2log (problem 3.3.6)

Average Error: 28.7 → 0.1

Time: 3.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.25}{{N}^{4}} - \mathsf{fma}\left(0.5, {N}^{-2}, \frac{-1}{N} - 0.3333333333333333 \cdot {N}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 2e-5)
   (-
    (/ -0.25 (pow N 4.0))
    (fma 0.5 (pow N -2.0) (- (/ -1.0 N) (* 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)) <= 2e-5) {
		tmp = (-0.25 / pow(N, 4.0)) - fma(0.5, pow(N, -2.0), ((-1.0 / N) - (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)) <= 2e-5)
		tmp = Float64(Float64(-0.25 / (N ^ 4.0)) - fma(0.5, (N ^ -2.0), Float64(Float64(-1.0 / N) - 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], 2e-5], N[(N[(-0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Power[N, -2.0], $MachinePrecision] + N[(N[(-1.0 / N), $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)) < 2.00000000000000016e-5

   1. Initial program 59.6

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

   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. Simplified 0.0

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

   4. Applied egg-rr 0.0

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

   if 2.00000000000000016e-5 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 0.1

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

   2. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.25}{{N}^{4}} - \mathsf{fma}\left(0.5, {N}^{-2}, \frac{-1}{N} - 0.3333333333333333 \cdot {N}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))