Average Error: 30.2 → 30.1
Time: 19.1s
Precision: 64
Internal Precision: 128
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[e^{\log \left(\left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\sqrt{\sqrt{1 + x}}}} \cdot \sqrt[3]{\sqrt{\sqrt{\sqrt{1 + x}}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}\right) - \sqrt{x}\right)}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.2
Target0.2
Herbie30.1
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.2

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

  3. Applied add-cube-cbrt30.3

    \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right) \cdot \sqrt[3]{\sqrt{x + 1}}} - \sqrt{x}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt30.2

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

  6. Applied cbrt-prod30.2

    \[\leadsto \left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt{\sqrt{x + 1}}} \cdot \sqrt[3]{\sqrt{\sqrt{x + 1}}}\right)} - \sqrt{x}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt30.2

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

  9. Applied cbrt-prod30.2

    \[\leadsto \left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt{\sqrt{x + 1}}}} \cdot \sqrt[3]{\sqrt{\sqrt{\sqrt{x + 1}}}}\right)} \cdot \sqrt[3]{\sqrt{\sqrt{x + 1}}}\right) - \sqrt{x}\]
  10. Using strategy rm

  11. Applied add-exp-log30.1

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

  12. Final simplification30.1

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

Reproduce

herbie shell --seed 2019026 
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))