Error

Derivation

if eps < -2.504564838281758e-10

1. Initial program 30.4

   \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm

3. Applied cos-sum 1.4

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

4. Applied associate--l- 1.5

   \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
5. Using strategy rm

6. Applied add-cbrt-cube 1.5

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

7. Applied add-cbrt-cube 1.6

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

8. Applied cbrt-unprod 1.5

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

if -2.504564838281758e-10 < eps < 3.6925394752786013e-23

1. Initial program 45.6

   \[\cos \left(x + \varepsilon\right) - \cos x\]

2. Applied taylor 6.7

   \[\leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\varepsilon \cdot x + \frac{1}{2} \cdot {\varepsilon}^2\right)\]

3. Taylor expanded around 0 6.7

   \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\varepsilon \cdot x + \frac{1}{2} \cdot {\varepsilon}^2\right)}\]

4. Applied simplify 6.7

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{2} + x\right)}\]

if 3.6925394752786013e-23 < eps

1. Initial program 31.7

   \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm

3. Applied cos-sum 4.0

   \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Using strategy rm

5. Applied flip3-- 4.1

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

6. Applied simplify 4.1

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

7. Applied simplify 4.2

   \[\leadsto \frac{{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right)}^3 - {\left(\cos x\right)}^3}{\color{blue}{{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right)}^2 + \left(\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon - \cos x\right)\right) \cdot \cos x}}\]
1. Removed slow pow expressions

Runtime

Total time: 50.1s Debug log

Please report a bug with the following info:

herbie --seed '#(2665525929 145650144 309546022 51433028 1710780485 3737191453)'
(FPCore (x eps)
  :name "NMSE problem 3.3.5"
  (- (cos (+ x eps)) (cos x)))