Average Error: 2.2 → 0.2
Time: 52.6s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 5.441980365048061 \cdot 10^{+29}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)} \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 - \frac{10}{k}}{\frac{k}{\frac{a}{k}}} + \frac{a \cdot 99}{{k}^{4}}\right) \cdot {k}^{m}\\ \end{array}\]

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 < 5.441980365048061e+29

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied *-un-lft-identity0.0

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

    5. Applied times-frac0.0

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

    6. Simplified0.0

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

    if 5.441980365048061e+29 < k

    1. Initial program 6.5

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

    2. Initial simplification6.5

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

    4. Applied *-un-lft-identity6.5

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

    5. Applied times-frac6.5

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

    6. Simplified6.5

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

    7. Taylor expanded around inf 6.5

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

    8. Simplified0.6

      \[\leadsto {k}^{m} \cdot \color{blue}{\left(\frac{1 - \frac{10}{k}}{\frac{k}{\frac{a}{k}}} + \frac{99 \cdot a}{{k}^{4}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 5.441980365048061 \cdot 10^{+29}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)} \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 - \frac{10}{k}}{\frac{k}{\frac{a}{k}}} + \frac{a \cdot 99}{{k}^{4}}\right) \cdot {k}^{m}\\ \end{array}\]

Runtime

Time bar (total: 52.6s)Debug logProfile

herbie shell --seed 2018230 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))