Average Error: 30.4 → 0.5
Time: 51.4s
Precision: 64
Ground Truth: 128
\[\frac{1 - \cos x}{\sin x}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -0.0006447367512968273:\\ \;\;\;\;\frac{{\left(\sin x\right)}^2}{1 + {\left(\cos x\right)}^3} \cdot \frac{\left(1 - \cos x\right) + {\left(\cos x\right)}^2}{\sin x}\\ \mathbf{if}\;x \le 0.0017960167992252625:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x}\\ \end{array}\]

Error

Bits error versus x

Target

Original30.4
Comparison0
Herbie0.5
\[ \tan \left(\frac{x}{2}\right) \]

Derivation

  1. Split input into 3 regimes.

  2. if x < -0.0006447367512968273

    1. Initial program 1.0

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

    3. Applied flip-- 1.4

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

    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 *-un-lft-identity 1.0

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

    7. Applied flip3-+ 1.0

      \[\leadsto \frac{\frac{\sin x \cdot \sin x}{\color{blue}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{{1}^2 + \left({\left(\cos x\right)}^2 - 1 \cdot \cos x\right)}}}}{1 \cdot \sin x}\]

    8. Applied associate-/r/ 1.0

      \[\leadsto \frac{\color{blue}{\frac{\sin x \cdot \sin x}{{1}^{3} + {\left(\cos x\right)}^{3}} \cdot \left({1}^2 + \left({\left(\cos x\right)}^2 - 1 \cdot \cos x\right)\right)}}{1 \cdot \sin x}\]

    9. Applied times-frac 1.0

      \[\leadsto \color{blue}{\frac{\frac{\sin x \cdot \sin x}{{1}^{3} + {\left(\cos x\right)}^{3}}}{1} \cdot \frac{{1}^2 + \left({\left(\cos x\right)}^2 - 1 \cdot \cos x\right)}{\sin x}}\]

    10. Applied simplify 1.0

      \[\leadsto \color{blue}{\frac{{\left(\sin x\right)}^2}{1 + {\left(\cos x\right)}^3}} \cdot \frac{{1}^2 + \left({\left(\cos x\right)}^2 - 1 \cdot \cos x\right)}{\sin x}\]

    11. Applied simplify 1.1

      \[\leadsto \frac{{\left(\sin x\right)}^2}{1 + {\left(\cos x\right)}^3} \cdot \color{blue}{\frac{\cos x \cdot \cos x + \left(1 - \cos x\right)}{\sin x}}\]

    12. Applied simplify 1.1

      \[\leadsto \frac{{\left(\sin x\right)}^2}{1 + {\left(\cos x\right)}^3} \cdot \frac{\color{blue}{\left(1 - \cos x\right) + {\left(\cos x\right)}^2}}{\sin x}\]

    if -0.0006447367512968273 < x < 0.0017960167992252625

    1. Initial program 60.1

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

    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 0.0017960167992252625 < x

    1. Initial program 0.9

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

    3. Applied flip-- 1.3

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

    4. Applied simplify 1.0

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

Runtime

Total time: 51.4s Debug log

Please include this information when filing a bug report:

herbie --seed '#(3571630856 2845139455 1429415848 2099849052 3567897525 1471759581)'
(FPCore (x)
  :name "NMSE example 3.4"
  :herbie-expected 1

  :target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))