2log (problem 3.3.6)

Average Error: 29.9 → 0.0

Time: 10.6s

Precision: binary64

Cost: 27204

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{-0.5}{N \cdot N} - \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (+
    (/ 1.0 N)
    (+
     (/ 0.3333333333333333 (pow N 3.0))
     (- (/ -0.5 (* N N)) (/ 0.25 (pow N 4.0)))))
   (- (log1p N) (log N))))

C

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

↓

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) + ((-0.5 / (N * N)) - (0.25 / pow(N, 4.0))));
	} else {
		tmp = log1p(N) - log(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 ((Math.log((N + 1.0)) - Math.log(N)) <= 0.001) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) + ((-0.5 / (N * N)) - (0.25 / Math.pow(N, 4.0))));
	} else {
		tmp = Math.log1p(N) - Math.log(N);
	}
	return tmp;
}

Python

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

↓

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.001:
		tmp = (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) + ((-0.5 / (N * N)) - (0.25 / math.pow(N, 4.0))))
	else:
		tmp = math.log1p(N) - math.log(N)
	return tmp

Julia

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

↓

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(-0.5 / Float64(N * N)) - Float64(0.25 / (N ^ 4.0)))));
	else
		tmp = Float64(log1p(N) - log(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[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.001], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1e-3

   1. Initial program 59.5

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

   2. Simplified 59.5

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

      [Start] 59.5	\[ \log \left(N + 1\right) - \log N \]
      +-commutative [=>] 59.5	\[ \log \color{blue}{\left(1 + N\right)} - \log N \]
      log1p-def [=>] 59.5	\[ \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Taylor expanded in N around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\right)} \]
      Proof

      [Start] 0.0	\[ \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]
      associate--l+ [=>] 0.0	\[ \color{blue}{\frac{1}{N} + \left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)} \]
      associate-*r/ [=>] 0.0	\[ \frac{1}{N} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \frac{1}{N} + \left(\frac{\color{blue}{0.3333333333333333}}{{N}^{3}} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]
      +-commutative [=>] 0.0	\[ \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \color{blue}{\left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)}\right) \]
      associate-*r/ [=>] 0.0	\[ \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{\color{blue}{0.5}}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right) \]
      unpow2 [=>] 0.0	\[ \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{\color{blue}{N \cdot N}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right) \]
      associate-*r/ [=>] 0.0	\[ \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{N \cdot N} + \color{blue}{\frac{0.25 \cdot 1}{{N}^{4}}}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{N \cdot N} + \frac{\color{blue}{0.25}}{{N}^{4}}\right)\right) \]

   if 1e-3 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \log \left(N + 1\right) - \log N \]
      +-commutative [=>] 0.1	\[ \log \color{blue}{\left(1 + N\right)} - \log N \]
      log1p-def [=>] 0.1	\[ \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{-0.5}{N \cdot N} - \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	19972

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-5}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{0.3333333333333333}{N} + -0.5}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 2
Error	0.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;N \leq 8000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{0.3333333333333333}{N} + -0.5}{N \cdot N}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	6724

\[\begin{array}{l} \mathbf{if}\;N \leq 0.85:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{0.3333333333333333}{N} + -0.5}{N \cdot N}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.58:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{0.3333333333333333}{N} + -0.5}{N \cdot N}\\ \end{array} \]

Alternative 5
Error	30.4
Cost	192

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

Alternative 6
Error	61.1
Cost	64

\[N \]

Reproduce

herbie shell --seed 2023054 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))