Average Error: 30.2 → 0.0
Time: 2.7s
Precision: binary64
\[\log \left(N + 1\right) - \log N \]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{-0.25}{{N}^{4}} + \frac{\frac{-0.5}{N}}{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]
(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)
    (+
     (/ 0.3333333333333333 (pow N 3.0))
     (+ (/ -0.25 (pow N 4.0)) (/ (/ -0.5 N) N))))
   (log (/ (+ N 1.0) N))))
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) + ((0.3333333333333333 / pow(N, 3.0)) + ((-0.25 / pow(N, 4.0)) + ((-0.5 / N) / N)));
	} else {
		tmp = log(((N + 1.0) / 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) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0001d0) then
        tmp = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) + (((-0.25d0) / (n ** 4.0d0)) + (((-0.5d0) / n) / n)))
    else
        tmp = log(((n + 1.0d0) / 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 tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0001) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) + ((-0.25 / Math.pow(N, 4.0)) + ((-0.5 / N) / N)));
	} else {
		tmp = Math.log(((N + 1.0) / N));
	}
	return tmp;
}

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) + ((0.3333333333333333 / math.pow(N, 3.0)) + ((-0.25 / math.pow(N, 4.0)) + ((-0.5 / N) / N)))
	else:
		tmp = math.log(((N + 1.0) / N))
	return tmp

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(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(-0.25 / (N ^ 4.0)) + Float64(Float64(-0.5 / N) / N))));
	else
		tmp = log(Float64(Float64(N + 1.0) / N));
	end
	return tmp
end

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) + ((0.3333333333333333 / (N ^ 3.0)) + ((-0.25 / (N ^ 4.0)) + ((-0.5 / N) / N)));
	else
		tmp = log(((N + 1.0) / N));
	end
	tmp_2 = tmp;
end

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[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]]

\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{0.3333333333333333}{{N}^{3}} + \left(\frac{-0.25}{{N}^{4}} + \frac{\frac{-0.5}{N}}{N}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{N + 1}{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)) < 1.00000000000000005e-4

    1. Initial program 59.5

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

    2. Simplified59.5

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

    3. Applied egg-rr59.3

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

    4. Taylor expanded in N around inf 0.0

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

    5. Simplified0.0

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

    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. Simplified0.1

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

    3. Applied egg-rr0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{-0.25}{{N}^{4}} + \frac{\frac{-0.5}{N}}{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022202 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))