Average Error: 2.2 → 0.2
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 2.659136990663816 \cdot 10^{+141}:\\ \;\;\;\;\frac{{k}^{m}}{(k \cdot \left(k + 10\right) + 1)_*} \cdot a\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{e^{m \cdot \log k}}{\frac{k}{a} \cdot k}}{k}\right) \cdot -10 + \left((99 \cdot \left(\frac{e^{m \cdot \log k}}{\frac{{k}^{4}}{a}}\right) + \left(\frac{e^{m \cdot \log k}}{\frac{k}{a} \cdot k}\right))_*\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 2.659136990663816e+141

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied associate-/r*0.1

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

    7. Applied *-un-lft-identity0.1

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

    8. Applied div-inv0.2

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

    9. Applied times-frac0.2

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

    10. Simplified0.2

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

    11. Simplified0.1

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

    if 2.659136990663816e+141 < k

    1. Initial program 10.0

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

    2. Simplified10.0

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

    4. Applied add-sqr-sqrt10.0

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

    5. Applied associate-/r*10.0

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

    6. Taylor expanded around inf 10.0

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

    7. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 2.659136990663816 \cdot 10^{+141}:\\ \;\;\;\;\frac{{k}^{m}}{(k \cdot \left(k + 10\right) + 1)_*} \cdot a\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{e^{m \cdot \log k}}{\frac{k}{a} \cdot k}}{k}\right) \cdot -10 + \left((99 \cdot \left(\frac{e^{m \cdot \log k}}{\frac{{k}^{4}}{a}}\right) + \left(\frac{e^{m \cdot \log k}}{\frac{k}{a} \cdot k}\right))_*\right))_*\\ \end{array}\]

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))))