Average Error: 20.5 → 0.4
Time: 1.2m
Precision: 64
Internal Precision: 1152
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x}}}{\sqrt{x + 1}}\]

Error

Bits error versus x

Derivation

  1. Initial program 20.5

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

  3. Applied frac-sub20.5

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

  4. Applied simplify20.5

    \[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
  5. Using strategy rm

  6. Applied flip--20.3

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

  7. Applied simplify0.4

    \[\leadsto \frac{\frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
  8. Using strategy rm

  9. Applied associate-/r*0.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x}}}{\sqrt{x + 1}}}\]

Runtime

Time bar (total: 1.2m)Debug log

herbie shell --seed '#(1743936871 1855164119 3668777427 1254258049 132811564 1366975197)' 
(FPCore (x)
  :name "NMSE example 3.6"
  :pre (>= x 0)
  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))