Average Error: 40.0 → 0.0
Time: 17.5s
Precision: 64
Internal Precision: 1344
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{(\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \left(\sqrt[3]{e^{x}}\right) + 1)_*}\]

Error

Bits error versus x

Derivation

  1. Initial program 40.0

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

  2. Initial simplification0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied fma-def0.0

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

  6. Final simplification0.0

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

Runtime

Time bar (total: 17.5s)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))