Average Error: 58.3 → 1.9
Time: 15.9s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\left(x + {x}^{5} \cdot \frac{2}{15}\right) - \frac{1}{3} \cdot {x}^{3}\]

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 58.3

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

  3. Applied tanh-undef0.0

    \[\leadsto \color{blue}{\tanh x}\]

  4. Taylor expanded around 0 1.9

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

  5. Final simplification1.9

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

Runtime

Time bar (total: 15.9s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.91.90.01.90%
herbie shell --seed 2018285 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))