?

Average Error: 29.4 → 0.1
Time: 1.4min
Precision: binary64
Cost: 33924

?

\[\log \left(N + 1\right) - \log N \]
↓
\[\begin{array}{l} t_0 := \log \left(N + 1\right)\\ \mathbf{if}\;t_0 - \log N \leq 0.0005:\\ \;\;\;\;\left(\frac{1}{N} + 1 \cdot \frac{0.3333333333333333}{{N}^{3}}\right) - \left(1 \cdot \frac{0.5}{{N}^{2}} + 1 \cdot \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 2 - \left(t_0 + \log N\right)\\ \end{array} \]
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (+ N 1.0))))
   (if (<= (- t_0 (log N)) 0.0005)
     (-
      (+ (/ 1.0 N) (* 1.0 (/ 0.3333333333333333 (pow N 3.0))))
      (+ (* 1.0 (/ 0.5 (pow N 2.0))) (* 1.0 (/ 0.25 (pow N 4.0)))))
     (- (* t_0 2.0) (+ t_0 (log N))))))
double code(double N) {
	return log((N + 1.0)) - log(N);
}
↓
double code(double N) {
	double t_0 = log((N + 1.0));
	double tmp;
	if ((t_0 - log(N)) <= 0.0005) {
		tmp = ((1.0 / N) + (1.0 * (0.3333333333333333 / pow(N, 3.0)))) - ((1.0 * (0.5 / pow(N, 2.0))) + (1.0 * (0.25 / pow(N, 4.0))));
	} else {
		tmp = (t_0 * 2.0) - (t_0 + log(N));
	}
	return tmp;
}

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))
    if ((t_0 - log(n)) <= 0.0005d0) then
        tmp = ((1.0d0 / n) + (1.0d0 * (0.3333333333333333d0 / (n ** 3.0d0)))) - ((1.0d0 * (0.5d0 / (n ** 2.0d0))) + (1.0d0 * (0.25d0 / (n ** 4.0d0))))
    else
        tmp = (t_0 * 2.0d0) - (t_0 + log(n))
    end if
    code = tmp
end function

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));
	double tmp;
	if ((t_0 - Math.log(N)) <= 0.0005) {
		tmp = ((1.0 / N) + (1.0 * (0.3333333333333333 / Math.pow(N, 3.0)))) - ((1.0 * (0.5 / Math.pow(N, 2.0))) + (1.0 * (0.25 / Math.pow(N, 4.0))));
	} else {
		tmp = (t_0 * 2.0) - (t_0 + Math.log(N));
	}
	return tmp;
}

def code(N):
	return math.log((N + 1.0)) - math.log(N)
↓
def code(N):
	t_0 = math.log((N + 1.0))
	tmp = 0
	if (t_0 - math.log(N)) <= 0.0005:
		tmp = ((1.0 / N) + (1.0 * (0.3333333333333333 / math.pow(N, 3.0)))) - ((1.0 * (0.5 / math.pow(N, 2.0))) + (1.0 * (0.25 / math.pow(N, 4.0))))
	else:
		tmp = (t_0 * 2.0) - (t_0 + math.log(N))
	return tmp

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end
↓
function code(N)
	t_0 = log(Float64(N + 1.0))
	tmp = 0.0
	if (Float64(t_0 - log(N)) <= 0.0005)
		tmp = Float64(Float64(Float64(1.0 / N) + Float64(1.0 * Float64(0.3333333333333333 / (N ^ 3.0)))) - Float64(Float64(1.0 * Float64(0.5 / (N ^ 2.0))) + Float64(1.0 * Float64(0.25 / (N ^ 4.0)))));
	else
		tmp = Float64(Float64(t_0 * 2.0) - Float64(t_0 + log(N)));
	end
	return tmp
end

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end
↓
function tmp_2 = code(N)
	t_0 = log((N + 1.0));
	tmp = 0.0;
	if ((t_0 - log(N)) <= 0.0005)
		tmp = ((1.0 / N) + (1.0 * (0.3333333333333333 / (N ^ 3.0)))) - ((1.0 * (0.5 / (N ^ 2.0))) + (1.0 * (0.25 / (N ^ 4.0))));
	else
		tmp = (t_0 * 2.0) - (t_0 + log(N));
	end
	tmp_2 = tmp;
end

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]
↓
code[N_] := Block[{t$95$0 = N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(N[(N[(1.0 / N), $MachinePrecision] + N[(1.0 * N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 * N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 * N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 2.0), $MachinePrecision] - N[(t$95$0 + N[Log[N], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
t_0 := \log \left(N + 1\right)\\
\mathbf{if}\;t_0 - \log N \leq 0.0005:\\
\;\;\;\;\left(\frac{1}{N} + 1 \cdot \frac{0.3333333333333333}{{N}^{3}}\right) - \left(1 \cdot \frac{0.5}{{N}^{2}} + 1 \cdot \frac{0.25}{{N}^{4}}\right)\\

\mathbf{else}:\\
\;\;\;\;t_0 \cdot 2 - \left(t_0 + \log N\right)\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.0000000000000001e-4

    1. Initial program 59.4

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

    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. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + 1 \cdot \frac{0.3333333333333333}{{N}^{3}}\right) - \left(1 \cdot \frac{0.5}{{N}^{2}} + 1 \cdot \frac{0.25}{{N}^{4}}\right)} \]
      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-55 [=>]0.0

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

      rational_best-simplify-3 [=>]0.0

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

      rational_best-simplify-55 [=>]0.0

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

      rational_best-simplify-55 [=>]0.0

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

    if 5.0000000000000001e-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-rr0.1

      \[\leadsto \color{blue}{\log \left(N + 1\right) \cdot 2 + \left(-\left(\log \left(N + 1\right) + \log N\right)\right)} \]

    3. Applied egg-rr0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\left(\frac{1}{N} + 1 \cdot \frac{0.3333333333333333}{{N}^{3}}\right) - \left(1 \cdot \frac{0.5}{{N}^{2}} + 1 \cdot \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(N + 1\right) \cdot 2 - \left(\log \left(N + 1\right) + \log N\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.1
Cost33540
\[\begin{array}{l} t_0 := \log \left(N + 1\right)\\ \mathbf{if}\;t_0 - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} - \left(\left(\frac{0.25}{{N}^{4}} + \frac{-1}{N}\right) + \frac{0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 2 - \left(t_0 + \log N\right)\\ \end{array} \]
Alternative 2
Error0.1
Cost32964
\[\begin{array}{l} t_0 := \log \left(N + 1\right)\\ \mathbf{if}\;t_0 - \log N \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \frac{0.5}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \log N \cdot -2\right) + \log N\\ \end{array} \]
Alternative 3
Error0.2
Cost26820
\[\begin{array}{l} t_0 := \log \left(N + 1\right) - \log N\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{1}{{N}^{2}}\right) + \frac{0.5}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error0.1
Cost26820
\[\begin{array}{l} t_0 := \log \left(N + 1\right) - \log N\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \frac{0.5}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error0.2
Cost26308
\[\begin{array}{l} t_0 := \log \left(N + 1\right) - \log N\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error0.1
Cost20036
\[\begin{array}{l} t_0 := \log \left(N + 1\right)\\ \mathbf{if}\;N \leq 4500:\\ \;\;\;\;t_0 \cdot 2 - \left(t_0 + \log N\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \frac{0.5}{{N}^{2}}\\ \end{array} \]
Alternative 7
Error1.0
Cost7044
\[\begin{array}{l} \mathbf{if}\;N \leq 1:\\ \;\;\;\;\left(N + N\right) + \left(-\left(N + \log N\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]
Alternative 8
Error0.7
Cost7044
\[\begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;\left(N + N\right) + \left(-\left(N + \log N\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{{N}^{2}}\\ \end{array} \]
Alternative 9
Error1.0
Cost6724
\[\begin{array}{l} \mathbf{if}\;N \leq 1:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]
Alternative 10
Error1.3
Cost6660
\[\begin{array}{l} \mathbf{if}\;N \leq 0.52:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]
Alternative 11
Error30.9
Cost192
\[\frac{1}{N} \]
Alternative 12
Error61.3
Cost64
\[0 \]
Alternative 13
Error61.1
Cost64
\[N \]

Error

Reproduce?

herbie shell --seed 2023099 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))