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

# Try it out

Results

 In Out
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# Derivation

1. Initial program 31.3

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

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

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