Average Error: 30.2 → 0.5
Time: 3.3m
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007466980376524548:\\ \;\;\;\;\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x}\\ \mathbf{if}\;x \le 0.0019551307389888218:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.007466980376524548 or 0.0019551307389888218 < x

    1. Initial program 1.0

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

    3. Applied flip--1.4

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

    4. Applied simplify1.0

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

    if -0.007466980376524548 < x < 0.0019551307389888218

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 3.3m)Debug log

herbie shell --seed '#(1743936871 1855164119 3668777427 1254258049 132811564 1366975197)' 
(FPCore (x)
  :name "NMSE example 3.4"
  :pre (!= x 0)
  (/ (- 1 (cos x)) (sin x)))