Average Error: 15.1 → 1.4
Time: 20.1s
Precision: 64
Internal Precision: 896
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\left(\frac{\sqrt[3]{-1}}{\sqrt[3]{(x \cdot x + x)_*}} \cdot \frac{\sqrt[3]{-1}}{\sqrt[3]{(x \cdot x + x)_*}}\right) \cdot \frac{\sqrt[3]{-1}}{\sqrt[3]{(x \cdot x + x)_*}}\]

Error

Bits error versus x

Derivation

  1. Initial program 15.1

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

  3. Applied frac-sub14.4

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

  4. Applied simplify14.4

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

  5. Applied simplify14.4

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

  7. Applied add-cube-cbrt15.2

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

  8. Applied add-cube-cbrt15.2

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

  9. Applied times-frac15.2

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

  10. Applied simplify15.2

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

  11. Applied simplify1.4

    \[\leadsto \left(\frac{\sqrt[3]{-1}}{\sqrt[3]{(x \cdot x + x)_*}} \cdot \frac{\sqrt[3]{-1}}{\sqrt[3]{(x \cdot x + x)_*}}\right) \cdot \color{blue}{\frac{\sqrt[3]{0 - 1}}{\sqrt[3]{(x \cdot x + x)_*}}}\]

  12. Applied simplify1.4

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

Runtime

Time bar (total: 20.1s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))