2log (problem 3.3.6)

Average Error: 29.3 → 0.1

Time: 2.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.00037743709491433464:\\ \;\;\;\;\frac{1}{N} + \left(\frac{-0.5}{N \cdot N} + \mathsf{fma}\left(-0.25, {N}^{-4}, 0.3333333333333333 \cdot {N}^{-3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.00037743709491433464)
   (+
    (/ 1.0 N)
    (+
     (/ -0.5 (* N N))
     (fma -0.25 (pow N -4.0) (* 0.3333333333333333 (pow N -3.0)))))
   (log (/ (+ N 1.0) N))))

C

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

↓

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.00037743709491433464) {
		tmp = (1.0 / N) + ((-0.5 / (N * N)) + fma(-0.25, pow(N, -4.0), (0.3333333333333333 * pow(N, -3.0))));
	} else {
		tmp = log(((N + 1.0) / N));
	}
	return tmp;
}

Julia

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)) <= 0.00037743709491433464)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(-0.5 / Float64(N * N)) + fma(-0.25, (N ^ -4.0), Float64(0.3333333333333333 * (N ^ -3.0)))));
	else
		tmp = log(Float64(Float64(N + 1.0) / N));
	end
	return tmp
end

Wolfram

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], 0.00037743709491433464], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(-0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[Power[N, -4.0], $MachinePrecision] + N[(0.3333333333333333 * N[Power[N, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]]

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 3.7743709491433e-4

   1. Initial program 59.5

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

   2. Simplified 59.5

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

   3. Applied egg-rr 59.3

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

   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. Simplified 0.0

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

   6. Applied egg-rr 0.0

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

   if 3.7743709491433e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   3. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.00037743709491433464:\\ \;\;\;\;\frac{1}{N} + \left(\frac{-0.5}{N \cdot N} + \mathsf{fma}\left(-0.25, {N}^{-4}, 0.3333333333333333 \cdot {N}^{-3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))