Average Error: 2.1 → 2.2
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\frac{{k}^{m} \cdot a}{\sqrt{1 + \left(k + 10\right) \cdot k}}}{\sqrt{1 + \left(k + 10\right) \cdot k}}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.1

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

  2. Initial simplification2.1

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

  4. Applied add-sqr-sqrt2.2

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

  5. Applied associate-/r*2.2

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

  6. Final simplification2.2

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018255 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))