Average Error: 31.0 → 0.4
Time: 1.7m
Precision: 64
Internal Precision: 2368
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{\frac{x}{\sin x}}}{1 + {\left(\cos x\right)}^{3}} \cdot \frac{\cos x \cdot \cos x - \left(\cos x - 1\right)}{x}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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 simplify15.5

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

  6. Applied flip3-+15.5

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

  7. Applied associate-/r/15.5

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

  8. Applied times-frac15.4

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

  9. Applied simplify0.4

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

  10. Applied simplify0.4

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2019053 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))