Average Error: 2.0 → 0.1
Time: 42.8s
Precision: 64
Internal Precision: 384
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 5.566067534285329 \cdot 10^{+125}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k} \cdot \left({k}^{m} - \frac{{k}^{m}}{\frac{k}{10}}\right) + \frac{{k}^{m} \cdot 99}{\frac{{k}^{4}}{a}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 5.566067534285329e+125

    1. Initial program 0.1

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

    if 5.566067534285329e+125 < k

    1. Initial program 8.4

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

    2. Taylor expanded around inf 8.4

      \[\leadsto \color{blue}{\left(\frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{2}} + 99 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{4}}\right) - 10 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{3}}}\]

    3. Applied simplify0.1

      \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k} \cdot \left({k}^{m} - \frac{{k}^{m}}{\frac{k}{10}}\right) + \frac{{k}^{m} \cdot 99}{\frac{{k}^{4}}{a}}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 42.8s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))