Average Error: 29.3 → 0.2
Time: 2.8s
Precision: binary64
\[\log \left(N + 1\right) - \log N \]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \end{array} \]
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 5e-9)
   (/ 1.0 (+ N 0.5))
   (log (+ 1.0 (/ 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)) <= 5e-9) {
		tmp = 1.0 / (N + 0.5);
	} else {
		tmp = log((1.0 + (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)) <= 5d-9) then
        tmp = 1.0d0 / (n + 0.5d0)
    else
        tmp = log((1.0d0 + (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)) <= 5e-9) {
		tmp = 1.0 / (N + 0.5);
	} else {
		tmp = Math.log((1.0 + (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)) <= 5e-9:
		tmp = 1.0 / (N + 0.5)
	else:
		tmp = math.log((1.0 + (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)) <= 5e-9)
		tmp = Float64(1.0 / Float64(N + 0.5));
	else
		tmp = log(Float64(1.0 + Float64(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)) <= 5e-9)
		tmp = 1.0 / (N + 0.5);
	else
		tmp = log((1.0 + (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], 5e-9], N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 + N[(1.0 / N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-9}:\\
\;\;\;\;\frac{1}{N + 0.5}\\

\mathbf{else}:\\
\;\;\;\;\log \left(1 + \frac{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)) < 5.0000000000000001e-9

    1. Initial program 60.1

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

    2. Simplified60.1

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

    3. Taylor expanded in N around inf 0.0

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]

    4. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{0.5}{N}}{N}} \]

    5. Applied egg-rr0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5}{N}}}} \]

    6. Taylor expanded in N around inf 0

      \[\leadsto \frac{1}{\color{blue}{N + 0.5}} \]

    if 5.0000000000000001e-9 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

    1. Initial program 0.4

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

    2. Simplified0.4

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

    3. Applied egg-rr0.4

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

    4. Taylor expanded in N around 0 0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))