Average Error: 39.8 → 0.1
Time: 1.2m
Precision: 64
Internal Precision: 1408
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{\sqrt[3]{{\left(1 + e^{x}\right)}^{3}}}{1}}\]

Error

Bits error versus x

Derivation

  1. Initial program 39.8

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

  2. Applied simplify0.0

    \[\leadsto \color{blue}{\sqrt{\frac{1 + e^{x}}{1}}}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.1

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

  5. Applied simplify0.1

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))