Average Error: 2.1 → 0.1
Time: 1.1m
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{1 + \frac{\left({10}^{3} + {k}^{3}\right) \cdot k}{10 \cdot 10 + \left(k \cdot k - 10 \cdot k\right)}} \le 3.0347748582371028 \cdot 10^{+161}:\\ \;\;\;\;\left({k}^{m} \cdot a\right) \cdot \frac{1}{1 + \left(10 + k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{\frac{k}{a}}{{k}^{m}} \cdot \left(10 + k\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (pow k m) a) (+ 1 (/ (* (+ (pow 10 3) (pow k 3)) k) (+ (* 10 10) (- (* k k) (* 10 k)))))) < 3.0347748582371028e+161

    1. Initial program 0.1

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

    2. Applied simplify0.0

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

    4. Applied div-inv0.1

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

    if 3.0347748582371028e+161 < (/ (* (pow k m) a) (+ 1 (/ (* (+ (pow 10 3) (pow k 3)) k) (+ (* 10 10) (- (* k k) (* 10 k))))))

    1. Initial program 7.3

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

    2. Applied simplify7.3

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

    4. Applied clear-num7.4

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

    5. Taylor expanded around inf 20.7

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

    6. Applied simplify0.4

      \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{\left(-m\right)}}{a} + \frac{\frac{k}{a}}{{k}^{m}} \cdot \left(10 + k\right)}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018193 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))