Average Error: 39.0 → 0.0
Time: 11.1s
Precision: 64
Internal Precision: 1344
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{\log_* (1 + e^{x})}}\]

Error

Bits error versus x

Derivation

  1. Initial program 39.0

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

  2. Applied simplify0.0

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

  4. Applied add-exp-log0.0

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

  5. Applied simplify0.0

    \[\leadsto \sqrt{e^{\color{blue}{\log_* (1 + e^{x})}}}\]

Runtime

Time bar (total: 11.1s)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))