Average Error: 31.0 → 31.0
Time: 53.4s
Precision: 64
Internal Precision: 2368
\[\frac{\left(\sqrt{x \cdot x - 1} + x\right) + 1}{2 \cdot \sqrt{\sqrt{x \cdot x - 1} + x}}\]
\[\frac{\sqrt{1 + \left(x + \sqrt{x \cdot x - 1}\right)}}{2} \cdot \frac{\sqrt{1 + \left(x + \sqrt{x \cdot x - 1}\right)}}{\sqrt{x + \left(\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

    \[\frac{\left(\sqrt{x \cdot x - 1} + x\right) + 1}{2 \cdot \sqrt{\sqrt{x \cdot x - 1} + x}}\]
  2. Using strategy rm
  3. Applied add-sqr-sqrt30.9

    \[\leadsto \frac{\color{blue}{\sqrt{\left(\sqrt{x \cdot x - 1} + x\right) + 1} \cdot \sqrt{\left(\sqrt{x \cdot x - 1} + x\right) + 1}}}{2 \cdot \sqrt{\sqrt{x \cdot x - 1} + x}}\]
  4. Applied times-frac30.8

    \[\leadsto \color{blue}{\frac{\sqrt{\left(\sqrt{x \cdot x - 1} + x\right) + 1}}{2} \cdot \frac{\sqrt{\left(\sqrt{x \cdot x - 1} + x\right) + 1}}{\sqrt{\sqrt{x \cdot x - 1} + x}}}\]
  5. Taylor expanded around inf 31.0

    \[\leadsto \frac{\sqrt{\left(\sqrt{x \cdot x - 1} + x\right) + 1}}{2} \cdot \frac{\sqrt{\left(\sqrt{x \cdot x - 1} + x\right) + 1}}{\sqrt{\color{blue}{\left(x - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)} + x}}\]
  6. Simplified31.0

    \[\leadsto \frac{\sqrt{\left(\sqrt{x \cdot x - 1} + x\right) + 1}}{2} \cdot \frac{\sqrt{\left(\sqrt{x \cdot x - 1} + x\right) + 1}}{\sqrt{\color{blue}{\left(\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)} + x}}\]
  7. Final simplification31.0

    \[\leadsto \frac{\sqrt{1 + \left(x + \sqrt{x \cdot x - 1}\right)}}{2} \cdot \frac{\sqrt{1 + \left(x + \sqrt{x \cdot x - 1}\right)}}{\sqrt{x + \left(\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}}\]

Runtime

Time bar (total: 53.4s)Debug log

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