Average Error: 2.1 → 0.3
Time: 27.4s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{{k}^{m}}{\frac{1}{a} + \left(10 + k\right) \cdot \frac{k}{a}}\]

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. Initial program 2.1

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

  2. Initial simplification2.1

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

  3. Taylor expanded around -inf 62.9

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

  4. Simplified2.1

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

  6. Applied associate-/l*2.2

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

  7. Taylor expanded around -inf 4.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Runtime

Time bar (total: 27.4s)Debug logProfile

herbie shell --seed 2018250 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))