2log (problem 3.3.6)

Average Error: 29.2 → 0.1

Time: 4.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 4112.20417112273844:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\log \left(N + 1\right)}, \sqrt{\log \left(N + 1\right)}, \log \left(\frac{1}{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

C

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

↓

double code(double N) {
	double VAR;
	if ((N <= 4112.204171122738)) {
		VAR = fma(sqrt(log((N + 1.0))), sqrt(log((N + 1.0))), log((1.0 / N)));
	} else {
		VAR = fma((1.0 / pow(N, 2.0)), ((0.3333333333333333 / N) - 0.5), (1.0 / N));
	}
	return VAR;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 4112.204171122738

   1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 0.1

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

   4. Applied fma-neg 0.1

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

   5. Simplified 0.1

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

   if 4112.204171122738 < N

   1. Initial program 59.5

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

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4112.20417112273844:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\log \left(N + 1\right)}, \sqrt{\log \left(N + 1\right)}, \log \left(\frac{1}{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))