Average Error: 36.6 → 14.4
Time: 1.4m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right) \le -6.817958129115741 \cdot 10^{-10}:\\ \;\;\;\;\frac{\cos x - \frac{1 - \tan x \cdot \tan \varepsilon}{\frac{\tan x + \tan \varepsilon}{\sin x}}}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon} \cdot \cos x}\\ \mathbf{if}\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right) \le 8.04931125323374 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.6
Target14.7
Herbie14.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < -6.817958129115741e-10

    1. Initial program 36.7

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

    3. Applied tan-sum10.0

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
    4. Using strategy rm

    5. Applied clear-num10.0

      \[\leadsto \color{blue}{\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}}} - \tan x\]
    6. Using strategy rm

    7. Applied tan-quot10.1

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

    8. Applied frac-sub10.1

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

    9. Applied simplify10.1

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

    if -6.817958129115741e-10 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < 8.04931125323374e-27

    1. Initial program 38.9

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

    2. Taylor expanded around 0 16.6

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

    if 8.04931125323374e-27 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x)))

    1. Initial program 35.0

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

    3. Applied tan-sum14.3

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
    4. Using strategy rm

    5. Applied flip--14.4

      \[\leadsto \color{blue}{\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018170 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))