Average Error: 31.0 → 0.5
Time: 47.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\frac{1}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}}}{\frac{\sqrt[3]{x}}{\sqrt[3]{\sin x}}} \cdot \frac{\sin x}{x}}{\cos x + 1}\]

Error

Bits error versus x

Derivation

  1. Initial program 31.0

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

  3. Applied flip--31.1

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

  4. Applied associate-/l/31.1

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

  5. Simplified15.4

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

  6. Taylor expanded around -inf 15.4

    \[\leadsto \color{blue}{\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \left(\cos x + 1\right)}}\]

  7. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x + 1}}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity0.3

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

  10. Applied associate-/l*0.3

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x}{\sin x}}} \cdot \frac{\sin x}{x}}{\cos x + 1}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt1.3

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

  13. Applied add-cube-cbrt0.5

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

  14. Applied times-frac0.5

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\sin x}}}} \cdot \frac{\sin x}{x}}{\cos x + 1}\]

  15. Applied associate-/r*0.5

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}}}{\frac{\sqrt[3]{x}}{\sqrt[3]{\sin x}}}} \cdot \frac{\sin x}{x}}{\cos x + 1}\]

  16. Final simplification0.5

    \[\leadsto \frac{\frac{\frac{1}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}}}{\frac{\sqrt[3]{x}}{\sqrt[3]{\sin x}}} \cdot \frac{\sin x}{x}}{\cos x + 1}\]

Reproduce

herbie shell --seed 2019100 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))