Average Error: 53.1 → 53.0
Time: 24.4s
Precision: 64
Internal Precision: 128
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[(\frac{1}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{6} + 1)_*\right))_* \cdot \sqrt[3]{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_* \cdot (\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\]

Error

Bits error versus x

Derivation

  1. Initial program 53.1

    \[\frac{2}{e^{x} + e^{-x}}\]

  2. Taylor expanded around 0 53.0

    \[\leadsto \color{blue}{\left(\frac{5}{24} \cdot {x}^{4} + 1\right) - \frac{1}{2} \cdot {x}^{2}}\]

  3. Simplified53.0

    \[\leadsto \color{blue}{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt53.0

    \[\leadsto \color{blue}{\left(\sqrt[3]{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*} \cdot \sqrt[3]{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\right) \cdot \sqrt[3]{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*}}\]

  6. Taylor expanded around 0 53.0

    \[\leadsto \left(\sqrt[3]{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*} \cdot \sqrt[3]{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\right) \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{4} + 1\right) - \frac{1}{6} \cdot {x}^{2}\right)}\]

  7. Simplified53.0

    \[\leadsto \left(\sqrt[3]{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*} \cdot \sqrt[3]{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\right) \cdot \color{blue}{(\frac{1}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{6} + 1)_*\right))_*}\]
  8. Using strategy rm

  9. Applied cbrt-unprod53.0

    \[\leadsto \color{blue}{\sqrt[3]{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_* \cdot (\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*}} \cdot (\frac{1}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{6} + 1)_*\right))_*\]

  10. Final simplification53.0

    \[\leadsto (\frac{1}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{6} + 1)_*\right))_* \cdot \sqrt[3]{(\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_* \cdot (\frac{5}{24} \cdot \left({x}^{4}\right) + \left((\left(x \cdot x\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\]

Runtime

Time bar (total: 24.4s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))