Average Error: 2.1 → 0.3
Time: 19.4s
Precision: 64
Internal Precision: 384
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{{k}^{m}}{(\left(\frac{k}{a}\right) \cdot \left(10 + k\right) + \left(\frac{1}{a}\right))_*}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Initial program 2.1

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

  2. Applied simplify2.1

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

  4. Applied associate-/l*2.2

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

  5. Taylor expanded around 0 4.0

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

  6. Applied simplify0.3

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

Runtime

Time bar (total: 19.4s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))