Average Error: 4.4 → 0.0
Time: 16.8s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{\log \left((\left(\sqrt{e^{x}}\right) \cdot \left(\sqrt{e^{x}}\right) + 1)_*\right)}}\]

Error

Bits error versus x

Derivation

  1. Initial program 4.4

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

  2. Simplified0.0

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

  4. Applied add-exp-log0.0

    \[\leadsto \sqrt{\color{blue}{e^{\log \left(e^{x} + 1\right)}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.0

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

  7. Applied fma-def0.0

    \[\leadsto \sqrt{e^{\log \color{blue}{\left((\left(\sqrt{e^{x}}\right) \cdot \left(\sqrt{e^{x}}\right) + 1)_*\right)}}}\]

  8. Final simplification0.0

    \[\leadsto \sqrt{e^{\log \left((\left(\sqrt{e^{x}}\right) \cdot \left(\sqrt{e^{x}}\right) + 1)_*\right)}}\]

Reproduce

herbie shell --seed 2019030 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))