Average Error: 57.9 → 0.7
Time: 34.7s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{(\left({\left(2 \cdot x\right)}^{\left(\frac{2}{3} + \frac{1}{3}\right)}\right) \cdot \left((\frac{1}{6} \cdot \left(x \cdot x\right) + \left(\frac{1}{120} \cdot {x}^{4}\right))_*\right) + \left({\left(2 \cdot x\right)}^{\left(\frac{2}{3} + \frac{1}{3}\right)}\right))_*}{2}\]

Error

Bits error versus x

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.7

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

  3. Applied simplify0.7

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

  5. Applied add-cube-cbrt2.2

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

  7. Applied pow1/333.6

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

  8. Applied pow1/334.1

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

  9. Applied pow-prod-up34.1

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

  10. Taylor expanded around 0 34.3

    \[\leadsto \frac{\color{blue}{\frac{1}{6} \cdot \left(e^{\frac{2}{3} \cdot \left(\log x + \log 2\right)} \cdot \left(e^{\frac{1}{3} \cdot \left(\log x + \log 2\right)} \cdot {x}^{2}\right)\right) + \left(\frac{1}{120} \cdot \left(e^{\frac{2}{3} \cdot \left(\log x + \log 2\right)} \cdot \left(e^{\frac{1}{3} \cdot \left(\log x + \log 2\right)} \cdot {x}^{4}\right)\right) + e^{\frac{2}{3} \cdot \left(\log x + \log 2\right)} \cdot e^{\frac{1}{3} \cdot \left(\log x + \log 2\right)}\right)}}{2}\]

  11. Applied simplify0.7

    \[\leadsto \color{blue}{\frac{(\left({\left(2 \cdot x\right)}^{\left(\frac{2}{3} + \frac{1}{3}\right)}\right) \cdot \left((\frac{1}{6} \cdot \left(x \cdot x\right) + \left(\frac{1}{120} \cdot {x}^{4}\right))_*\right) + \left({\left(2 \cdot x\right)}^{\left(\frac{2}{3} + \frac{1}{3}\right)}\right))_*}{2}}\]

Runtime

Time bar (total: 34.7s)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))