Average Error: 30.8 → 0.5
Time: 25.3s
Precision: 64
Internal precision: 2432
\[\frac{1 - \cos x}{\sin x}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -6.667433711053936 \cdot 10^{-08}:\\ \;\;\;\;\frac{\sin x \cdot \sin x}{\frac{\sin x}{\frac{1}{1 + \cos x}}}\\ \mathbf{if}\;x \le 1183236732512159.2:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes.

  2. if x < -6.667433711053936e-08

    1. Initial program 1.4

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

    3. Applied flip-- 1.9

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

    4. Applied simplify 1.0

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

    6. Applied div-inv 1.1

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

    7. Applied associate-/l* 1.0

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

    if -6.667433711053936e-08 < x < 1183236732512159.2

    1. Initial program 60.0

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

    2. Applied taylor 0.0

      \[\leadsto \frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\]

    3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]

    if 1183236732512159.2 < x

    1. Initial program 0.9

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

    3. Applied clear-num 1.0

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

Runtime

Time bar (total: 25.3s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (x)
  :name "NMSE example 3.4"
  :pre (!= x 0)
  (/ (- 1 (cos x)) (sin x)))