Average Error: 2.0 → 2.1
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[a \cdot \frac{\frac{{k}^{m}}{\sqrt{(\left(k + 10\right) \cdot k + 1)_*}}}{\sqrt{(\left(k + 10\right) \cdot k + 1)_*}}\]

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 *-un-lft-identity2.0

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

  4. Applied times-frac2.0

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

  5. Simplified2.0

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

  6. Simplified2.0

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

  8. Applied add-sqr-sqrt2.1

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

  9. Applied associate-/r*2.1

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

  10. Final simplification2.1

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018278 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))