Average Error: 62.0 → 56.2
Time: 1.7m
Precision: 64
Internal Precision: 128
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\sqrt{(\left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \sqrt[3]{\frac{1}{x \cdot x}}\right) \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(3 - \frac{2}{x}\right))_*}\]

Error

Bits error versus x

Derivation

  1. Initial program 62.0

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

  3. Applied add-sqr-sqrt61.6

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

  5. Applied sqrt-unprod61.2

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

  6. Taylor expanded around 0 56.2

    \[\leadsto \sqrt{\color{blue}{\left(\frac{1}{{x}^{2}} + 3\right) - 2 \cdot \frac{1}{x}}}\]

  7. Simplified56.2

    \[\leadsto \sqrt{\color{blue}{\frac{1}{x \cdot x} + \left(3 - \frac{2}{x}\right)}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt56.2

    \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \sqrt[3]{\frac{1}{x \cdot x}}\right) \cdot \sqrt[3]{\frac{1}{x \cdot x}}} + \left(3 - \frac{2}{x}\right)}\]

  10. Applied fma-def56.2

    \[\leadsto \sqrt{\color{blue}{(\left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \sqrt[3]{\frac{1}{x \cdot x}}\right) \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(3 - \frac{2}{x}\right))_*}}\]

  11. Final simplification56.2

    \[\leadsto \sqrt{(\left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \sqrt[3]{\frac{1}{x \cdot x}}\right) \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(3 - \frac{2}{x}\right))_*}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))