Average Error: 30.9 → 0.2
Time: 20.2s
Precision: 64
Internal Precision: 1408
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{(\left(\sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x + 1}}\right) + \left(\sqrt{x}\right))_*}\]

Error

Bits error versus x

Target

Original30.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.9

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

  3. Applied flip--30.7

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

  4. Applied simplify0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied sqrt-prod0.3

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

  8. Applied fma-def0.2

    \[\leadsto \frac{1}{\color{blue}{(\left(\sqrt{\sqrt{x + 1}}\right) \cdot \left(\sqrt{\sqrt{x + 1}}\right) + \left(\sqrt{x}\right))_*}}\]

Runtime

Time bar (total: 20.2s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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