Average Error: 57.8 → 0.7
Time: 28.7s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{(x \cdot \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right) + \left(\frac{1}{60} \cdot {x}^{5}\right))_*}{2}\]

Error

Bits error versus x

Derivation

  1. Initial program 57.8

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

  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

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

Runtime

Time bar (total: 28.7s)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))