2log (problem 3.3.6)

Average Accuracy: 54.2% → 100.0%

Time: 7.3s

Precision: binary64

Cost: 6592

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 54.2%

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

2. Applied egg-rr 54.3%

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

   [Start] 54.2	\[ \log \left(N + 1\right) - \log N \]
   diff-log [=>] 54.3	\[ \color{blue}{\log \left(\frac{N + 1}{N}\right)} \]

3. Applied egg-rr 54.2%

   \[\leadsto \color{blue}{0 + \left(\mathsf{log1p}\left(N\right) - \log N\right)} \]
   Proof

   [Start] 54.3	\[ \log \left(\frac{N + 1}{N}\right) \]
   *-un-lft-identity [=>] 54.3	\[ \log \color{blue}{\left(1 \cdot \frac{N + 1}{N}\right)} \]
   log-prod [=>] 54.3	\[ \color{blue}{\log 1 + \log \left(\frac{N + 1}{N}\right)} \]
   metadata-eval [=>] 54.3	\[ \color{blue}{0} + \log \left(\frac{N + 1}{N}\right) \]
   log-div [=>] 54.2	\[ 0 + \color{blue}{\left(\log \left(N + 1\right) - \log N\right)} \]
   +-commutative [=>] 54.2	\[ 0 + \left(\log \color{blue}{\left(1 + N\right)} - \log N\right) \]
   log1p-udef [<=] 54.2	\[ 0 + \left(\color{blue}{\mathsf{log1p}\left(N\right)} - \log N\right) \]

4. Simplified 100.0%

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

   [Start] 54.2	\[ 0 + \left(\mathsf{log1p}\left(N\right) - \log N\right) \]
   +-lft-identity [=>] 54.2	\[ \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   log1p-def [<=] 54.2	\[ \color{blue}{\log \left(1 + N\right)} - \log N \]
   +-commutative [<=] 54.2	\[ \log \color{blue}{\left(N + 1\right)} - \log N \]
   log-div [<=] 54.3	\[ \color{blue}{\log \left(\frac{N + 1}{N}\right)} \]
   *-lft-identity [<=] 54.3	\[ \log \left(\frac{\color{blue}{1 \cdot \left(N + 1\right)}}{N}\right) \]
   associate-*l/ [<=] 54.1	\[ \log \color{blue}{\left(\frac{1}{N} \cdot \left(N + 1\right)\right)} \]
   distribute-lft-in [=>] 54.1	\[ \log \color{blue}{\left(\frac{1}{N} \cdot N + \frac{1}{N} \cdot 1\right)} \]
   lft-mult-inverse [=>] 54.3	\[ \log \left(\color{blue}{1} + \frac{1}{N} \cdot 1\right) \]
   *-rgt-identity [=>] 54.3	\[ \log \left(1 + \color{blue}{\frac{1}{N}}\right) \]
   log1p-def [=>] 100.0	\[ \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	98.7%
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.28:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 2
Accuracy	56.7%
Cost	320

\[\frac{1}{N + 0.5} \]

Alternative 3
Accuracy	51.6%
Cost	192

\[\frac{1}{N} \]

Reproduce

herbie shell --seed 2023147 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))