Average Error: 31.3 → 0.0
Time: 26.1s
Precision: 64
Internal Precision: 576
\[\frac{1 + \sqrt{{x}^{2} - 1}}{\sqrt{{x}^{2} - 1}}\]
↓
\[\frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

    \[\frac{1 + \sqrt{{x}^{2} - 1}}{\sqrt{{x}^{2} - 1}}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt31.3

    \[\leadsto \frac{1 + \sqrt{\color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{{x}^{2}}} - 1}}{\sqrt{{x}^{2} - 1}}\]

  4. Applied difference-of-sqr-131.3

    \[\leadsto \frac{1 + \sqrt{\color{blue}{\left(\sqrt{{x}^{2}} + 1\right) \cdot \left(\sqrt{{x}^{2}} - 1\right)}}}{\sqrt{{x}^{2} - 1}}\]

  5. Applied sqrt-prod31.6

    \[\leadsto \frac{1 + \color{blue}{\sqrt{\sqrt{{x}^{2}} + 1} \cdot \sqrt{\sqrt{{x}^{2}} - 1}}}{\sqrt{{x}^{2} - 1}}\]

  6. Applied simplify31.6

    \[\leadsto \frac{1 + \color{blue}{\sqrt{\left|x\right| + 1}} \cdot \sqrt{\sqrt{{x}^{2}} - 1}}{\sqrt{{x}^{2} - 1}}\]

  7. Applied simplify31.6

    \[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \color{blue}{\sqrt{\left|x\right| - 1}}}{\sqrt{{x}^{2} - 1}}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt31.6

    \[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\sqrt{\color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{{x}^{2}}} - 1}}\]

  10. Applied difference-of-sqr-131.6

    \[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\sqrt{\color{blue}{\left(\sqrt{{x}^{2}} + 1\right) \cdot \left(\sqrt{{x}^{2}} - 1\right)}}}\]

  11. Applied sqrt-prod31.3

    \[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\color{blue}{\sqrt{\sqrt{{x}^{2}} + 1} \cdot \sqrt{\sqrt{{x}^{2}} - 1}}}\]

  12. Applied simplify31.3

    \[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\color{blue}{\sqrt{\left|x\right| + 1}} \cdot \sqrt{\sqrt{{x}^{2}} - 1}}\]

  13. Applied simplify0.0

    \[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\sqrt{\left|x\right| + 1} \cdot \color{blue}{\sqrt{\left|x\right| - 1}}}\]

Runtime

Time bar (total: 26.1s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "(1+sqrt(x^2-1))/sqrt(x^2-1)"
  (/ (+ 1 (sqrt (- (pow x 2) 1))) (sqrt (- (pow x 2) 1))))