Input Error: 30.3b
Output Error: 0.5b
Time: 31.8s
Precision: 64b
Ground Truth: 128b
\[\frac{1 - \cos x}{\sin x}\]
⬇
\[\begin{cases} \frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x} & \text{when } x \le -7.621017336353511 \cdot 10^{-15} \\ \frac{1}{24} \cdot {x}^3 + \left({x}^{5} \cdot \frac{1}{240} + x \cdot \frac{1}{2}\right) & \text{when } x \le 3.265136463998974 \cdot 10^{-32} \\ \frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x} & \text{otherwise} \end{cases}\]

Error

Bits error versus x

Derivation

    if x < -7.621017336353511e-15

    1. Initial program 3.0b

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

    3. Applied flip-- 3.5b

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

    4. Applied simplify 0.9b

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

    if -7.621017336353511e-15 < x < 3.265136463998974e-32

    1. Initial program 60.5b

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

    2. Applied taylor 0.0b

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

    3. Taylor expanded around 0 0.0b

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

    4. Applied simplify 0.0b

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

    if 3.265136463998974e-32 < x

    1. Initial program 5.6b

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

    3. Applied flip-- 6.1b

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

    4. Applied simplify 1.0b

      \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{\sin x}\]
  1. Removed slow pow expressions

Runtime

Total time: 31.8s Debug log

herbie --seed '#(1889567388 1459648381 1530898341 3110027691 2396490459 3766870378)'
(FPCore (x)
  :name "NMSE example 3.4"
  
  :target
  (tan (/ x 2))(/ (- 1 (cos x)) (sin x)))