Average Error: 2.2 → 0.3
Time: 42.4s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 2.4138843895853474 \cdot 10^{-07}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{\sqrt{1}}}{\left|\sqrt[3]{(\left(k + 10\right) \cdot k + 1)_*}\right|} \cdot \frac{\frac{a}{\sqrt{\sqrt[3]{(k \cdot \left(k + 10\right) + 1)_*}}}}{\sqrt{(k \cdot \left(k + 10\right) + 1)_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{k}{{k}^{m}}\right) \cdot \left(\frac{10}{a} + \frac{k}{a}\right) + \left(\frac{{k}^{\left(-m\right)}}{a}\right))_*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 2.4138843895853474e-07

    1. Initial program 0.1

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

    2. Initial simplification0.0

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied associate-/r*0.1

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

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

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

    8. Applied sqrt-prod0.1

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

    9. Applied add-cube-cbrt0.1

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

    10. Applied sqrt-prod0.1

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

    11. Applied times-frac0.1

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

    12. Applied times-frac0.1

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

    13. Simplified0.1

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

    if 2.4138843895853474e-07 < k

    1. Initial program 5.5

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

    2. Initial simplification5.5

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

    4. Applied clear-num5.7

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

    5. Taylor expanded around inf 5.7

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

    6. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 2.4138843895853474 \cdot 10^{-07}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{\sqrt{1}}}{\left|\sqrt[3]{(\left(k + 10\right) \cdot k + 1)_*}\right|} \cdot \frac{\frac{a}{\sqrt{\sqrt[3]{(k \cdot \left(k + 10\right) + 1)_*}}}}{\sqrt{(k \cdot \left(k + 10\right) + 1)_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{(\left(\frac{k}{{k}^{m}}\right) \cdot \left(\frac{10}{a} + \frac{k}{a}\right) + \left(\frac{{k}^{\left(-m\right)}}{a}\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 42.4s)Debug logProfile

herbie shell --seed 2018219 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))