Average Error: 2.1 → 0.1
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 4863966.65727832:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\sqrt{1 + \left(k + 10\right) \cdot k}}}{\sqrt{1 + \left(k + 10\right) \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(99 \cdot \frac{{k}^{m}}{{k}^{4}}\right) + \frac{\left(a - \frac{10}{k} \cdot a\right) \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if k < 4863966.65727832

    1. Initial program 0.1

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

    2. Initial simplification0.0

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied associate-/r*0.1

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

    if 4863966.65727832 < k

    1. Initial program 5.4

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

    2. Initial simplification5.4

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

    3. Taylor expanded around inf 5.4

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

    4. Simplified0.3

      \[\leadsto \color{blue}{\frac{{k}^{m}}{k} \cdot \left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) + \frac{{k}^{m}}{{k}^{4}} \cdot \left(99 \cdot a\right)}\]
    5. Using strategy rm

    6. Applied associate-*r/0.3

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

    7. Applied sub-div0.3

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

    8. Applied associate-*r/0.3

      \[\leadsto \color{blue}{\frac{\frac{{k}^{m}}{k} \cdot \left(a - \frac{10}{k} \cdot a\right)}{k}} + \frac{{k}^{m}}{{k}^{4}} \cdot \left(99 \cdot a\right)\]
    9. Using strategy rm

    10. Applied associate-*r*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 4863966.65727832:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\sqrt{1 + \left(k + 10\right) \cdot k}}}{\sqrt{1 + \left(k + 10\right) \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(99 \cdot \frac{{k}^{m}}{{k}^{4}}\right) + \frac{\left(a - \frac{10}{k} \cdot a\right) \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018220 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))