2log (problem 3.3.6)

Average Error: 29.2 → 0.0

Time: 9.7s

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 29.2

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

2. Applied egg-rr 29.1

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

3. Applied egg-rr 29.1

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

4. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{N} + 0\right)} \]
   Proof

   [Start] 29.1	\[ \mathsf{log1p}\left(\frac{N + 1}{N} - 1\right) \]
   *-lft-identity [<=] 29.1	\[ \mathsf{log1p}\left(\color{blue}{1 \cdot \frac{N + 1}{N}} - 1\right) \]
   associate-*r/ [=>] 29.1	\[ \mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(N + 1\right)}{N}} - 1\right) \]
   associate-*l/ [<=] 29.2	\[ \mathsf{log1p}\left(\color{blue}{\frac{1}{N} \cdot \left(N + 1\right)} - 1\right) \]
   distribute-rgt-in [=>] 29.2	\[ \mathsf{log1p}\left(\color{blue}{\left(N \cdot \frac{1}{N} + 1 \cdot \frac{1}{N}\right)} - 1\right) \]
   +-commutative [=>] 29.2	\[ \mathsf{log1p}\left(\color{blue}{\left(1 \cdot \frac{1}{N} + N \cdot \frac{1}{N}\right)} - 1\right) \]
   rgt-mult-inverse [=>] 29.1	\[ \mathsf{log1p}\left(\left(1 \cdot \frac{1}{N} + \color{blue}{1}\right) - 1\right) \]
   *-lft-identity [=>] 29.1	\[ \mathsf{log1p}\left(\left(\color{blue}{\frac{1}{N}} + 1\right) - 1\right) \]
   associate--l+ [=>] 0.0	\[ \mathsf{log1p}\left(\color{blue}{\frac{1}{N} + \left(1 - 1\right)}\right) \]
   metadata-eval [=>] 0.0	\[ \mathsf{log1p}\left(\frac{1}{N} + \color{blue}{0}\right) \]

5. Applied egg-rr 29.1

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

6. Simplified 0.0

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

   [Start] 29.1	\[ \left(1 + \mathsf{log1p}\left(\frac{1}{N}\right)\right) - 1 \]
   +-commutative [=>] 29.1	\[ \color{blue}{\left(\mathsf{log1p}\left(\frac{1}{N}\right) + 1\right)} - 1 \]
   associate--l+ [=>] 0.0	\[ \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right) + \left(1 - 1\right)} \]
   metadata-eval [=>] 0.0	\[ \mathsf{log1p}\left(\frac{1}{N}\right) + \color{blue}{0} \]
   +-rgt-identity [=>] 0.0	\[ \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]

7. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.8
Cost	6660

\[\begin{array}{l} t_0 := \frac{0.08333333333333333}{N} + 0.5\\ \mathbf{if}\;N \leq 0.35:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.25}{t_0} + \left(N + \frac{\frac{-0.006944444444444444}{N \cdot N}}{t_0}\right)}\\ \end{array} \]

Alternative 2
Error	28.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;N \leq 0.5:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 3
Error	27.9
Cost	320

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

Alternative 4
Error	57.6
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023066 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))