Average Error: 29.8 → 29.8
Time: 17.7s
Precision: 64
Internal Precision: 128
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[(\left(\sqrt{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right)\right)}\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt{x}\right))_*\]

Error

Bits error versus x

Target

Original29.8
Target0.2
Herbie29.8
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.8

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

  3. Applied add-cube-cbrt29.8

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

  4. Applied sqrt-prod29.8

    \[\leadsto \color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}} - \sqrt{x}\]

  5. Applied fma-neg29.8

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

  7. Applied add-cube-cbrt29.8

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

  8. Final simplification29.8

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

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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