Average Error: 4.3 → 0.1
Time: 31.7s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{{\left(e^{x}\right)}^{3} + 1}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\]

Error

Bits error versus x

Derivation

  1. Initial program 4.3

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

  2. Simplified0.1

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

  4. Applied flip3-+0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019093 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))