Average Error: 60.0 → 53.1
Time: 15.7s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \log \left(\sqrt{\log \left(e^{e^{{x}^{5} \cdot \frac{1}{60}}}\right)} \cdot \sqrt{\log \left(e^{e^{{x}^{5} \cdot \frac{1}{60}}}\right)}\right)\right)}{2}\]

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 60.0

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

  2. Taylor expanded around 0 53.0

    \[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]
  3. Using strategy rm

  4. Applied add-log-exp53.1

    \[\leadsto \frac{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \color{blue}{\log \left(e^{\frac{1}{60} \cdot {x}^{5}}\right)}\right)}{2}\]
  5. Using strategy rm

  6. Applied add-log-exp53.1

    \[\leadsto \frac{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \log \color{blue}{\left(\log \left(e^{e^{\frac{1}{60} \cdot {x}^{5}}}\right)\right)}\right)}{2}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt53.1

    \[\leadsto \frac{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \log \color{blue}{\left(\sqrt{\log \left(e^{e^{\frac{1}{60} \cdot {x}^{5}}}\right)} \cdot \sqrt{\log \left(e^{e^{\frac{1}{60} \cdot {x}^{5}}}\right)}\right)}\right)}{2}\]

  9. Final simplification53.1

    \[\leadsto \frac{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \log \left(\sqrt{\log \left(e^{e^{{x}^{5} \cdot \frac{1}{60}}}\right)} \cdot \sqrt{\log \left(e^{e^{{x}^{5} \cdot \frac{1}{60}}}\right)}\right)\right)}{2}\]

Runtime

Time bar (total: 15.7s)Debug logProfile

herbie shell --seed 2018255 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))