Average Error: 4.4 → 0.1
Time: 18.0s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{(\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt[3]{\sqrt[3]{e^{x}}}\right) + 1)_*}\]

Error

Bits error versus x

Derivation

  1. Initial program 4.4

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

  2. Simplified0.1

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

  4. Applied add-cube-cbrt0.1

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

  5. Applied fma-def0.1

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

  7. Applied add-cube-cbrt0.1

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

  8. Applied cbrt-prod0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))