Average Error: 2.0 → 2.1
Time: 1.2m
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{a}{\sqrt{(\left(k + 10\right) \cdot k + 1)_*}} \cdot \frac{{k}^{m}}{\sqrt{(\left(10 + k\right) \cdot k + 1)_*}}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Initial program 2.0

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

  3. Applied add-sqr-sqrt2.1

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

  4. Applied times-frac2.1

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

  5. Applied simplify2.1

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

  6. Applied simplify2.1

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018199 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))