2log (problem 3.3.6)

Average Error: 29.6 → 0.1

Time: 6.7s

Precision: binary64

Cost: 26436

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \leq 954.8915609233318:\\ \;\;\;\;0.5 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\\ \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (if (<= N 954.8915609233318)
   (+ (* 0.5 (- (log1p N) (log N))) (log (sqrt (/ (+ N 1.0) N))))
   (+ (/ 0.3333333333333333 (pow N 3.0)) (/ (+ 1.0 (/ -0.5 N)) N))))

C

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

↓

double code(double N) {
	double tmp;
	if (N <= 954.8915609233318) {
		tmp = (0.5 * (log1p(N) - log(N))) + log(sqrt(((N + 1.0) / N)));
	} else {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N);
	}
	return tmp;
}

Java

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

↓

public static double code(double N) {
	double tmp;
	if (N <= 954.8915609233318) {
		tmp = (0.5 * (Math.log1p(N) - Math.log(N))) + Math.log(Math.sqrt(((N + 1.0) / N)));
	} else {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N);
	}
	return tmp;
}

Python

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

↓

def code(N):
	tmp = 0
	if N <= 954.8915609233318:
		tmp = (0.5 * (math.log1p(N) - math.log(N))) + math.log(math.sqrt(((N + 1.0) / N)))
	else:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N)
	return tmp

Julia

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

↓

function code(N)
	tmp = 0.0
	if (N <= 954.8915609233318)
		tmp = Float64(Float64(0.5 * Float64(log1p(N) - log(N))) + log(sqrt(Float64(Float64(N + 1.0) / N))));
	else
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 + Float64(-0.5 / N)) / N));
	end
	return tmp
end

Wolfram

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

↓

code[N_] := If[LessEqual[N, 954.8915609233318], N[(N[(0.5 * N[(N[Log[1 + N], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[N[Sqrt[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 954.89156092333178

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   3. Applied egg-rr 0.1

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

   4. Applied egg-rr 0.1

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

   5. Applied egg-rr 0.1

      \[\leadsto \color{blue}{0.5 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)} + \log \left(\sqrt{\frac{N + 1}{N}}\right) \]

   if 954.89156092333178 < N

   1. Initial program 59.3

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

   2. Simplified 59.3

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

   3. Taylor expanded in N around inf 0.1

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

   4. Simplified 0.1

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

   5. Applied egg-rr 0.1

      \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\frac{1 + \frac{-0.5}{N}}{N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 954.8915609233318:\\ \;\;\;\;0.5 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7300

\[\begin{array}{l} \mathbf{if}\;N \leq 954.8915609233318:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;N \leq 506056.9125754291:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	6724

\[\begin{array}{l} \mathbf{if}\;N \leq 0.39211478482209317:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 4
Error	1.2
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.39211478482209317:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 5
Error	30.8
Cost	192

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

Alternative 6
Error	61.1
Cost	64

\[N \]

Reproduce

herbie shell --seed 2022228 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))