Average Error: 39.5 → 0.1
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\frac{\sqrt{{\left(e^{x}\right)}^{3} + {1}^{3}}}{\sqrt{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\]

Error

Bits error versus x

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Your Program's Arguments

Results

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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. Final simplification0.1

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

Runtime

Time bar (total: 1.1m)Debug logProfile

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