Average Error: 31.3 → 0.3
Time: 45.0s
Precision: 64
Internal Precision: 2368
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03354579266000314:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \le 1.3047674561131453 \cdot 10^{-06}:\\ \;\;\;\;(\frac{1}{720} \cdot \left({x}^{4}\right) + \frac{1}{2})_* - \left(x \cdot x\right) \cdot \frac{1}{24}\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}} - 1)^*)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.03354579266000314

    1. Initial program 1.0

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

    3. Applied associate-/r*0.4

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

    if -0.03354579266000314 < x < 1.3047674561131453e-06

    1. Initial program 61.6

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

    3. Applied flip--61.6

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

    4. Applied associate-/l/61.6

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

    5. Simplified29.5

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

    7. Applied log1p-expm1-u29.5

      \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}} - 1)^*)}\]
    8. Using strategy rm

    9. Applied times-frac30.0

      \[\leadsto \log_* (1 + (e^{\color{blue}{\frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x}}} - 1)^*)\]

    10. Simplified30.0

      \[\leadsto \log_* (1 + (e^{\frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}} - 1)^*)\]

    11. Taylor expanded around 0 0.0

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

    12. Simplified0.0

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

    if 1.3047674561131453e-06 < x

    1. Initial program 1.2

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

    3. Applied flip--1.5

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

    4. Applied associate-/l/1.5

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

    5. Simplified1.1

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

    7. Applied log1p-expm1-u1.1

      \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}} - 1)^*)}\]
    8. Using strategy rm

    9. Applied times-frac1.1

      \[\leadsto \log_* (1 + (e^{\color{blue}{\frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x}}} - 1)^*)\]

    10. Simplified0.8

      \[\leadsto \log_* (1 + (e^{\frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}} - 1)^*)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03354579266000314:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \le 1.3047674561131453 \cdot 10^{-06}:\\ \;\;\;\;(\frac{1}{720} \cdot \left({x}^{4}\right) + \frac{1}{2})_* - \left(x \cdot x\right) \cdot \frac{1}{24}\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}} - 1)^*)\\ \end{array}\]

Runtime

Time bar (total: 45.0s)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))