+ N 1.0))) (log 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)) <= 1.9136352946702573e-07) {
		tmp = (1.0 / N) * (1.0 + ((-0.5 + (0.3333333333333333 / N)) / N));
	} else {
		tmp = log(cbrt(N + 1.0) * cbrt(N + 1.0)) + (log(cbrt(N + 1.0)) - log(N));
	}
	return tmp;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1.913635295e-7

    1. Initial program 59.9

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

    3. Applied diff-log_binary6459.8

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

    4. Simplified59.8

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

    5. Taylor expanded around inf 0.0

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

    6. Simplified0.0

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

    if 1.913635295e-7 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt_binary640.3

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

    4. Applied log-prod_binary640.3

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

    5. Applied associate--l+_binary640.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 1.9136352946702573 \cdot 10^{-07}:\\ \;\;\;\;\frac{1}{N} \cdot \left(1 + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt[3]{N + 1} \cdot \sqrt[3]{N + 1}\right) + \left(\log \left(\sqrt[3]{N + 1}\right) - \log N\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))