Error

Derivation

1. Split input into 2 regimes

2. if (* (sqrt (/ (/ (* (sin x) (sin x)) (+ 1 (cos x))) (* x x))) (sqrt (/ (/ (* (sin x) (sin x)) (+ 1 (cos x))) (* x x)))) < 0.47658470044976553

   1. Initial program 1.0

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

   3. Applied flip--1.2

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

   4. Applied simplify1.1

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

   6. Applied *-un-lft-identity1.1

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

   7. Applied times-frac0.6

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

   8. Applied simplify0.6

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

   if 0.47658470044976553 < (* (sqrt (/ (/ (* (sin x) (sin x)) (+ 1 (cos x))) (* x x))) (sqrt (/ (/ (* (sin x) (sin x)) (+ 1 (cos x))) (* x x))))

   1. Initial program 61.0

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

   2. Taylor expanded around 0 0.1

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

   3. Applied simplify0.1

      \[\leadsto \color{blue}{(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - x \cdot \left(\frac{1}{24} \cdot x\right)}\]
3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))