Average Error: 2.0 → 0.3
Time: 38.6s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{a + \log k \cdot \left(a \cdot m\right)}{k \cdot k + \left(k \cdot 10 + 1\right)} \le -8.308973878466061 \cdot 10^{-180}:\\ \;\;\;\;\frac{a}{\sqrt{1 + k \cdot \left(10 + k\right)}} \cdot \frac{{k}^{m}}{\sqrt{1 + k \cdot \left(10 + k\right)}}\\ \mathbf{elif}\;\frac{a + \log k \cdot \left(a \cdot m\right)}{k \cdot k + \left(k \cdot 10 + 1\right)} \le 1.9634168765731 \cdot 10^{-320}:\\ \;\;\;\;\frac{1}{\left(10 + k\right) \cdot \frac{\frac{k}{a}}{{k}^{m}} + \frac{{k}^{\left(-m\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{1 + k \cdot \left(10 + k\right)}} \cdot \frac{{k}^{m}}{\sqrt{1 + k \cdot \left(10 + k\right)}}\\ \end{array}\]

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 (/ (+ (* (log k) (* a m)) a) (+ (+ 1 (* 10 k)) (* k k))) < -8.308973878466061e-180 or 1.9634168765731e-320 < (/ (+ (* (log k) (* a m)) a) (+ (+ 1 (* 10 k)) (* k k)))

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

    6. Simplified0.1

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

    if -8.308973878466061e-180 < (/ (+ (* (log k) (* a m)) a) (+ (+ 1 (* 10 k)) (* k k))) < 1.9634168765731e-320

    1. Initial program 7.6

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

    3. Applied clear-num7.8

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

    4. Taylor expanded around inf 8.1

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

    5. Simplified0.8

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a + \log k \cdot \left(a \cdot m\right)}{k \cdot k + \left(k \cdot 10 + 1\right)} \le -8.308973878466061 \cdot 10^{-180}:\\ \;\;\;\;\frac{a}{\sqrt{1 + k \cdot \left(10 + k\right)}} \cdot \frac{{k}^{m}}{\sqrt{1 + k \cdot \left(10 + k\right)}}\\ \mathbf{elif}\;\frac{a + \log k \cdot \left(a \cdot m\right)}{k \cdot k + \left(k \cdot 10 + 1\right)} \le 1.9634168765731 \cdot 10^{-320}:\\ \;\;\;\;\frac{1}{\left(10 + k\right) \cdot \frac{\frac{k}{a}}{{k}^{m}} + \frac{{k}^{\left(-m\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{1 + k \cdot \left(10 + k\right)}} \cdot \frac{{k}^{m}}{\sqrt{1 + k \cdot \left(10 + k\right)}}\\ \end{array}\]

Runtime

Time bar (total: 38.6s)Debug logProfile

herbie shell --seed 2018215 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))