Average Error: 31.1 → 0.5
Time: 1.9m
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.7387217572195604 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin x \cdot \sin x}{1 + \cos x} \cdot \frac{1}{x \cdot x}\\ \mathbf{if}\;x \le 2500116.352981572:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x}{\sqrt[3]{1 + \cos x} \cdot \sqrt[3]{1 + \cos x}}}{x} \cdot \frac{\frac{\sin x}{\sqrt[3]{1 + \cos x}}}{x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -1.7387217572195604e-16

    1. Initial program 3.3

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

    3. Applied flip--3.5

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

    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 div-inv1.1

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

    if -1.7387217572195604e-16 < x < 2500116.352981572

    1. Initial program 61.6

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

    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 2500116.352981572 < x

    1. Initial program 1.1

      \[\frac{1 - \cos x}{x \cdot x}\]
    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 simplify1.2

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

    6. Applied add-cube-cbrt1.5

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

    7. Applied times-frac1.5

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

    8. Applied times-frac0.9

      \[\leadsto \color{blue}{\frac{\frac{\sin x}{\sqrt[3]{1 + \cos x} \cdot \sqrt[3]{1 + \cos x}}}{x} \cdot \frac{\frac{\sin x}{\sqrt[3]{1 + \cos x}}}{x}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1062803647 245428163 493620569 3595423923 1908391097 2390014376)' 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))