2log (problem 3.3.6)

Average Error: 29.9 → 0.1

Time: 7.1s

Precision: binary64

Cost: 33540

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(N + 1\right) - \log N\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (let* ((t_0 (- (log (+ N 1.0)) (log N))))
   (if (<= t_0 4e-5)
     (-
      (+ (/ 1.0 N) (/ 0.3333333333333333 (pow N 3.0)))
      (+ (/ 0.5 (pow N 2.0)) (/ 0.25 (pow N 4.0))))
     t_0)))

C

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

↓

double code(double N) {
	double t_0 = log((N + 1.0)) - log(N);
	double tmp;
	if (t_0 <= 4e-5) {
		tmp = ((1.0 / N) + (0.3333333333333333 / pow(N, 3.0))) - ((0.5 / pow(N, 2.0)) + (0.25 / pow(N, 4.0)));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = log((n + 1.0d0)) - log(n)
    if (t_0 <= 4d-5) then
        tmp = ((1.0d0 / n) + (0.3333333333333333d0 / (n ** 3.0d0))) - ((0.5d0 / (n ** 2.0d0)) + (0.25d0 / (n ** 4.0d0)))
    else
        tmp = t_0
    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 t_0 = Math.log((N + 1.0)) - Math.log(N);
	double tmp;
	if (t_0 <= 4e-5) {
		tmp = ((1.0 / N) + (0.3333333333333333 / Math.pow(N, 3.0))) - ((0.5 / Math.pow(N, 2.0)) + (0.25 / Math.pow(N, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(N):
	t_0 = math.log((N + 1.0)) - math.log(N)
	tmp = 0
	if t_0 <= 4e-5:
		tmp = ((1.0 / N) + (0.3333333333333333 / math.pow(N, 3.0))) - ((0.5 / math.pow(N, 2.0)) + (0.25 / math.pow(N, 4.0)))
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(N)
	t_0 = Float64(log(Float64(N + 1.0)) - log(N))
	tmp = 0.0
	if (t_0 <= 4e-5)
		tmp = Float64(Float64(Float64(1.0 / N) + Float64(0.3333333333333333 / (N ^ 3.0))) - Float64(Float64(0.5 / (N ^ 2.0)) + Float64(0.25 / (N ^ 4.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(N)
	t_0 = log((N + 1.0)) - log(N);
	tmp = 0.0;
	if (t_0 <= 4e-5)
		tmp = ((1.0 / N) + (0.3333333333333333 / (N ^ 3.0))) - ((0.5 / (N ^ 2.0)) + (0.25 / (N ^ 4.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[N_] := Block[{t$95$0 = N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-5], N[(N[(N[(1.0 / N), $MachinePrecision] + N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 4.00000000000000033e-5

   1. Initial program 59.7

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

   2. Taylor expanded in N around inf 0.0

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

   3. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]
      rational_best-simplify-62 [=>] 0.0	\[ \left(\frac{1}{N} + \color{blue}{\frac{1 \cdot 0.3333333333333333}{{N}^{3}}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]
      metadata-eval [=>] 0.0	\[ \left(\frac{1}{N} + \frac{\color{blue}{0.3333333333333333}}{{N}^{3}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]
      rational_best-simplify-3 [=>] 0.0	\[ \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \color{blue}{\left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)} \]
      rational_best-simplify-62 [=>] 0.0	\[ \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\color{blue}{\frac{1 \cdot 0.5}{{N}^{2}}} + 0.25 \cdot \frac{1}{{N}^{4}}\right) \]
      metadata-eval [=>] 0.0	\[ \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{\color{blue}{0.5}}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right) \]
      rational_best-simplify-62 [=>] 0.0	\[ \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{{N}^{2}} + \color{blue}{\frac{1 \cdot 0.25}{{N}^{4}}}\right) \]
      metadata-eval [=>] 0.0	\[ \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{{N}^{2}} + \frac{\color{blue}{0.25}}{{N}^{4}}\right) \]

   if 4.00000000000000033e-5 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(N + 1\right) - \log N\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	26820

\[\begin{array}{l} t_0 := \log \left(N + 1\right) - \log N\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \frac{0.5}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.3
Cost	26308

\[\begin{array}{l} t_0 := \log \left(N + 1\right) - \log N\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.7
Cost	7364

\[\begin{array}{l} \mathbf{if}\;N \leq 0.91:\\ \;\;\;\;\left(\left(-0.5 \cdot N - \log N\right) + N \cdot 0.75\right) + N \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{{N}^{2}}\\ \end{array} \]

Alternative 4
Error	0.7
Cost	7044

\[\begin{array}{l} \mathbf{if}\;N \leq 0.91:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{{N}^{2}}\\ \end{array} \]

Alternative 5
Error	1.0
Cost	6724

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

Alternative 6
Error	1.3
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.54:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 7
Error	30.4
Cost	192

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

Alternative 8
Error	61.1
Cost	64

\[N \]

Reproduce

herbie shell --seed 2023101 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))