Average Error: 2.1 → 0.3
Time: 44.8s
Precision: 64
Internal Precision: 384
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -2.9090854064384 \cdot 10^{-309}:\\ \;\;\;\;\left(\sqrt[3]{\frac{{k}^{m} \cdot a}{1 + k \cdot \left(10 + k\right)}} \cdot \sqrt[3]{\frac{{k}^{m} \cdot a}{1 + k \cdot \left(10 + k\right)}}\right) \cdot \sqrt[3]{\frac{{k}^{m} \cdot a}{\left(10 + k\right) \cdot k + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{k}{{k}^{m}} \cdot \left(\frac{k}{a} + \frac{10}{a}\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < -2.9090854064384e-309

    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 add-cube-cbrt0.1

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

    4. Applied simplify0.1

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

    5. Applied simplify0

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

    if -2.9090854064384e-309 < k

    1. Initial program 2.7

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

    3. Applied clear-num2.9

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

    4. Applied simplify2.9

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

    5. Taylor expanded around inf 8.0

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

    6. Applied simplify0.4

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

Runtime

Time bar (total: 44.8s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))