Average Error: 2.2 → 0.2
Time: 44.9s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 2.172783873877796 \cdot 10^{+143}:\\ \;\;\;\;\frac{{k}^{m}}{(\left(k + 10\right) \cdot k + 1)_*} \cdot a\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{{k}^{4}}{a}}\right) \cdot 99 + \left((\left(\frac{-10}{k}\right) \cdot \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right) + \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right))_*\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 2.172783873877796e+143

    1. Initial program 0.1

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

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

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

    6. Simplified0.1

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

    if 2.172783873877796e+143 < k

    1. Initial program 10.5

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

    3. Applied *-un-lft-identity10.5

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

    4. Applied times-frac10.5

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

    5. Simplified10.5

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

    6. Simplified10.5

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

    7. Taylor expanded around -inf 63.0

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

    8. Simplified0.6

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 2.172783873877796 \cdot 10^{+143}:\\ \;\;\;\;\frac{{k}^{m}}{(\left(k + 10\right) \cdot k + 1)_*} \cdot a\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{{k}^{4}}{a}}\right) \cdot 99 + \left((\left(\frac{-10}{k}\right) \cdot \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right) + \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{\frac{k}{\frac{a}{k}}}\right))_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 44.9s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes2.20.20.12.098.3%
herbie shell --seed 2018353 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))