Average Error: 2.2 → 0.7
Time: 38.2s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\frac{\sqrt{{k}^{m}}}{\sqrt[3]{(\left(\frac{k}{a}\right) \cdot \left(10 + k\right) + \left(\frac{1}{a}\right))_*}}}{\sqrt[3]{(\left(\frac{k}{a}\right) \cdot \left(10 + k\right) + \left(\frac{1}{a}\right))_*}} \cdot \frac{\sqrt{{k}^{m}}}{\sqrt[3]{(\left(\frac{k}{a}\right) \cdot \left(10 + k\right) + \left(\frac{1}{a}\right))_*}}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Initial program 2.2

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

  2. Applied simplify2.1

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

  4. Applied associate-/l*2.3

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

  5. Taylor expanded around 0 4.0

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

  6. Applied simplify2.0

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

  8. Applied add-cube-cbrt2.4

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

  9. Applied add-sqr-sqrt2.4

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

  10. Applied times-frac2.4

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

  11. Applied simplify2.4

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

  12. Applied simplify0.7

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

Runtime

Time bar (total: 38.2s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))