Average Error: 2.2 → 0.1
Time: 1.5m
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 3757789.9836495663:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\sqrt{(\left(k + 10\right) \cdot k + 1)_*}}}{\sqrt{(\left(k + 10\right) \cdot k + 1)_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k} + (\left(\frac{99}{k \cdot k}\right) \cdot \left(\frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k}\right) + \left(-10 \cdot \frac{\frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k}}{k}\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 3757789.9836495663

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied associate-/r*0.1

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

    if 3757789.9836495663 < k

    1. Initial program 5.6

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

    2. Simplified5.6

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

    4. Applied associate-/l*5.8

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

    5. Taylor expanded around -inf 62.9

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

    6. Simplified0.1

      \[\leadsto \color{blue}{(\left(\frac{99}{k \cdot k}\right) \cdot \left(\frac{\frac{a}{k} \cdot e^{m \cdot \left(0 + \log k\right)}}{k}\right) + \left(-10 \cdot \frac{\frac{\frac{a}{k} \cdot e^{m \cdot \left(0 + \log k\right)}}{k}}{k}\right))_* + \frac{\frac{a}{k} \cdot e^{m \cdot \left(0 + \log k\right)}}{k}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 3757789.9836495663:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\sqrt{(\left(k + 10\right) \cdot k + 1)_*}}}{\sqrt{(\left(k + 10\right) \cdot k + 1)_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k} + (\left(\frac{99}{k \cdot k}\right) \cdot \left(\frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k}\right) + \left(-10 \cdot \frac{\frac{\frac{a}{k} \cdot e^{\log k \cdot m}}{k}}{k}\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))