2log (problem 3.3.6)

Average Error: 29.5 → 0.0

Time: 3.4s

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 29.5

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

2. Simplified 29.5

   \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   Proof
   (-.f64 (log1p.f64 N) (log.f64 N)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 N))) (log.f64 N)): 0 points increase in error, 0 points decrease in error
   (-.f64 (log.f64 (Rewrite<= +-commutative_binary64 (+.f64 N 1))) (log.f64 N)): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 29.3

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

4. Applied egg-rr 29.5

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

5. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]
   Proof
   (log1p.f64 (/.f64 1 N)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (/.f64 1 N)))): 131 points increase in error, 0 points decrease in error
   (log.f64 (+.f64 (Rewrite<= lft-mult-inverse_binary64 (*.f64 (/.f64 1 N) N)) (/.f64 1 N))): 13 points increase in error, 0 points decrease in error
   (log.f64 (+.f64 (*.f64 (/.f64 1 N) N) (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 1 N) 1)))): 0 points increase in error, 0 points decrease in error
   (log.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 (/.f64 1 N) (+.f64 N 1)))): 3 points increase in error, 0 points decrease in error
   (log.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 (+.f64 N 1)) N))): 0 points increase in error, 14 points decrease in error
   (log.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 (+.f64 N 1)) N)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> log-div_binary64 (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))): 7 points increase in error, 1 points decrease in error
   (-.f64 (log.f64 (Rewrite=> +-commutative_binary64 (+.f64 1 N))) (log.f64 N)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite=> log1p-def_binary64 (log1p.f64 N)) (log.f64 N)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-lft-identity_binary64 (+.f64 0 (-.f64 (log1p.f64 N) (log.f64 N)))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.8
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.66:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5}{N}}{N}\\ \end{array} \]

Alternative 2
Error	27.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;N \leq 0.25:\\ \;\;\;\;\frac{2}{N} \cdot \left(N + N\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 3
Error	27.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;N \leq 0.6:\\ \;\;\;\;\frac{2}{N} \cdot \left(N + N\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5}{N}}{N}\\ \end{array} \]

Alternative 4
Error	28.0
Cost	452

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

Alternative 5
Error	30.9
Cost	192

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

Alternative 6
Error	61.3
Cost	64

\[0 \]

Alternative 7
Error	61.1
Cost	64

\[N \]

Reproduce

herbie shell --seed 2022329 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))