Error: 45.1 → 7.6
Time: 21.8s
Precision: 64
Ground Truth: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
⬇
\[\begin{cases} \sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^3}}{e^{x} - 1}} & \text{when } x \le -1.8773586721409905 \cdot 10^{-08} \\ \sqrt{\frac{1}{2} \cdot {x}^2 + \left(2 + x\right)} & \text{otherwise} \end{cases}\]

Error

Bits error versus x

Derivation

    if x < -1.8773586721409905e-08

    1. Initial program 0.3

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube 0.3

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

    if -1.8773586721409905e-08 < x

    1. Initial program 60.4

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

    2. Applied taylor 10.2

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

    3. Taylor expanded around 0 10.2

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

Runtime

Total time: 21.8s Debug log

Please report a bug with the following info:

herbie --seed '#(3022428339 3568377940 1907857276 3908843386 3369637826 71831183)'
(FPCore (x)
  :name "NMSE problem 3.4.4"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))