Average Error: 39.6 → 0.1
Time: 26.1s
Precision: 64
Internal Precision: 1408
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{\sqrt[3]{e^{\log \left(e^{x} + 1\right) \cdot 3}}}{1}}\]

Error

Bits error versus x

Derivation

  1. Initial program 39.6

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

  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

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

  7. Applied add-exp-log0.1

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

  8. Applied pow-exp0.1

    \[\leadsto \sqrt{\frac{\sqrt[3]{\color{blue}{e^{\log \left(e^{x} + 1\right) \cdot 3}}}}{1}}\]
  9. Removed slow pow expressions.

Runtime

Time bar (total: 26.1s)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))