Error: 36.2 → 9.5
Time: 52.4s
Precision: 64
Ground Truth: 128
\[\tan \left(x + \varepsilon\right) - \tan x\]
⬇
\[\begin{cases} \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \tan x & \text{when } \varepsilon \le -2.5569065403496266 \cdot 10^{-24} \\ \left(\varepsilon + {\varepsilon}^{4} \cdot {x}^3\right) + \left(x \cdot x\right) \cdot {\varepsilon}^3 & \text{when } \varepsilon \le 1.0388221479750895 \cdot 10^{-47} \\ \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \tan x & \text{otherwise} \end{cases}\]

Error

Bits error versus x

Bits error versus eps

Derivation

    if eps < -2.5569065403496266e-24 or 1.0388221479750895e-47 < eps

    1. Initial program 29.9

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

    3. Applied tan-quot 29.9

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

    5. Applied sin-sum 28.0

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

    7. Applied cos-sum 2.9

      \[\leadsto \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\color{blue}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} - \tan x\]

    if -2.5569065403496266e-24 < eps < 1.0388221479750895e-47

    1. Initial program 44.6

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

    2. Applied taylor 18.1

      \[\leadsto \varepsilon + \left({\varepsilon}^{4} \cdot {x}^{3} + {\varepsilon}^{3} \cdot {x}^2\right)\]

    3. Taylor expanded around 0 18.1

      \[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{4} \cdot {x}^{3} + {\varepsilon}^{3} \cdot {x}^2\right)}\]

    4. Applied simplify 18.1

      \[\leadsto \color{blue}{\left(\varepsilon + {\varepsilon}^{4} \cdot {x}^3\right) + \left(x \cdot x\right) \cdot {\varepsilon}^3}\]
  1. Removed slow pow expressions

Runtime

Total time: 52.4s Debug log

Please report a bug with the following info:

herbie --seed '#(3170425915 1086398764 2767494986 895451451 2982682639 2650094026)'
(FPCore (x eps)
  :name "NMSE problem 3.3.2"
  
  :target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))(- (tan (+ x eps)) (tan x)))