Average Error: 19.4 → 0.4
Time: 47.3s
Precision: 64
Internal Precision: 1088
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{x + 1}}}\]

Error

Bits error versus x

Target

Original19.4
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.4

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

  3. Applied frac-sub19.4

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

  4. Applied simplify19.4

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

  6. Applied flip--19.2

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

  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 add-sqr-sqrt0.4

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

  10. Applied sqrt-prod0.4

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

  11. Applied associate-*r*0.4

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

Runtime

Time bar (total: 47.3s)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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