Average Error: 2.1 → 2.1
Time: 39.5s
Precision: 64
Internal Precision: 576
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\frac{{k}^{m} \cdot a}{\sqrt{(k \cdot \left(10 + k\right) + 1)_*}}}{\sqrt{(k \cdot \left(10 + k\right) + 1)_*}}\]

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. Initial simplification2.1

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

  4. Applied add-sqr-sqrt2.1

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

  5. Applied associate-/r*2.1

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

  6. Final simplification2.1

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

Runtime

Time bar (total: 39.5s)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))