Average Error: 2.0 → 0.1
Time: 49.7s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\log k \cdot \left(a \cdot m\right) + a}{1 + \left(10 + k\right) \cdot k} \le -1.332932409217441 \cdot 10^{-302}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;\frac{\log k \cdot \left(a \cdot m\right) + a}{1 + \left(10 + k\right) \cdot k} \le 4.096510593335 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\frac{a}{k}}{k} \cdot \left(\frac{10}{k} + 1\right) + \frac{99 \cdot a}{{k}^{4}}\right) \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + \left(10 + k\right) \cdot k}\\ \end{array}\]

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 3 regimes

  2. if (/ (+ (* (log k) (* a m)) a) (+ 1 (* (+ 10 k) k))) < -1.332932409217441e-302

    1. Initial program 0.1

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

    if -1.332932409217441e-302 < (/ (+ (* (log k) (* a m)) a) (+ 1 (* (+ 10 k) k))) < 4.096510593335e-310

    1. Initial program 9.2

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

    2. Applied simplify9.2

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

    4. Applied *-un-lft-identity9.2

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

    5. Applied times-frac9.2

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

    6. Applied simplify9.2

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

    7. Taylor expanded around -inf 9.5

      \[\leadsto {k}^{m} \cdot \color{blue}{\left(99 \cdot \frac{a}{{k}^{4}} + \left(\frac{a}{{k}^{2}} + 10 \cdot \frac{a}{{k}^{3}}\right)\right)}\]

    8. Applied simplify0.4

      \[\leadsto \color{blue}{\left(\frac{\frac{a}{k}}{k} \cdot \left(\frac{10}{k} + 1\right) + \frac{99 \cdot a}{{k}^{4}}\right) \cdot {k}^{m}}\]

    if 4.096510593335e-310 < (/ (+ (* (log k) (* a m)) a) (+ 1 (* (+ 10 k) k)))

    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 *-un-lft-identity0.0

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

    5. Applied times-frac0.0

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

    6. Applied simplify0.0

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

Runtime

Time bar (total: 49.7s)Debug logProfile

herbie shell --seed 2018195 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))