2log (problem 3.3.6)

Percentage Accurate: 54.5% → 100.0%

Time: 7.5s

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 53.4%

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

2. Simplified 53.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   Step-by-step derivation

   [Start] 53.4%	\[ \log \left(N + 1\right) - \log N \]
   +-commutative [=>] 53.4%	\[ \log \color{blue}{\left(1 + N\right)} - \log N \]
   log1p-def [=>] 53.4%	\[ \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

3. Applied egg-rr 53.6%

   \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)} \]
   Step-by-step derivation

   [Start] 53.4%	\[ \mathsf{log1p}\left(N\right) - \log N \]
   log1p-udef [=>] 53.4%	\[ \color{blue}{\log \left(1 + N\right)} - \log N \]
   diff-log [=>] 53.6%	\[ \color{blue}{\log \left(\frac{1 + N}{N}\right)} \]
   +-commutative [=>] 53.6%	\[ \log \left(\frac{\color{blue}{N + 1}}{N}\right) \]

4. Applied egg-rr 53.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{N + 1}{N} - 1\right)} \]
   Step-by-step derivation

   [Start] 53.6%	\[ \log \left(\frac{N + 1}{N}\right) \]
   log1p-expm1-u [=>] 53.6%	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{N + 1}{N}\right)\right)\right)} \]
   expm1-udef [=>] 53.6%	\[ \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{N + 1}{N}\right)} - 1}\right) \]
   add-exp-log [<=] 53.6%	\[ \mathsf{log1p}\left(\color{blue}{\frac{N + 1}{N}} - 1\right) \]

5. Taylor expanded in N around 0 100.0%

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

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6592

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

Alternative 2
Accuracy	98.8%
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.56:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{N \cdot \left(N \cdot N\right)} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 3
Accuracy	51.2%
Cost	192

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

Reproduce

herbie shell --seed 2023272 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))