Input Error: 44.8b

Output Error: 7.3b

Time: 16.8s

Precision: 64b

Ground Truth: 128b

\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

⬇

\[\begin{cases} \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} & \text{when } x \le -2.110554417076555 \cdot 10^{-07} \\ \sqrt{x + \left(2 + \frac{1}{2} \cdot {x}^2\right)} & \text{otherwise} \end{cases}\]

Error

The X axis may use a short exponential scale — Bits error versus `x`

Derivation

`if x < -2.110554417076555e-07`

Initial program 0.2b
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

`if -2.110554417076555e-07 < x`

Initial program 60.5b
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
Applied taylor 9.8b
\[\leadsto \sqrt{x + \left(2 + \frac{1}{2} \cdot {x}^2\right)}\]
Taylor expanded around 0 9.8b

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

Removed slow pow expressions

Runtime

herbie --seed '#(2983337528 3758187870 3729518623 3767075579 1842131948 2540937932)'
(FPCore (x)
  :name "NMSE problem 3.4.4"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))