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Derivation

1. Split input into 2 regimes

2. if x < 2.0634884741685034e-07

   1. Initial program 58.8

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

   2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]

   if 2.0634884741685034e-07 < x

   1. Initial program 29.6

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

   3. Applied tanh-undef0.1

      \[\leadsto \color{blue}{\tanh x}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt0.3

      \[\leadsto \color{blue}{\sqrt{\tanh x} \cdot \sqrt{\tanh x}}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.0634884741685034 \cdot 10^{-07}:\\ \;\;\;\;\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\tanh x} \cdot \sqrt{\tanh x}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

Details

Time bar (total: 9.4s)Debug log