Average Error: 2.1 → 0.1
Time: 1.9m
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 8.89891869659912 \cdot 10^{+118}:\\ \;\;\;\;\frac{\sqrt{{k}^{m}} \cdot \left(a \cdot \sqrt{{k}^{m}}\right)}{k \cdot \left(10 + k\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(\frac{-1}{\frac{-1}{k}}\right)}^{m}}{\frac{k}{\frac{a}{k}}} + \frac{99 \cdot {\left(\frac{-1}{\frac{-1}{k}}\right)}^{m}}{\frac{{k}^{4}}{a}}\right) - \frac{{\left(\frac{-1}{\frac{-1}{k}}\right)}^{m} \cdot 10}{\frac{k \cdot k}{\frac{a}{k}}}\\ \end{array}\]

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 < 8.89891869659912e+118

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied associate-*l*0.1

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

    if 8.89891869659912e+118 < k

    1. Initial program 8.6

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

    2. Initial simplification8.6

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

    3. Taylor expanded around -inf 63.0

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

    4. Simplified0.4

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 8.89891869659912 \cdot 10^{+118}:\\ \;\;\;\;\frac{\sqrt{{k}^{m}} \cdot \left(a \cdot \sqrt{{k}^{m}}\right)}{k \cdot \left(10 + k\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(\frac{-1}{\frac{-1}{k}}\right)}^{m}}{\frac{k}{\frac{a}{k}}} + \frac{99 \cdot {\left(\frac{-1}{\frac{-1}{k}}\right)}^{m}}{\frac{{k}^{4}}{a}}\right) - \frac{{\left(\frac{-1}{\frac{-1}{k}}\right)}^{m} \cdot 10}{\frac{k \cdot k}{\frac{a}{k}}}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))