Average Error: 29.6 → 0.2
Time: 3.9s
Precision: binary64
\[\log \left(N + 1\right) - \log N \]
↓
\[\begin{array}{l} t_0 := 0.5 \cdot \mathsf{log1p}\left(N\right)\\ \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8.861960054673546 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(t_0 - \log N\right)\\ \end{array} \]
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (let* ((t_0 (* 0.5 (log1p N))))
   (if (<= (- (log (+ N 1.0)) (log N)) 8.861960054673546e-10)
     (- (/ 1.0 N) (/ 0.5 (* N N)))
     (+ t_0 (- t_0 (log N))))))
double code(double N) {
	return log((N + 1.0)) - log(N);
}
↓
double code(double N) {
	double t_0 = 0.5 * log1p(N);
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 8.861960054673546e-10) {
		tmp = (1.0 / N) - (0.5 / (N * N));
	} else {
		tmp = t_0 + (t_0 - log(N));
	}
	return tmp;
}

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}
↓
public static double code(double N) {
	double t_0 = 0.5 * Math.log1p(N);
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 8.861960054673546e-10) {
		tmp = (1.0 / N) - (0.5 / (N * N));
	} else {
		tmp = t_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 = 0.5 * math.log1p(N)
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 8.861960054673546e-10:
		tmp = (1.0 / N) - (0.5 / (N * N))
	else:
		tmp = t_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 = Float64(0.5 * log1p(N))
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 8.861960054673546e-10)
		tmp = Float64(Float64(1.0 / N) - Float64(0.5 / Float64(N * N)));
	else
		tmp = Float64(t_0 + Float64(t_0 - log(N)));
	end
	return 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[(0.5 * N[Log[1 + N], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 8.861960054673546e-10], N[(N[(1.0 / N), $MachinePrecision] - N[(0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(t$95$0 - N[Log[N], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
t_0 := 0.5 \cdot \mathsf{log1p}\left(N\right)\\
\mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8.861960054673546 \cdot 10^{-10}:\\
\;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot N}\\

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


\end{array}

Error

Bits error versus N

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)) < 8.8619601e-10

    1. Initial program 60.1

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

    2. Taylor expanded in N around inf 0.0

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

    3. Simplified0.0

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

    if 8.8619601e-10 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

    1. Initial program 0.5

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

    2. Applied add-sqr-sqrt_binary640.5

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

    3. Applied log-prod_binary640.5

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

    4. Applied associate--l+_binary640.5

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

    5. Simplified0.5

      \[\leadsto \log \left(\sqrt{N + 1}\right) + \color{blue}{\left(0.5 \cdot \mathsf{log1p}\left(N\right) - \log N\right)} \]

    6. Applied pow1/2_binary640.5

      \[\leadsto \log \color{blue}{\left({\left(N + 1\right)}^{0.5}\right)} + \left(0.5 \cdot \mathsf{log1p}\left(N\right) - \log N\right) \]

    7. Applied log-pow_binary640.5

      \[\leadsto \color{blue}{0.5 \cdot \log \left(N + 1\right)} + \left(0.5 \cdot \mathsf{log1p}\left(N\right) - \log N\right) \]

    8. Simplified0.5

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(N\right)} + \left(0.5 \cdot \mathsf{log1p}\left(N\right) - \log N\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8.861960054673546 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(N\right) + \left(0.5 \cdot \mathsf{log1p}\left(N\right) - \log N\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))