Average Error: 0.0 → 0.0
Time: 20.8s
Precision: 64
Internal Precision: 128
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\left(\sqrt{\sqrt{\frac{1}{e^{x} + e^{-x}}} \cdot \sqrt{2}} \cdot \sqrt{\sqrt{\frac{1}{e^{x} + e^{-x}}} \cdot \sqrt{2}}\right) \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]

Error

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Results

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Derivation

  1. Initial program 0.0

    \[\frac{2}{e^{x} + e^{-x}}\]
  2. Using strategy rm

  3. Applied add-log-exp0.1

    \[\leadsto \color{blue}{\log \left(e^{\frac{2}{e^{x} + e^{-x}}}\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.1

    \[\leadsto \log \left(e^{\color{blue}{\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}}}\right)\]

  6. Applied exp-prod0.1

    \[\leadsto \log \color{blue}{\left({\left(e^{\sqrt{\frac{2}{e^{x} + e^{-x}}}}\right)}^{\left(\sqrt{\frac{2}{e^{x} + e^{-x}}}\right)}\right)}\]

  7. Applied log-pow0.1

    \[\leadsto \color{blue}{\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \log \left(e^{\sqrt{\frac{2}{e^{x} + e^{-x}}}}\right)}\]

  8. Taylor expanded around inf 0.5

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \color{blue}{\left(\sqrt{\frac{1}{e^{x} + e^{-x}}} \cdot \sqrt{2}\right)}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt0.0

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

  11. Final simplification0.0

    \[\leadsto \left(\sqrt{\sqrt{\frac{1}{e^{x} + e^{-x}}} \cdot \sqrt{2}} \cdot \sqrt{\sqrt{\frac{1}{e^{x} + e^{-x}}} \cdot \sqrt{2}}\right) \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]

Runtime

Time bar (total: 20.8s)Debug logProfile

herbie shell --seed 2018346 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))