Average Error: 2.2 → 0.1
Time: 57.4s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 3.8534943108273086 \cdot 10^{+128}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{1 + \left(10 + k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{\frac{k}{a}}{{k}^{m}} \cdot \left(10 + k\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 3.8534943108273086e+128

    1. Initial program 0.1

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

    2. Applied simplify0.0

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

    if 3.8534943108273086e+128 < k

    1. Initial program 9.6

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

    2. Applied simplify9.6

      \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{1 + \left(10 + k\right) \cdot k}}\]
    3. Using strategy rm

    4. Applied clear-num9.6

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

    5. Taylor expanded around inf 9.6

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

    6. Applied simplify0.5

      \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{\frac{k}{a}}{{k}^{m}} \cdot \left(10 + k\right)}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 57.4s)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))