Average Error: 0.0 → 0.1
Time: 46.8s
Precision: 64
Internal Precision: 320
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt[3]{\frac{2}{e^{x} + e^{-x}} \cdot \left(\frac{2}{e^{x} + e^{-x}} \cdot \frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}\right)}\]

Error

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Results

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Derivation

  1. Initial program 0.0

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

  2. Initial simplification0.0

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

  4. Applied add-cbrt-cube0.1

    \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{2}{e^{x} + e^{-x}} \cdot \frac{2}{e^{x} + e^{-x}}\right) \cdot \frac{2}{e^{x} + e^{-x}}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.6

    \[\leadsto \sqrt[3]{\left(\frac{2}{\color{blue}{\sqrt{e^{x} + e^{-x}} \cdot \sqrt{e^{x} + e^{-x}}}} \cdot \frac{2}{e^{x} + e^{-x}}\right) \cdot \frac{2}{e^{x} + e^{-x}}}\]

  7. Applied associate-/r*0.1

    \[\leadsto \sqrt[3]{\left(\color{blue}{\frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}} \cdot \frac{2}{e^{x} + e^{-x}}\right) \cdot \frac{2}{e^{x} + e^{-x}}}\]

  8. Final simplification0.1

    \[\leadsto \sqrt[3]{\frac{2}{e^{x} + e^{-x}} \cdot \left(\frac{2}{e^{x} + e^{-x}} \cdot \frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}\right)}\]

Runtime

Time bar (total: 46.8s)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))