Average Error: 1.9 → 0.2
Time: 1.2m
Precision: 64
Internal Precision: 384
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 2.2029610971913975 \cdot 10^{+145}:\\ \;\;\;\;\frac{a}{\sqrt{k \cdot \left(k + 10\right) + 1}} \cdot \frac{{k}^{m}}{\sqrt{1 + k \cdot \left(10 + k\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{\frac{k}{a}}{{k}^{m}} \cdot \left(10 + k\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 2.2029610971913975e+145

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied times-frac0.1

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

    5. Applied simplify0.1

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

    6. Applied simplify0.1

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

    if 2.2029610971913975e+145 < k

    1. Initial program 9.4

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

    3. Applied clear-num9.5

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

    4. Applied simplify9.5

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

    5. Taylor expanded around inf 9.5

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

    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: 1.2m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))