2log (problem 3.3.6)

Average Error: 29.9 → 0.0

Time: 7.7s

Precision: binary64

Cost: 21124

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{1}{N} \cdot \frac{0.3333333333333333}{N \cdot N} + \left(\frac{-0.5}{N \cdot N} + \frac{-0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0001)
   (+
    (/ 1.0 N)
    (+
     (* (/ 1.0 N) (/ 0.3333333333333333 (* N N)))
     (+ (/ -0.5 (* N N)) (/ -0.25 (pow N 4.0)))))
   (log (+ 1.0 (/ 1.0 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)) <= 0.0001) {
		tmp = (1.0 / N) + (((1.0 / N) * (0.3333333333333333 / (N * N))) + ((-0.5 / (N * N)) + (-0.25 / pow(N, 4.0))));
	} else {
		tmp = log((1.0 + (1.0 / N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = log((n + 1.0d0)) - log(n)
end function

↓

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0001d0) then
        tmp = (1.0d0 / n) + (((1.0d0 / n) * (0.3333333333333333d0 / (n * n))) + (((-0.5d0) / (n * n)) + ((-0.25d0) / (n ** 4.0d0))))
    else
        tmp = log((1.0d0 + (1.0d0 / n)))
    end if
    code = tmp
end function

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)) <= 0.0001) {
		tmp = (1.0 / N) + (((1.0 / N) * (0.3333333333333333 / (N * N))) + ((-0.5 / (N * N)) + (-0.25 / Math.pow(N, 4.0))));
	} else {
		tmp = Math.log((1.0 + (1.0 / 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)) <= 0.0001:
		tmp = (1.0 / N) + (((1.0 / N) * (0.3333333333333333 / (N * N))) + ((-0.5 / (N * N)) + (-0.25 / math.pow(N, 4.0))))
	else:
		tmp = math.log((1.0 + (1.0 / 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)) <= 0.0001)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(Float64(1.0 / N) * Float64(0.3333333333333333 / Float64(N * N))) + Float64(Float64(-0.5 / Float64(N * N)) + Float64(-0.25 / (N ^ 4.0)))));
	else
		tmp = log(Float64(1.0 + Float64(1.0 / N)));
	end
	return tmp
end

MATLAB

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end

↓

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0001)
		tmp = (1.0 / N) + (((1.0 / N) * (0.3333333333333333 / (N * N))) + ((-0.5 / (N * N)) + (-0.25 / (N ^ 4.0))));
	else
		tmp = log((1.0 + (1.0 / N)));
	end
	tmp_2 = 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], 0.0001], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(N[(1.0 / N), $MachinePrecision] * N[(0.3333333333333333 / N[(N * N), $MachinePrecision]), $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[Log[N[(1.0 + N[(1.0 / N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1.00000000000000005e-4

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

   5. Applied egg-rr 0.0

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

   if 1.00000000000000005e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   3. Taylor expanded in N around 0 0.1

      \[\leadsto \log \color{blue}{\left(1 + \frac{1}{N}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{1}{N} \cdot \frac{0.3333333333333333}{N \cdot N} + \left(\frac{-0.5}{N \cdot N} + \frac{-0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;N \leq 8500:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{0.3333333333333333}{N} + -0.5}{N \cdot N}\\ \end{array} \]

Alternative 2
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 3
Error	0.7
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.6:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{0.3333333333333333}{N} + -0.5}{N \cdot N}\\ \end{array} \]

Alternative 4
Error	30.5
Cost	192

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

Reproduce

herbie shell --seed 2022364 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))