Error

Derivation

1. Split input into 3 regimes.

2. if x < -7.829158115972637e-05

   1. Initial program 1.3

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

   3. Applied square-mult 1.3

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

   4. Applied *-un-lft-identity 1.3

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

   5. Applied times-frac 0.6

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

   if -7.829158115972637e-05 < x < 2.576641619206354e-19

   1. Initial program 61.9

      \[\frac{1 - \cos x}{{x}^2}\]

   2. Applied taylor 0

      \[\leadsto \left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^2\]

   3. Taylor expanded around 0 0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^2}\]

   if 2.576641619206354e-19 < x

   1. Initial program 3.3

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

   3. Applied flip-- 3.5

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

   4. Applied simplify 1.1

      \[\leadsto \frac{\frac{\color{blue}{{\left(\sin x\right)}^2}}{1 + \cos x}}{{x}^2}\]
   5. Using strategy rm

   6. Applied square-mult 1.1

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

   7. Applied *-un-lft-identity 1.1

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

   8. Applied *-un-lft-identity 1.1

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

   9. Applied square-prod 1.1

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

   10. Applied times-frac 1.1

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

   11. Applied times-frac 0.5

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

   12. Applied simplify 0.5

       \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x}\]
3. Recombined 3 regimes into one program.
4. Removed slow pow expressions

Runtime

Total time: 1.7m Debug log

Please include this information when filing a bug report:

herbie --seed '#(2844525053 3990250133 247123594 353958486 2001714664 3183081771)'
(FPCore (x)
  :name "NMSE problem 3.4.1"
  (/ (- 1 (cos x)) (sqr x)))