Average Error: 2.0 → 1.1
Time: 1.5m
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{99 \cdot a}{{k}^{4}} \cdot {k}^{m} - \frac{{k}^{m}}{k} \cdot \left(\frac{10}{k} \cdot \frac{a}{k} - \frac{a}{k}\right) \le -7.998481467270114 \cdot 10^{-108}:\\ \;\;\;\;\frac{\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \left(\sqrt[3]{{k}^{m}} \cdot a\right)}{1 + k \cdot \left(10 + k\right)}\\ \mathbf{if}\;\frac{99 \cdot a}{{k}^{4}} \cdot {k}^{m} - \frac{{k}^{m}}{k} \cdot \left(\frac{10}{k} \cdot \frac{a}{k} - \frac{a}{k}\right) \le 9.264873509848405 \cdot 10^{-262}:\\ \;\;\;\;\frac{99 \cdot a}{{k}^{4}} \cdot {k}^{m} - \frac{{k}^{m}}{k} \cdot \left(\frac{10}{k} \cdot \frac{a}{k} - \frac{a}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \left(\sqrt[3]{{k}^{m}} \cdot a\right)}{1 + k \cdot \left(10 + k\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (- (* (/ (* 99 a) (pow k 4)) (pow k m)) (* (/ (pow k m) k) (- (* (/ 10 k) (/ a k)) (/ a k)))) < -7.998481467270114e-108 or 9.264873509848405e-262 < (- (* (/ (* 99 a) (pow k 4)) (pow k m)) (* (/ (pow k m) k) (- (* (/ 10 k) (/ a k)) (/ a k))))

    1. Initial program 1.5

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

    2. Applied simplify1.5

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

    4. Applied add-cube-cbrt1.6

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

    5. Applied associate-*l*1.6

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

    if -7.998481467270114e-108 < (- (* (/ (* 99 a) (pow k 4)) (pow k m)) (* (/ (pow k m) k) (- (* (/ 10 k) (/ a k)) (/ a k)))) < 9.264873509848405e-262

    1. Initial program 2.5

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

    2. Applied simplify2.5

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

    3. Taylor expanded around inf 22.2

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

    4. Applied simplify0.5

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2020178 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))