Average Error: 2.2 → 0.3
Time: 55.0s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{{k}^{m}}{\frac{k}{a} \cdot \left(10 + k\right) + \frac{1}{a}}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Initial program 2.2

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

  2. Applied simplify2.1

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

  4. Applied associate-/l*2.3

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

  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

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

Runtime

Time bar (total: 55.0s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))