Average Error: 2.0 → 0.2
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{{k}^{m}}{\frac{1}{a} + \frac{k}{a} \cdot \left(10 + k\right)}\]

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. Simplified2.0

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

  4. Applied associate-/l*2.1

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

  5. Taylor expanded around -inf 4.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019090 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))