Average Error: 31.5 → 0.4
Time: 2.0m
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0024305039006771983:\\ \;\;\;\;\frac{\frac{\sin x \cdot \sin x}{x}}{1 + {\left(\cos x\right)}^{3}} \cdot \frac{\left(1 - \cos x\right) + \cos x \cdot \cos x}{x}\\ \mathbf{if}\;x \le 2859.9798077968026:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{x \cdot x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.0024305039006771983

    1. Initial program 1.0

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

    3. Applied flip--1.3

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

    4. Applied simplify1.1

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

    6. Applied flip3-+1.1

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

    7. Applied associate-/r/1.1

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

    8. Applied times-frac0.6

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

    9. Applied simplify0.6

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

    10. Applied simplify0.6

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

    if -0.0024305039006771983 < x < 2859.9798077968026

    1. Initial program 61.2

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    if 2859.9798077968026 < x

    1. Initial program 0.9

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

    3. Applied add-exp-log0.9

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{x \cdot x}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))