Average Error: 39.5 → 0.1
Time: 25.2s
Precision: 64
Internal Precision: 1344
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\frac{\sqrt{1 + \left({\left(\sqrt[3]{\sqrt{e^{x}}}\right)}^{3} \cdot e^{x}\right) \cdot {\left(\sqrt{e^{x}}\right)}^{3}}}{\sqrt{\left(1 - e^{x}\right) + e^{x} \cdot e^{x}}}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.5

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

  2. Initial simplification0.0

    \[\leadsto \sqrt{e^{x} + 1}\]
  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. Applied sqrt-div0.1

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

  6. Simplified0.1

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

  8. Applied add-sqr-sqrt0.1

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

  9. Applied unpow-prod-down0.1

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

  11. Applied add-cube-cbrt0.1

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

  12. Applied unpow-prod-down0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Runtime

Time bar (total: 25.2s)Debug logProfile

herbie shell --seed 2018230 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))