Average Error: 1.8 → 1.4
Time: 58.8s
Precision: 64
Internal Precision: 384
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left({k}^{m} - \frac{{k}^{m}}{\frac{k}{10}}\right) \cdot \frac{\frac{a}{k}}{k} + \frac{99 \cdot {k}^{m}}{\frac{{k}^{4}}{a}} \le -1.1867684620269203 \cdot 10^{-52}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;\left({k}^{m} - \frac{{k}^{m}}{\frac{k}{10}}\right) \cdot \frac{\frac{a}{k}}{k} + \frac{99 \cdot {k}^{m}}{\frac{{k}^{4}}{a}} \le 1.5268999844280633 \cdot 10^{-56}:\\ \;\;\;\;\left({k}^{m} - \frac{{k}^{m}}{\frac{k}{10}}\right) \cdot \frac{\frac{a}{k}}{k} + \frac{99 \cdot {k}^{m}}{\frac{{k}^{4}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\left(k \cdot \left(10 + k\right)\right) \cdot \left(k \cdot \left(10 + k\right)\right) - k \cdot \left(10 + k\right)\right)\right) \cdot \frac{{k}^{m} \cdot a}{{\left(k \cdot \left(10 + k\right)\right)}^{3} + 1}\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 3 regimes

  2. if (+ (* (/ (/ a k) k) (- (pow k m) (/ (pow k m) (/ k 10)))) (/ (* (pow k m) 99) (/ (pow k 4) a))) < -1.1867684620269203e-52

    1. Initial program 0.8

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

    if -1.1867684620269203e-52 < (+ (* (/ (/ a k) k) (- (pow k m) (/ (pow k m) (/ k 10)))) (/ (* (pow k m) 99) (/ (pow k 4) a))) < 1.5268999844280633e-56

    1. Initial program 3.4

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

    2. Applied simplify3.3

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

    3. Taylor expanded around inf 22.9

      \[\leadsto \color{blue}{\left(\frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{2}} + 99 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{4}}\right) - 10 \cdot \frac{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}{{k}^{3}}}\]

    4. Applied simplify1.5

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

    if 1.5268999844280633e-56 < (+ (* (/ (/ a k) k) (- (pow k m) (/ (pow k m) (/ k 10)))) (/ (* (pow k m) 99) (/ (pow k 4) a)))

    1. Initial program 0.1

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

    2. Applied simplify0.1

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

    4. Applied flip3-+1.4

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

    5. Applied associate-/r/1.4

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

    6. Applied simplify1.4

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

  4. Applied simplify1.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left({k}^{m} - \frac{{k}^{m}}{\frac{k}{10}}\right) \cdot \frac{\frac{a}{k}}{k} + \frac{99 \cdot {k}^{m}}{\frac{{k}^{4}}{a}} \le -1.1867684620269203 \cdot 10^{-52}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;\left({k}^{m} - \frac{{k}^{m}}{\frac{k}{10}}\right) \cdot \frac{\frac{a}{k}}{k} + \frac{99 \cdot {k}^{m}}{\frac{{k}^{4}}{a}} \le 1.5268999844280633 \cdot 10^{-56}:\\ \;\;\;\;\left({k}^{m} - \frac{{k}^{m}}{\frac{k}{10}}\right) \cdot \frac{\frac{a}{k}}{k} + \frac{99 \cdot {k}^{m}}{\frac{{k}^{4}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\left(k \cdot \left(10 + k\right)\right) \cdot \left(k \cdot \left(10 + k\right)\right) - k \cdot \left(10 + k\right)\right)\right) \cdot \frac{{k}^{m} \cdot a}{{\left(k \cdot \left(10 + k\right)\right)}^{3} + 1}\\ \end{array}}\]

Runtime

Time bar (total: 58.8s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))